Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf ·...

201
Pointwise convergence to initial data Keith Rogers August 28, 2016

Transcript of Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf ·...

Page 1: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Pointwise convergence to initial data

Keith Rogers

August 28, 2016

Page 2: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Summary

I Part 1: Set-up and introduction to the PDEs.

I Part 2: Convergence for the heat equation.

I Part 3: Convergence for the Schrodinger equation.

I Part 4: Counterexample for the Schrodinger equation.

I Part 5: Decay of the Fourier transform of fractal measures.

I Part 6: Convergence for the wave equation.

Page 3: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Summary

I Part 1: Set-up and introduction to the PDEs.

I Part 2: Convergence for the heat equation.

I Part 3: Convergence for the Schrodinger equation.

I Part 4: Counterexample for the Schrodinger equation.

I Part 5: Decay of the Fourier transform of fractal measures.

I Part 6: Convergence for the wave equation.

Page 4: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Summary

I Part 1: Set-up and introduction to the PDEs.

I Part 2: Convergence for the heat equation.

I Part 3: Convergence for the Schrodinger equation.

I Part 4: Counterexample for the Schrodinger equation.

I Part 5: Decay of the Fourier transform of fractal measures.

I Part 6: Convergence for the wave equation.

Page 5: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Summary

I Part 1: Set-up and introduction to the PDEs.

I Part 2: Convergence for the heat equation.

I Part 3: Convergence for the Schrodinger equation.

I Part 4: Counterexample for the Schrodinger equation.

I Part 5: Decay of the Fourier transform of fractal measures.

I Part 6: Convergence for the wave equation.

Page 6: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Summary

I Part 1: Set-up and introduction to the PDEs.

I Part 2: Convergence for the heat equation.

I Part 3: Convergence for the Schrodinger equation.

I Part 4: Counterexample for the Schrodinger equation.

I Part 5: Decay of the Fourier transform of fractal measures.

I Part 6: Convergence for the wave equation.

Page 7: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Summary

I Part 1: Set-up and introduction to the PDEs.

I Part 2: Convergence for the heat equation.

I Part 3: Convergence for the Schrodinger equation.

I Part 4: Counterexample for the Schrodinger equation.

I Part 5: Decay of the Fourier transform of fractal measures.

I Part 6: Convergence for the wave equation.

Page 8: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Summary

I Part 1: Set-up and introduction to the PDEs.

I Part 2: Convergence for the heat equation.

I Part 3: Convergence for the Schrodinger equation.

I Part 4: Counterexample for the Schrodinger equation.

I Part 5: Decay of the Fourier transform of fractal measures.

I Part 6: Convergence for the wave equation.

Page 9: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Part 1:

Set-up and introduction to thePDEs

Page 10: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

The heat equation

∂tu = ∆u in Rn × [0,∞)u = u0 in Rn × 0.

Taking the Fourier transform of the equation we obtain∂t u(ξ) = −|ξ|2u(ξ)u(ξ) = u0(ξ).

Solving the ODE this yields

u(ξ) = e−t|ξ|2u0(ξ) .

Inverting the Fourier transform, we write

u(x , t) = et∆u0(x) := 1(2π)n/2

´Rn e

ix ·ξe−t|ξ|2u0(ξ) dξ .

Page 11: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

The heat equation

∂tu = ∆u in Rn × [0,∞)u = u0 in Rn × 0.

Taking the Fourier transform of the equation we obtain∂t u(ξ) = −|ξ|2u(ξ)u(ξ) = u0(ξ).

Solving the ODE this yields

u(ξ) = e−t|ξ|2u0(ξ) .

Inverting the Fourier transform, we write

u(x , t) = et∆u0(x) := 1(2π)n/2

´Rn e

ix ·ξe−t|ξ|2u0(ξ) dξ .

Page 12: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

The heat equation

∂tu = ∆u in Rn × [0,∞)u = u0 in Rn × 0.

Taking the Fourier transform of the equation we obtain∂t u(ξ) = −|ξ|2u(ξ)u(ξ) = u0(ξ).

Solving the ODE this yields

u(ξ) = e−t|ξ|2u0(ξ) .

Inverting the Fourier transform, we write

u(x , t) = et∆u0(x) := 1(2π)n/2

´Rn e

ix ·ξe−t|ξ|2u0(ξ) dξ .

Page 13: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

The heat equation

∂tu = ∆u in Rn × [0,∞)u = u0 in Rn × 0.

Taking the Fourier transform of the equation we obtain∂t u(ξ) = −|ξ|2u(ξ)u(ξ) = u0(ξ).

Solving the ODE this yields

u(ξ) = e−t|ξ|2u0(ξ) .

Inverting the Fourier transform, we write

u(x , t) = et∆u0(x) := 1(2π)n/2

´Rn e

ix ·ξe−t|ξ|2u0(ξ) dξ .

Page 14: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

The heat equation

∂tu = ∆u in Rn × [0,∞)u = u0 in Rn × 0.

Taking the Fourier transform of the equation we obtain∂t u(ξ) = −|ξ|2u(ξ)u(ξ) = u0(ξ).

Solving the ODE this yields

u(ξ) = e−t|ξ|2u0(ξ) .

Inverting the Fourier transform, we write

u(x , t) = et∆u0(x) := 1(2π)n/2

´Rn e

ix ·ξe−t|ξ|2u0(ξ) dξ .

Page 15: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

The Schrodinger equation

∂tu = i∆u in Rn × Ru = u0 in Rn × 0.

Taking the Fourier transform of the equation we obtain∂t u(ξ) = −i |ξ|2u(ξ)u(ξ) = u0(ξ).

Solving the ODE this yields

u(ξ) = e−it|ξ|2u0(ξ) .

Inverting the Fourier transform, we write

u(x , t) = e it∆u0(x) := 1(2π)n/2

´Rn e

ix ·ξe−it|ξ|2u0(ξ) dξ .

Page 16: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

The Schrodinger equation

∂tu = i∆u in Rn × Ru = u0 in Rn × 0.

Taking the Fourier transform of the equation we obtain∂t u(ξ) = −i |ξ|2u(ξ)u(ξ) = u0(ξ).

Solving the ODE this yields

u(ξ) = e−it|ξ|2u0(ξ) .

Inverting the Fourier transform, we write

u(x , t) = e it∆u0(x) := 1(2π)n/2

´Rn e

ix ·ξe−it|ξ|2u0(ξ) dξ .

Page 17: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

The Schrodinger equation

∂tu = i∆u in Rn × Ru = u0 in Rn × 0.

Taking the Fourier transform of the equation we obtain∂t u(ξ) = −i |ξ|2u(ξ)u(ξ) = u0(ξ).

Solving the ODE this yields

u(ξ) = e−it|ξ|2u0(ξ) .

Inverting the Fourier transform, we write

u(x , t) = e it∆u0(x) := 1(2π)n/2

´Rn e

ix ·ξe−it|ξ|2u0(ξ) dξ .

Page 18: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

The Schrodinger equation

∂tu = i∆u in Rn × Ru = u0 in Rn × 0.

Taking the Fourier transform of the equation we obtain∂t u(ξ) = −i |ξ|2u(ξ)u(ξ) = u0(ξ).

Solving the ODE this yields

u(ξ) = e−it|ξ|2u0(ξ) .

Inverting the Fourier transform, we write

u(x , t) = e it∆u0(x) := 1(2π)n/2

´Rn e

ix ·ξe−it|ξ|2u0(ξ) dξ .

Page 19: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

The Schrodinger equation

∂tu = i∆u in Rn × Ru = u0 in Rn × 0.

Taking the Fourier transform of the equation we obtain∂t u(ξ) = −i |ξ|2u(ξ)u(ξ) = u0(ξ).

Solving the ODE this yields

u(ξ) = e−it|ξ|2u0(ξ) .

Inverting the Fourier transform, we write

u(x , t) = e it∆u0(x) := 1(2π)n/2

´Rn e

ix ·ξe−it|ξ|2u0(ξ) dξ .

Page 20: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

The wave equation∂ttu = ∆u in Rn × R

u = u0 in Rn × 0∂tu = u1 in Rn × 0.

Taking the Fourier transform of the equation we obtain∂tt u(ξ) = −|ξ|2u(ξ)

u(ξ) = u0(ξ)∂t u(ξ) = u1(ξ).

Solving the ODE this yields

u(ξ) = cos(t|ξ|)u0(ξ) + sin(t|ξ|)|ξ| u1(ξ) .

Inverting the Fourier transform, we write

u(·, t) = cos(t√−∆)u0 + sin(t

√−∆)√−∆

u1.

Page 21: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

The wave equation∂ttu = ∆u in Rn × R

u = u0 in Rn × 0∂tu = u1 in Rn × 0.

Taking the Fourier transform of the equation we obtain∂tt u(ξ) = −|ξ|2u(ξ)

u(ξ) = u0(ξ)∂t u(ξ) = u1(ξ).

Solving the ODE this yields

u(ξ) = cos(t|ξ|)u0(ξ) + sin(t|ξ|)|ξ| u1(ξ) .

Inverting the Fourier transform, we write

u(·, t) = cos(t√−∆)u0 + sin(t

√−∆)√−∆

u1.

Page 22: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

The wave equation∂ttu = ∆u in Rn × R

u = u0 in Rn × 0∂tu = u1 in Rn × 0.

Taking the Fourier transform of the equation we obtain∂tt u(ξ) = −|ξ|2u(ξ)

u(ξ) = u0(ξ)∂t u(ξ) = u1(ξ).

Solving the ODE this yields

u(ξ) = cos(t|ξ|)u0(ξ) + sin(t|ξ|)|ξ| u1(ξ) .

Inverting the Fourier transform, we write

u(·, t) = cos(t√−∆)u0 + sin(t

√−∆)√−∆

u1.

Page 23: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

The wave equation∂ttu = ∆u in Rn × R

u = u0 in Rn × 0∂tu = u1 in Rn × 0.

Taking the Fourier transform of the equation we obtain∂tt u(ξ) = −|ξ|2u(ξ)

u(ξ) = u0(ξ)∂t u(ξ) = u1(ξ).

Solving the ODE this yields

u(ξ) = cos(t|ξ|)u0(ξ) + sin(t|ξ|)|ξ| u1(ξ) .

Inverting the Fourier transform, we write

u(·, t) = cos(t√−∆)u0 + sin(t

√−∆)√−∆

u1.

Page 24: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

The wave equation∂ttu = ∆u in Rn × R

u = u0 in Rn × 0∂tu = u1 in Rn × 0.

Taking the Fourier transform of the equation we obtain∂tt u(ξ) = −|ξ|2u(ξ)

u(ξ) = u0(ξ)∂t u(ξ) = u1(ξ).

Solving the ODE this yields

u(ξ) = cos(t|ξ|)u0(ξ) + sin(t|ξ|)|ξ| u1(ξ) .

Inverting the Fourier transform, we write

u(·, t) = cos(t√−∆)u0 + sin(t

√−∆)√−∆

u1.

Page 25: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

The initial dataWe take the initial data u0 in the Bessel potential space

Hs(Rn) := (1−∆)−s/2L2(Rn)

:= f : f = (1 + | · |2)−s/2g , g ∈ L2(Rn)

with norm

‖f ‖Hs =

(ˆRn

(1 + |ξ|2)s |f (ξ)|2 dξ)1/2

= ‖g‖L2(Rn) .

or in the Riesz potential space

Hs(Rn) := (−∆)−s/2L2(Rn)

:= f : f = | · |−s g , g ∈ L2(Rn) ,

with norm

‖f ‖Hs =

(ˆRn

|ξ|2s |f (ξ)|2 dξ)1/2

= ‖g‖L2(Rn) .

Page 26: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

The initial dataWe take the initial data u0 in the Bessel potential space

Hs(Rn) := (1−∆)−s/2L2(Rn)

:= f : f = (1 + | · |2)−s/2g , g ∈ L2(Rn)

with norm

‖f ‖Hs =

(ˆRn

(1 + |ξ|2)s |f (ξ)|2 dξ)1/2

= ‖g‖L2(Rn) .

or in the Riesz potential space

Hs(Rn) := (−∆)−s/2L2(Rn)

:= f : f = | · |−s g , g ∈ L2(Rn) ,

with norm

‖f ‖Hs =

(ˆRn

|ξ|2s |f (ξ)|2 dξ)1/2

= ‖g‖L2(Rn) .

Page 27: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

The initial dataWe take the initial data u0 in the Bessel potential space

Hs(Rn) := (1−∆)−s/2L2(Rn)

:= f : f = (1 + | · |2)−s/2g , g ∈ L2(Rn)

with norm

‖f ‖Hs =

(ˆRn

(1 + |ξ|2)s |f (ξ)|2 dξ)1/2

= ‖g‖L2(Rn) .

or in the Riesz potential space

Hs(Rn) := (−∆)−s/2L2(Rn)

:= f : f = | · |−s g , g ∈ L2(Rn) ,

with norm

‖f ‖Hs =

(ˆRn

|ξ|2s |f (ξ)|2 dξ)1/2

= ‖g‖L2(Rn) .

Page 28: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

The initial dataWe take the initial data u0 in the Bessel potential space

Hs(Rn) := (1−∆)−s/2L2(Rn)

:= f : f = (1 + | · |2)−s/2g , g ∈ L2(Rn)

with norm

‖f ‖Hs =

(ˆRn

(1 + |ξ|2)s |f (ξ)|2 dξ)1/2

= ‖g‖L2(Rn) .

or in the Riesz potential space

Hs(Rn) := (−∆)−s/2L2(Rn)

:= f : f = | · |−s g , g ∈ L2(Rn) ,

with norm

‖f ‖Hs =

(ˆRn

|ξ|2s |f (ξ)|2 dξ)1/2

= ‖g‖L2(Rn) .

Page 29: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

The initial dataWe take the initial data u0 in the Bessel potential space

Hs(Rn) := (1−∆)−s/2L2(Rn)

:= f : f = (1 + | · |2)−s/2g , g ∈ L2(Rn)

with norm

‖f ‖Hs =

(ˆRn

(1 + |ξ|2)s |f (ξ)|2 dξ)1/2

= ‖g‖L2(Rn) .

or in the Riesz potential space

Hs(Rn) := (−∆)−s/2L2(Rn)

:= f : f = | · |−s g , g ∈ L2(Rn) ,

with norm

‖f ‖Hs =

(ˆRn

|ξ|2s |f (ξ)|2 dξ)1/2

= ‖g‖L2(Rn) .

Page 30: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

The initial dataWe take the initial data u0 in the Bessel potential space

Hs(Rn) := (1−∆)−s/2L2(Rn)

:= f : f = (1 + | · |2)−s/2g , g ∈ L2(Rn)

with norm

‖f ‖Hs =

(ˆRn

(1 + |ξ|2)s |f (ξ)|2 dξ)1/2

= ‖g‖L2(Rn) .

or in the Riesz potential space

Hs(Rn) := (−∆)−s/2L2(Rn)

:= f : f = | · |−s g , g ∈ L2(Rn) ,

with norm

‖f ‖Hs =

(ˆRn

|ξ|2s |f (ξ)|2 dξ)1/2

= ‖g‖L2(Rn) .

Page 31: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Lemma (Pointwise convergence for smooth data)

Let u solve the heat or Schrodinger equation with u0 ∈ Hs(Rn)with n/2 < s < n/2 + 2. Then

limt→0

u(x , t) = u0(x) for all x ∈ Rn.

Proof: u0 = | · |−s g with g ∈ L2

(2π)n/2|et∆u0(x)− u0(x)| =

∣∣∣∣∣ˆ

g(ξ) e ix ·ξ(e−t|ξ|2 − 1)

|ξ|sdξ

∣∣∣∣∣≤ ‖g ‖2

(ˆ |e−t|ξ|2 − 1|2

|ξ|2sdξ)1/2

= ts/2−n/4‖g‖2

(ˆ |e−|y |2 − 1|2

|y |2sdy)1/2

= ts/2−n/4‖f ‖Hs

(ˆ min|y |2, 12

|y |2sdy)1/2

≤ Csts/2−n/4‖f ‖Hs .

The same calculation works for the Schrodinger equation.

Page 32: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Lemma (Pointwise convergence for smooth data)

Let u solve the heat or Schrodinger equation with u0 ∈ Hs(Rn)with n/2 < s < n/2 + 2. Then

limt→0

u(x , t) = u0(x) for all x ∈ Rn.

Proof: u0 = | · |−s g with g ∈ L2

(2π)n/2|et∆u0(x)− u0(x)| =

∣∣∣∣∣ˆ

g(ξ) e ix ·ξ(e−t|ξ|2 − 1)

|ξ|sdξ

∣∣∣∣∣≤ ‖g ‖2

(ˆ |e−t|ξ|2 − 1|2

|ξ|2sdξ)1/2

= ts/2−n/4‖g‖2

(ˆ |e−|y |2 − 1|2

|y |2sdy)1/2

= ts/2−n/4‖f ‖Hs

(ˆ min|y |2, 12

|y |2sdy)1/2

≤ Csts/2−n/4‖f ‖Hs .

The same calculation works for the Schrodinger equation.

Page 33: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Lemma (Pointwise convergence for smooth data)

Let u solve the heat or Schrodinger equation with u0 ∈ Hs(Rn)with n/2 < s < n/2 + 2. Then

limt→0

u(x , t) = u0(x) for all x ∈ Rn.

Proof: u0 = | · |−s g with g ∈ L2

(2π)n/2|et∆u0(x)− u0(x)| =

∣∣∣∣∣ˆ

g(ξ) e ix ·ξ(e−t|ξ|2 − 1)

|ξ|sdξ

∣∣∣∣∣≤ ‖g ‖2

(ˆ |e−t|ξ|2 − 1|2

|ξ|2sdξ)1/2

= ts/2−n/4‖g‖2

(ˆ |e−|y |2 − 1|2

|y |2sdy)1/2

= ts/2−n/4‖f ‖Hs

(ˆ min|y |2, 12

|y |2sdy)1/2

≤ Csts/2−n/4‖f ‖Hs .

The same calculation works for the Schrodinger equation.

Page 34: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Lemma (Pointwise convergence for smooth data)

Let u solve the heat or Schrodinger equation with u0 ∈ Hs(Rn)with n/2 < s < n/2 + 2. Then

limt→0

u(x , t) = u0(x) for all x ∈ Rn.

Proof: u0 = | · |−s g with g ∈ L2

(2π)n/2|et∆u0(x)− u0(x)| =

∣∣∣∣∣ˆ

g(ξ) e ix ·ξ(e−t|ξ|2 − 1)

|ξ|sdξ

∣∣∣∣∣≤ ‖g ‖2

(ˆ |e−t|ξ|2 − 1|2

|ξ|2sdξ)1/2

= ts/2−n/4‖g‖2

(ˆ |e−|y |2 − 1|2

|y |2sdy)1/2

= ts/2−n/4‖f ‖Hs

(ˆ min|y |2, 12

|y |2sdy)1/2

≤ Csts/2−n/4‖f ‖Hs .

The same calculation works for the Schrodinger equation.

Page 35: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Lemma (Pointwise convergence for smooth data)

Let u solve the heat or Schrodinger equation with u0 ∈ Hs(Rn)with n/2 < s < n/2 + 2. Then

limt→0

u(x , t) = u0(x) for all x ∈ Rn.

Proof: u0 = | · |−s g with g ∈ L2

(2π)n/2|et∆u0(x)− u0(x)| =

∣∣∣∣∣ˆ

g(ξ) e ix ·ξ(e−t|ξ|2 − 1)

|ξ|sdξ

∣∣∣∣∣≤ ‖g ‖2

(ˆ |e−t|ξ|2 − 1|2

|ξ|2sdξ)1/2

= ts/2−n/4‖g‖2

(ˆ |e−|y |2 − 1|2

|y |2sdy)1/2

= ts/2−n/4‖f ‖Hs

(ˆ min|y |2, 12

|y |2sdy)1/2

≤ Csts/2−n/4‖f ‖Hs .

The same calculation works for the Schrodinger equation.

Page 36: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Lemma (Pointwise convergence for smooth data)

Let u solve the heat or Schrodinger equation with u0 ∈ Hs(Rn)with n/2 < s < n/2 + 2. Then

limt→0

u(x , t) = u0(x) for all x ∈ Rn.

Proof: u0 = | · |−s g with g ∈ L2

(2π)n/2|et∆u0(x)− u0(x)| =

∣∣∣∣∣ˆ

g(ξ) e ix ·ξ(e−t|ξ|2 − 1)

|ξ|sdξ

∣∣∣∣∣≤ ‖g ‖2

(ˆ |e−t|ξ|2 − 1|2

|ξ|2sdξ)1/2

= ts/2−n/4‖g‖2

(ˆ |e−|y |2 − 1|2

|y |2sdy)1/2

= ts/2−n/4‖f ‖Hs

(ˆ min|y |2, 12

|y |2sdy)1/2

≤ Csts/2−n/4‖f ‖Hs .

The same calculation works for the Schrodinger equation.

Page 37: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Lemma (Pointwise convergence for smooth data)

Let u solve the heat or Schrodinger equation with u0 ∈ Hs(Rn)with n/2 < s < n/2 + 2. Then

limt→0

u(x , t) = u0(x) for all x ∈ Rn.

Proof: u0 = | · |−s g with g ∈ L2

(2π)n/2|et∆u0(x)− u0(x)| =

∣∣∣∣∣ˆ

g(ξ) e ix ·ξ(e−t|ξ|2 − 1)

|ξ|sdξ

∣∣∣∣∣≤ ‖g ‖2

(ˆ |e−t|ξ|2 − 1|2

|ξ|2sdξ)1/2

= ts/2−n/4‖g‖2

(ˆ |e−|y |2 − 1|2

|y |2sdy)1/2

= ts/2−n/4‖f ‖Hs

(ˆ min|y |2, 12

|y |2sdy)1/2

≤ Csts/2−n/4‖f ‖Hs .

The same calculation works for the Schrodinger equation.

Page 38: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Lemma (Pointwise convergence for smooth data)

Let u solve the heat or Schrodinger equation with u0 ∈ Hs(Rn)with n/2 < s < n/2 + 2. Then

limt→0

u(x , t) = u0(x) for all x ∈ Rn.

Proof: u0 = | · |−s g with g ∈ L2

(2π)n/2|et∆u0(x)− u0(x)| =

∣∣∣∣∣ˆ

g(ξ) e ix ·ξ(e−t|ξ|2 − 1)

|ξ|sdξ

∣∣∣∣∣≤ ‖g ‖2

(ˆ |e−t|ξ|2 − 1|2

|ξ|2sdξ)1/2

= ts/2−n/4‖g‖2

(ˆ |e−|y |2 − 1|2

|y |2sdy)1/2

= ts/2−n/4‖f ‖Hs

(ˆ min|y |2, 12

|y |2sdy)1/2

≤ Csts/2−n/4‖f ‖Hs .

The same calculation works for the Schrodinger equation.

Page 39: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Lemma (Pointwise convergence for smooth data)

Let u solve the heat or Schrodinger equation with u0 ∈ Hs(Rn)with n/2 < s < n/2 + 2. Then

limt→0

u(x , t) = u0(x) for all x ∈ Rn.

Proof: u0 = | · |−s g with g ∈ L2

(2π)n/2|et∆u0(x)− u0(x)| =

∣∣∣∣∣ˆ

g(ξ) e ix ·ξ(e−t|ξ|2 − 1)

|ξ|sdξ

∣∣∣∣∣≤ ‖g ‖2

(ˆ |e−t|ξ|2 − 1|2

|ξ|2sdξ)1/2

= ts/2−n/4‖g‖2

(ˆ |e−|y |2 − 1|2

|y |2sdy)1/2

= ts/2−n/4‖f ‖Hs

(ˆ min|y |2, 12

|y |2sdy)1/2

≤ Csts/2−n/4‖f ‖Hs .

The same calculation works for the Schrodinger equation.

Page 40: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Lebesgue a.e. convergence for data in L2

Recall that the Hardy–Littlewood maximal operator M is defined by

Mf = supr>0

1|B(0,r)|1B(0,r) ∗ |f |,

and that it is bounded from L2(Rn) to L2(Rn).

This allows one to conclude that

limr→0

1|B(0,r)|1B(0,r) ∗ f (x)→ f (x), a.e. x ∈ Rn,

for all f ∈ L2(Rn).

Later, I will remind you how to prove this using the L2-bound.

Page 41: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Lebesgue a.e. convergence for data in L2

Recall that the Hardy–Littlewood maximal operator M is defined by

Mf = supr>0

1|B(0,r)|1B(0,r) ∗ |f |,

and that it is bounded from L2(Rn) to L2(Rn).

This allows one to conclude that

limr→0

1|B(0,r)|1B(0,r) ∗ f (x)→ f (x), a.e. x ∈ Rn,

for all f ∈ L2(Rn).

Later, I will remind you how to prove this using the L2-bound.

Page 42: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Lebesgue a.e. convergence for data in L2

Recall that the Hardy–Littlewood maximal operator M is defined by

Mf = supr>0

1|B(0,r)|1B(0,r) ∗ |f |,

and that it is bounded from L2(Rn) to L2(Rn).

This allows one to conclude that

limr→0

1|B(0,r)|1B(0,r) ∗ f (x)→ f (x), a.e. x ∈ Rn,

for all f ∈ L2(Rn).

Later, I will remind you how to prove this using the L2-bound.

Page 43: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Lebesgue a.e. convergence for data in L2

Recall that the Hardy–Littlewood maximal operator M is defined by

Mf = supr>0

1|B(0,r)|1B(0,r) ∗ |f |,

and that it is bounded from L2(Rn) to L2(Rn).

This allows one to conclude that

limr→0

1|B(0,r)|1B(0,r) ∗ f (x)→ f (x), a.e. x ∈ Rn,

for all f ∈ L2(Rn).

Later, I will remind you how to prove this using the L2-bound.

Page 44: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Now e−|·|2 ≤

∑j≥0

2−j(n+1)1B(0,2j )

so thate−|·|

2/t ≤∑j≥0

2−j(n+1)1B(0,t1/22j )

so that

1

tn/2e−|·|

2/t ≤∑j≥0

2−j1

|B(0, t1/22j)|1B(0,t1/22j ).

Thus

supt>0|et∆f | = sup

t>0| 1

tn/2e |·|

2/t ∗ f | ≤∑j≥0

2−jMf ≤ 2Mf .

So the L2-bound for M gives an L2 maximal estimate for the heatequation which allows us to conclude that

limt→0

et∆f (x) = f (x), a.e. x ∈ Rn,

using the same argument, which I will remind you of soon.

Page 45: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Now e−|·|2 ≤

∑j≥0

2−j(n+1)1B(0,2j )

so thate−|·|

2/t ≤∑j≥0

2−j(n+1)1B(0,t1/22j )

so that

1

tn/2e−|·|

2/t ≤∑j≥0

2−j1

|B(0, t1/22j)|1B(0,t1/22j ).

Thus

supt>0|et∆f | = sup

t>0| 1

tn/2e |·|

2/t ∗ f | ≤∑j≥0

2−jMf ≤ 2Mf .

So the L2-bound for M gives an L2 maximal estimate for the heatequation which allows us to conclude that

limt→0

et∆f (x) = f (x), a.e. x ∈ Rn,

using the same argument, which I will remind you of soon.

Page 46: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Now e−|·|2 ≤

∑j≥0

2−j(n+1)1B(0,2j )

so thate−|·|

2/t ≤∑j≥0

2−j(n+1)1B(0,t1/22j )

so that

1

tn/2e−|·|

2/t ≤∑j≥0

2−j1

|B(0, t1/22j)|1B(0,t1/22j ).

Thus

supt>0|et∆f | = sup

t>0| 1

tn/2e |·|

2/t ∗ f | ≤∑j≥0

2−jMf ≤ 2Mf .

So the L2-bound for M gives an L2 maximal estimate for the heatequation which allows us to conclude that

limt→0

et∆f (x) = f (x), a.e. x ∈ Rn,

using the same argument, which I will remind you of soon.

Page 47: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Now e−|·|2 ≤

∑j≥0

2−j(n+1)1B(0,2j )

so thate−|·|

2/t ≤∑j≥0

2−j(n+1)1B(0,t1/22j )

so that

1

tn/2e−|·|

2/t ≤∑j≥0

2−j1

|B(0, t1/22j)|1B(0,t1/22j ).

Thus

supt>0|et∆f | = sup

t>0| 1

tn/2e |·|

2/t ∗ f | ≤∑j≥0

2−jMf ≤ 2Mf .

So the L2-bound for M gives an L2 maximal estimate for the heatequation which allows us to conclude that

limt→0

et∆f (x) = f (x), a.e. x ∈ Rn,

using the same argument, which I will remind you of soon.

Page 48: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Now e−|·|2 ≤

∑j≥0

2−j(n+1)1B(0,2j )

so thate−|·|

2/t ≤∑j≥0

2−j(n+1)1B(0,t1/22j )

so that

1

tn/2e−|·|

2/t ≤∑j≥0

2−j1

|B(0, t1/22j)|1B(0,t1/22j ).

Thus

supt>0|et∆f | = sup

t>0| 1

tn/2e |·|

2/t ∗ f | ≤∑j≥0

2−jMf ≤ 2Mf .

So the L2-bound for M gives an L2 maximal estimate for the heatequation which allows us to conclude that

limt→0

et∆f (x) = f (x), a.e. x ∈ Rn,

using the same argument, which I will remind you of soon.

Page 49: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Hausdorff measure

Let A ⊆ Rn be a borel set, 0 < α < n and

Hαδ (A) := inf∑

i

δαi : A ⊂⋃i

B(xi , δi ), δi < δ.

Definition

The α-Hausdorff measure of A is

Hα(A) := limδ→0Hαδ (A).

Page 50: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Hausdorff measure

Let A ⊆ Rn be a borel set, 0 < α < n and

Hαδ (A) := inf∑

i

δαi : A ⊂⋃i

B(xi , δi ), δi < δ.

Definition

The α-Hausdorff measure of A is

Hα(A) := limδ→0Hαδ (A).

Page 51: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Hausdorff dimension

Remark

There exists a unique α0 such that

Hα(A) =

∞ if α < α0

0 if α > α0.

Definition

α0 is the Hausdorff dimension of the set A:

dim(A) := α0.

Page 52: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Hausdorff dimension

Remark

There exists a unique α0 such that

Hα(A) =

∞ if α < α0

0 if α > α0.

Definition

α0 is the Hausdorff dimension of the set A:

dim(A) := α0.

Page 53: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Definition (Frostman measures)

We say that a positive Borel measure µ with supp(µ) ⊂ B(0, 1) isα-dimensional if

cα(µ) := supx∈Rn

r>0

µ(B(x , r))

rα<∞.

Eα′(µ) :=

ˆ ˆdµ(x)dµ(y)

|x − y |α′=

ˆ ∞∑j=0

ˆA(y ,2−j )

dµ(x)

|x − y |α′dµ(y)

≤ˆ ∞∑

j=0

cα(µ)2−jα2jα′dµ(y)

. cα(µ)‖µ‖ <∞ if α > α′.

Lemma (Frostman)

Let A ⊂ Rn be a Borel set. The following are equivalent:

I Hα(A) > 0;

I there is an α-dimensional measure µ such that µ(A) > 0.

Page 54: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Definition (Frostman measures)

We say that a positive Borel measure µ with supp(µ) ⊂ B(0, 1) isα-dimensional if

cα(µ) := supx∈Rn

r>0

µ(B(x , r))

rα<∞.

Eα′(µ) :=

ˆ ˆdµ(x)dµ(y)

|x − y |α′=

ˆ ∞∑j=0

ˆA(y ,2−j )

dµ(x)

|x − y |α′dµ(y)

≤ˆ ∞∑

j=0

cα(µ)2−jα2jα′dµ(y)

. cα(µ)‖µ‖ <∞ if α > α′.

Lemma (Frostman)

Let A ⊂ Rn be a Borel set. The following are equivalent:

I Hα(A) > 0;

I there is an α-dimensional measure µ such that µ(A) > 0.

Page 55: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Definition (Frostman measures)

We say that a positive Borel measure µ with supp(µ) ⊂ B(0, 1) isα-dimensional if

cα(µ) := supx∈Rn

r>0

µ(B(x , r))

rα<∞.

Eα′(µ) :=

ˆ ˆdµ(x)dµ(y)

|x − y |α′=

ˆ ∞∑j=0

ˆA(y ,2−j )

dµ(x)

|x − y |α′dµ(y)

≤ˆ ∞∑

j=0

cα(µ)2−jα2jα′dµ(y)

. cα(µ)‖µ‖ <∞ if α > α′.

Lemma (Frostman)

Let A ⊂ Rn be a Borel set. The following are equivalent:

I Hα(A) > 0;

I there is an α-dimensional measure µ such that µ(A) > 0.

Page 56: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Definition (Frostman measures)

We say that a positive Borel measure µ with supp(µ) ⊂ B(0, 1) isα-dimensional if

cα(µ) := supx∈Rn

r>0

µ(B(x , r))

rα<∞.

Eα′(µ) :=

ˆ ˆdµ(x)dµ(y)

|x − y |α′=

ˆ ∞∑j=0

ˆA(y ,2−j )

dµ(x)

|x − y |α′dµ(y)

≤ˆ ∞∑

j=0

cα(µ)2−jα2jα′dµ(y)

. cα(µ)‖µ‖ <∞ if α > α′.

Lemma (Frostman)

Let A ⊂ Rn be a Borel set. The following are equivalent:

I Hα(A) > 0;

I there is an α-dimensional measure µ such that µ(A) > 0.

Page 57: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Definition (Frostman measures)

We say that a positive Borel measure µ with supp(µ) ⊂ B(0, 1) isα-dimensional if

cα(µ) := supx∈Rn

r>0

µ(B(x , r))

rα<∞.

Eα′(µ) :=

ˆ ˆdµ(x)dµ(y)

|x − y |α′=

ˆ ∞∑j=0

ˆA(y ,2−j )

dµ(x)

|x − y |α′dµ(y)

≤ˆ ∞∑

j=0

cα(µ)2−jα2jα′dµ(y)

. cα(µ)‖µ‖ <∞ if α > α′.

Lemma (Frostman)

Let A ⊂ Rn be a Borel set. The following are equivalent:

I Hα(A) > 0;

I there is an α-dimensional measure µ such that µ(A) > 0.

Page 58: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Control of singularities

Lemma

Let 0 < s < n/2 and α > n − 2s. Then, for all α-dimensional µ,

‖f ‖L1(dµ) ≤ Cµ‖f ‖Hs .

Proof: f = Is ∗ g with g ∈ L2 and Is = | · |−s . Suffices to prove

‖Is ∗ g‖L1(dµ) .√En−2s(µ) ‖g‖L2(Rn).

By Fubini’s theorem and the Cauchy–Schwarz inequality,

‖Is ∗ g‖L1(dµ) ≤ˆ ˆ

Is(x − y) dµ(x) |g(y)| dy

≤ ‖Is ∗ µ‖L2(Rn)‖g‖L2(Rn).

Page 59: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Control of singularities

Lemma

Let 0 < s < n/2 and α > n − 2s. Then, for all α-dimensional µ,

‖f ‖L1(dµ) ≤ Cµ‖f ‖Hs .

Proof: f = Is ∗ g with g ∈ L2 and Is = | · |−s . Suffices to prove

‖Is ∗ g‖L1(dµ) .√En−2s(µ) ‖g‖L2(Rn).

By Fubini’s theorem and the Cauchy–Schwarz inequality,

‖Is ∗ g‖L1(dµ) ≤ˆ ˆ

Is(x − y) dµ(x) |g(y)| dy

≤ ‖Is ∗ µ‖L2(Rn)‖g‖L2(Rn).

Page 60: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Control of singularities

Lemma

Let 0 < s < n/2 and α > n − 2s. Then, for all α-dimensional µ,

‖f ‖L1(dµ) ≤ Cµ‖f ‖Hs .

Proof: f = Is ∗ g with g ∈ L2 and Is = | · |−s . Suffices to prove

‖Is ∗ g‖L1(dµ) .√En−2s(µ) ‖g‖L2(Rn).

By Fubini’s theorem and the Cauchy–Schwarz inequality,

‖Is ∗ g‖L1(dµ) ≤ˆ ˆ

Is(x − y) dµ(x) |g(y)| dy

≤ ‖Is ∗ µ‖L2(Rn)‖g‖L2(Rn).

Page 61: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Control of singularities

Lemma

Let 0 < s < n/2 and α > n − 2s. Then, for all α-dimensional µ,

‖f ‖L1(dµ) ≤ Cµ‖f ‖Hs .

Proof: f = Is ∗ g with g ∈ L2 and Is = | · |−s . Suffices to prove

‖Is ∗ g‖L1(dµ) .√En−2s(µ) ‖g‖L2(Rn).

By Fubini’s theorem and the Cauchy–Schwarz inequality,

‖Is ∗ g‖L1(dµ) ≤ˆ ˆ

Is(x − y) dµ(x) |g(y)| dy

≤ ‖Is ∗ µ‖L2(Rn)‖g‖L2(Rn).

Page 62: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Thus it remains to prove that

‖Is ∗ µ‖2L2(Rn) . En−2s(µ).

By Plancherel’s theorem,

‖Is ∗ µ‖2L2(Rn) = ‖Is µ‖2

L2(Rn) =

ˆµ(ξ) µ(ξ) I2s(ξ) dξ.

Recalling that I2s(x) = Cn,s |x |−(n−2s),

‖Is ∗ µ‖2L2(Rn) =

ˆµ ∗ I2s(y) dµ(y)

= Cn,s

ˆ ˆdµ(x)dµ(y)

|x − y |n−2s= Cn,sEn−2s(µ),

and we are done.

Page 63: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Thus it remains to prove that

‖Is ∗ µ‖2L2(Rn) . En−2s(µ).

By Plancherel’s theorem,

‖Is ∗ µ‖2L2(Rn) = ‖Is µ‖2

L2(Rn) =

ˆµ(ξ) µ(ξ) I2s(ξ) dξ.

Recalling that I2s(x) = Cn,s |x |−(n−2s),

‖Is ∗ µ‖2L2(Rn) =

ˆµ ∗ I2s(y) dµ(y)

= Cn,s

ˆ ˆdµ(x)dµ(y)

|x − y |n−2s= Cn,sEn−2s(µ),

and we are done.

Page 64: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Thus it remains to prove that

‖Is ∗ µ‖2L2(Rn) . En−2s(µ).

By Plancherel’s theorem,

‖Is ∗ µ‖2L2(Rn) = ‖Is µ‖2

L2(Rn) =

ˆµ(ξ) µ(ξ) I2s(ξ) dξ.

Recalling that I2s(x) = Cn,s |x |−(n−2s),

‖Is ∗ µ‖2L2(Rn) =

ˆµ ∗ I2s(y) dµ(y)

= Cn,s

ˆ ˆdµ(x)dµ(y)

|x − y |n−2s= Cn,sEn−2s(µ),

and we are done.

Page 65: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Thus it remains to prove that

‖Is ∗ µ‖2L2(Rn) . En−2s(µ).

By Plancherel’s theorem,

‖Is ∗ µ‖2L2(Rn) = ‖Is µ‖2

L2(Rn) =

ˆµ(ξ) µ(ξ) I2s(ξ) dξ.

Recalling that I2s(x) = Cn,s |x |−(n−2s),

‖Is ∗ µ‖2L2(Rn) =

ˆµ ∗ I2s(y) dµ(y)

= Cn,s

ˆ ˆdµ(x)dµ(y)

|x − y |n−2s= Cn,sEn−2s(µ),

and we are done.

Page 66: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Optimality of the control of singularities lemma

If dim(A) = α with α < n − 2s, then we can take a γ such that

α < γ < n − 2s.

Then1B(0,1)dist(·,A)−γ ∈ L2(Rn)

but on the other hand

u0 := Is ∗[1B(0,1)dist(·,A)−γ

]=∞ on A.

So there is initial data u0 ∈ Hs(Rn) which is singular on a set ofdimension α < n − 2s.

Page 67: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Optimality of the control of singularities lemma

If dim(A) = α with α < n − 2s, then we can take a γ such that

α < γ < n − 2s.

Then1B(0,1)dist(·,A)−γ ∈ L2(Rn)

but on the other hand

u0 := Is ∗[1B(0,1)dist(·,A)−γ

]=∞ on A.

So there is initial data u0 ∈ Hs(Rn) which is singular on a set ofdimension α < n − 2s.

Page 68: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Optimality of the control of singularities lemma

If dim(A) = α with α < n − 2s, then we can take a γ such that

α < γ < n − 2s.

Then1B(0,1)dist(·,A)−γ ∈ L2(Rn)

but on the other hand

u0 := Is ∗[1B(0,1)dist(·,A)−γ

]=∞ on A.

So there is initial data u0 ∈ Hs(Rn) which is singular on a set ofdimension α < n − 2s.

Page 69: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Optimality of the control of singularities lemma

If dim(A) = α with α < n − 2s, then we can take a γ such that

α < γ < n − 2s.

Then1B(0,1)dist(·,A)−γ ∈ L2(Rn)

but on the other hand

u0 := Is ∗[1B(0,1)dist(·,A)−γ

]=∞ on A.

So there is initial data u0 ∈ Hs(Rn) which is singular on a set ofdimension α < n − 2s.

Page 70: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Optimality of the control of singularities lemma

If dim(A) = α with α < n − 2s, then we can take a γ such that

α < γ < n − 2s.

Then1B(0,1)dist(·,A)−γ ∈ L2(Rn)

but on the other hand

u0 := Is ∗[1B(0,1)dist(·,A)−γ

]=∞ on A.

So there is initial data u0 ∈ Hs(Rn) which is singular on a set ofdimension α < n − 2s.

Page 71: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Proposition (Maximal estimates imply convergence)

Let α > α0 ≥ n − 2s. Suppose that, for all α-dimensional µ,∥∥∥ sup0<t<1

|u(·, t)|∥∥∥L1(dµ)

≤ Cµ‖u0‖Hs .

Then, for all u0 ∈ Hs ,

dimx ∈ Rn lim

t→0u(t, x) 6= u0(x)

≤ α0.

Proof: We are required to prove that for all α > α0,

Hαx ∈ Rn lim

t→0u(t, x) 6= u0(x)

= 0

whenever u0 ∈ Hs . By Frostman’s lemma, this follows by showing

µx ∈ Rn lim

t→0u(t, x) 6= u0(x)

= 0

whenever µ is α-dimensional.

Page 72: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Proposition (Maximal estimates imply convergence)

Let α > α0 ≥ n − 2s. Suppose that, for all α-dimensional µ,∥∥∥ sup0<t<1

|u(·, t)|∥∥∥L1(dµ)

≤ Cµ‖u0‖Hs .

Then, for all u0 ∈ Hs ,

dimx ∈ Rn lim

t→0u(t, x) 6= u0(x)

≤ α0.

Proof: We are required to prove that for all α > α0,

Hαx ∈ Rn lim

t→0u(t, x) 6= u0(x)

= 0

whenever u0 ∈ Hs . By Frostman’s lemma, this follows by showing

µx ∈ Rn lim

t→0u(t, x) 6= u0(x)

= 0

whenever µ is α-dimensional.

Page 73: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Proposition (Maximal estimates imply convergence)

Let α > α0 ≥ n − 2s. Suppose that, for all α-dimensional µ,∥∥∥ sup0<t<1

|u(·, t)|∥∥∥L1(dµ)

≤ Cµ‖u0‖Hs .

Then, for all u0 ∈ Hs ,

dimx ∈ Rn lim

t→0u(t, x) 6= u0(x)

≤ α0.

Proof: We are required to prove that for all α > α0,

Hαx ∈ Rn lim

t→0u(t, x) 6= u0(x)

= 0

whenever u0 ∈ Hs . By Frostman’s lemma, this follows by showing

µx ∈ Rn lim

t→0u(t, x) 6= u0(x)

= 0

whenever µ is α-dimensional.

Page 74: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Proposition (Maximal estimates imply convergence)

Let α > α0 ≥ n − 2s. Suppose that, for all α-dimensional µ,∥∥∥ sup0<t<1

|u(·, t)|∥∥∥L1(dµ)

≤ Cµ‖u0‖Hs .

Then, for all u0 ∈ Hs ,

dimx ∈ Rn lim

t→0u(t, x) 6= u0(x)

≤ α0.

Proof: We are required to prove that for all α > α0,

Hαx ∈ Rn lim

t→0u(t, x) 6= u0(x)

= 0

whenever u0 ∈ Hs . By Frostman’s lemma, this follows by showing

µx ∈ Rn lim

t→0u(t, x) 6= u0(x)

= 0

whenever µ is α-dimensional.

Page 75: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Proposition (Maximal estimates imply convergence)

Let α > α0 ≥ n − 2s. Suppose that, for all α-dimensional µ,∥∥∥ sup0<t<1

|u(·, t)|∥∥∥L1(dµ)

≤ Cµ‖u0‖Hs .

Then, for all u0 ∈ Hs ,

dimx ∈ Rn lim

t→0u(t, x) 6= u0(x)

≤ α0.

Proof: We are required to prove that for all α > α0,

Hαx ∈ Rn lim

t→0u(t, x) 6= u0(x)

= 0

whenever u0 ∈ Hs . By Frostman’s lemma, this follows by showing

µx ∈ Rn lim

t→0u(t, x) 6= u0(x)

= 0

whenever µ is α-dimensional.

Page 76: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Take h ∈ Hn/2+1 such that ‖u0 − h‖Hs < ε, and note that

|u(·, t)− u0| ≤ |u(·, t)− uh(·, t)|+ |uh(·, t)− h|+ |h − u0|,

where uh denotes the solution with initial data h.

Page 77: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Take h ∈ Hn/2+1 such that ‖u0 − h‖Hs < ε, and note that

|u(·, t)− u0| ≤ |uu0−h(·, t)|+ |uh(·, t)− h|+ |h − u0|.

Then, µ x : lim supt→0

|u(x , t)− u0(x)| > λ

≤ µ x : lim supt→0

|uu0−h(x , t)| > λ/3

+ µ x : lim supt→0

|uh(x , t)− h| > λ/3

+ µ x : lim supt→0

|h(x)− u0(x)| > λ/3 .

Page 78: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Take h ∈ Hn/2+1 such that ‖u0 − h‖Hs < ε, and note that

|u(·, t)− u0| ≤ |uu0−h(·, t)|+ |uh(·, t)− h|+ |h − u0|.

Then, µ x : lim supt→0

|u(x , t)− u0(x)| > λ

≤ µ x : lim supt→0

|uu0−h(x , t)| > λ/3

+ µ x : lim supt→0

|uh(x , t)− h| > λ/3

+ µ x : lim supt→0

|h(x)− u0(x)| > λ/3 .

Page 79: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Take h ∈ Hn/2+1 such that ‖u0 − h‖Hs < ε, and note that

|u(·, t)− u0| ≤ |uu0−h(·, t)|+ |uh(·, t)− h|+ |h − u0|.

Then, µ x : lim supt→0

|u(x , t)− u0(x)| > λ

≤ µ x : sup0<t<1

|uu0−h(x , t)| > λ/3

+ µ x : lim supt→0

|uh(x , t)− h| > λ/3

+ µ x : |h(x)− u0(x)| > λ/3 .

We use the maximal estimate for the first term, the second term iszero by the smooth data lemma, and the third term can bebounded by the control of singularities lemma so that

µ x : limt→0|u(x , t)− u0(x)| > λ ≤ λ−1Cµ ‖u0 − h‖Hs(Rn) ≤ λ

−1Cµ ε.

Letting ε tend to zero, then λ tend to zero, we are done.

Page 80: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Take h ∈ Hn/2+1 such that ‖u0 − h‖Hs < ε, and note that

|u(·, t)− u0| ≤ |uu0−h(·, t)|+ |uh(·, t)− h|+ |h − u0|.

Then, µ x : lim supt→0

|u(x , t)− u0(x)| > λ

≤ µ x : sup0<t<1

|uu0−h(x , t)| > λ/3

+ µ x : lim supt→0

|uh(x , t)− h| > λ/3

+ µ x : |h(x)− u0(x)| > λ/3 .

We use the maximal estimate for the first term, the second term iszero by the smooth data lemma, and the third term can bebounded by the control of singularities lemma so that

µ x : limt→0|u(x , t)− u0(x)| > λ ≤ λ−1Cµ ‖u0 − h‖Hs(Rn) ≤ λ

−1Cµ ε.

Letting ε tend to zero, then λ tend to zero, we are done.

Page 81: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Take h ∈ Hn/2+1 such that ‖u0 − h‖Hs < ε, and note that

|u(·, t)− u0| ≤ |uu0−h(·, t)|+ |uh(·, t)− h|+ |h − u0|.

Then, µ x : lim supt→0

|u(x , t)− u0(x)| > λ

≤ µ x : sup0<t<1

|uu0−h(x , t)| > λ/3

+ µ x : lim supt→0

|uh(x , t)− h| > λ/3

+ µ x : |h(x)− u0(x)| > λ/3 .

We use the maximal estimate for the first term, the second term iszero by the smooth data lemma, and the third term can bebounded by the control of singularities lemma so that

µ x : limt→0|u(x , t)− u0(x)| > λ ≤ λ−1Cµ ‖u0 − h‖Hs(Rn) ≤ λ

−1Cµ ε.

Letting ε tend to zero, then λ tend to zero, we are done.

Page 82: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Take h ∈ Hn/2+1 such that ‖u0 − h‖Hs < ε, and note that

|u(·, t)− u0| ≤ |uu0−h(·, t)|+ |uh(·, t)− h|+ |h − u0|.

Then, µ x : lim supt→0

|u(x , t)− u0(x)| > λ

≤ µ x : sup0<t<1

|uu0−h(x , t)| > λ/3

+ µ x : lim supt→0

|uh(x , t)− h| > λ/3

+ µ x : |h(x)− u0(x)| > λ/3 .

We use the maximal estimate for the first term, the second term iszero by the smooth data lemma, and the third term can bebounded by the control of singularities lemma so that

µ x : limt→0|u(x , t)− u0(x)| > λ ≤ λ−1Cµ ‖u0 − h‖Hs(Rn) ≤ λ

−1Cµ ε.

Letting ε tend to zero, then λ tend to zero, we are done.

Page 83: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Take h ∈ Hn/2+1 such that ‖u0 − h‖Hs < ε, and note that

|u(·, t)− u0| ≤ |uu0−h(·, t)|+ |uh(·, t)− h|+ |h − u0|.

Then, µ x : lim supt→0

|u(x , t)− u0(x)| > λ

≤ µ x : sup0<t<1

|uu0−h(x , t)| > λ/3

+ µ x : lim supt→0

|uh(x , t)− h| > λ/3

+ µ x : |h(x)− u0(x)| > λ/3 .

We use the maximal estimate for the first term, the second term iszero by the smooth data lemma, and the third term can bebounded by the control of singularities lemma so that

µ x : limt→0|u(x , t)− u0(x)| > λ ≤ λ−1Cµ ‖u0 − h‖Hs(Rn) ≤ λ

−1Cµ ε.

Letting ε tend to zero, then λ tend to zero, we are done.

Page 84: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Take h ∈ Hn/2+1 such that ‖u0 − h‖Hs < ε, and note that

|u(·, t)− u0| ≤ |uu0−h(·, t)|+ |uh(·, t)− h|+ |h − u0|.

Then, µ x : lim supt→0

|u(x , t)− u0(x)| > λ

≤ µ x : sup0<t<1

|uu0−h(x , t)| > λ/3

+ µ x : lim supt→0

|uh(x , t)− h| > λ/3

+ µ x : |h(x)− u0(x)| > λ/3 .

We use the maximal estimate for the first term, the second term iszero by the smooth data lemma, and the third term can bebounded by the control of singularities lemma so that

µ x : limt→0|u(x , t)− u0(x)| > λ ≤ λ−1Cµ ‖u0 − h‖Hs(Rn) ≤ λ

−1Cµ ε.

Letting ε tend to zero, then λ tend to zero, we are done.

Page 85: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Part 2:

Convergence for the heatequation

Page 86: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Theorem (Maximal estimate for the heat equation)

Let 0 < s < n/2 and α > n − 2s. Then, for all α-dimensional µ,∥∥∥ sup0<t<1

|et∆f |∥∥∥L1(dµ)

≤ Cµ‖f ‖Hs .

Proof: By linearising the operator, it will suffice to prove∣∣∣∣ˆ ˆ e ix ·ξe−t(x)|ξ|2 f (ξ) dξ w(x) dµ(x)

∣∣∣∣2 . En−2s(µ) ‖f ‖2Hs ,

whenever t : Rn → (0,∞) and w : Rn → S1 are measurable. Now,by Fubini and Cauchy–Schwarz, the LHS is bounded by

ˆ|f (ξ)|2|ξ|2sdξ

ˆ ∣∣∣∣ˆ e ix ·ξe−t(x)|ξ|2w(x) dµ(x)

∣∣∣∣2 dξ

|ξ|2s.

Squaring out the integral, it will suffice to show thatˆ ˆ ˆe i(x−y)·ξe−(t(x)+t(y))|ξ|2 dξ

|ξ|2sw(x)w(y) dµ(x)dµ(y) . En−2s(µ).

Page 87: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Theorem (Maximal estimate for the heat equation)

Let 0 < s < n/2 and α > n − 2s. Then, for all α-dimensional µ,∥∥∥ sup0<t<1

|et∆f |∥∥∥L1(dµ)

≤ Cµ‖f ‖Hs .

Proof: By linearising the operator, it will suffice to prove∣∣∣∣ˆ ˆ e ix ·ξe−t(x)|ξ|2 f (ξ) dξ w(x) dµ(x)

∣∣∣∣2 . En−2s(µ) ‖f ‖2Hs ,

whenever t : Rn → (0,∞) and w : Rn → S1 are measurable. Now,by Fubini and Cauchy–Schwarz, the LHS is bounded by

ˆ|f (ξ)|2|ξ|2sdξ

ˆ ∣∣∣∣ˆ e ix ·ξe−t(x)|ξ|2w(x) dµ(x)

∣∣∣∣2 dξ

|ξ|2s.

Squaring out the integral, it will suffice to show thatˆ ˆ ˆe i(x−y)·ξe−(t(x)+t(y))|ξ|2 dξ

|ξ|2sw(x)w(y) dµ(x)dµ(y) . En−2s(µ).

Page 88: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Theorem (Maximal estimate for the heat equation)

Let 0 < s < n/2 and α > n − 2s. Then, for all α-dimensional µ,∥∥∥ sup0<t<1

|et∆f |∥∥∥L1(dµ)

≤ Cµ‖f ‖Hs .

Proof: By linearising the operator, it will suffice to prove∣∣∣∣ˆ ˆ e ix ·ξe−t(x)|ξ|2 f (ξ) dξ w(x) dµ(x)

∣∣∣∣2 . En−2s(µ) ‖f ‖2Hs ,

whenever t : Rn → (0,∞) and w : Rn → S1 are measurable. Now,by Fubini and Cauchy–Schwarz, the LHS is bounded by

ˆ|f (ξ)|2|ξ|2sdξ

ˆ ∣∣∣∣ˆ e ix ·ξe−t(x)|ξ|2w(x) dµ(x)

∣∣∣∣2 dξ

|ξ|2s.

Squaring out the integral, it will suffice to show thatˆ ˆ ˆe i(x−y)·ξe−(t(x)+t(y))|ξ|2 dξ

|ξ|2sw(x)w(y) dµ(x)dµ(y) . En−2s(µ).

Page 89: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Theorem (Maximal estimate for the heat equation)

Let 0 < s < n/2 and α > n − 2s. Then, for all α-dimensional µ,∥∥∥ sup0<t<1

|et∆f |∥∥∥L1(dµ)

≤ Cµ‖f ‖Hs .

Proof: By linearising the operator, it will suffice to prove∣∣∣∣ˆ ˆ e ix ·ξe−t(x)|ξ|2 f (ξ) dξ w(x) dµ(x)

∣∣∣∣2 . En−2s(µ) ‖f ‖2Hs ,

whenever t : Rn → (0,∞) and w : Rn → S1 are measurable. Now,by Fubini and Cauchy–Schwarz, the LHS is bounded by

ˆ|f (ξ)|2|ξ|2sdξ

ˆ ∣∣∣∣ˆ e ix ·ξe−t(x)|ξ|2w(x) dµ(x)

∣∣∣∣2 dξ

|ξ|2s.

Squaring out the integral, it will suffice to show thatˆ ˆ ˆe i(x−y)·ξe−(t(x)+t(y))|ξ|2 dξ

|ξ|2sw(x)w(y) dµ(x)dµ(y) . En−2s(µ).

Page 90: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Theorem (Maximal estimate for the heat equation)

Let 0 < s < n/2 and α > n − 2s. Then, for all α-dimensional µ,∥∥∥ sup0<t<1

|et∆f |∥∥∥L1(dµ)

≤ Cµ‖f ‖Hs .

Proof: By linearising the operator, it will suffice to prove∣∣∣∣ˆ ˆ e ix ·ξe−t(x)|ξ|2 f (ξ) dξ w(x) dµ(x)

∣∣∣∣2 . En−2s(µ) ‖f ‖2Hs ,

whenever t : Rn → (0,∞) and w : Rn → S1 are measurable. Now,by Fubini and Cauchy–Schwarz, the LHS is bounded by

ˆ|f (ξ)|2|ξ|2sdξ

ˆ ∣∣∣∣ˆ e ix ·ξe−t(x)|ξ|2w(x) dµ(x)

∣∣∣∣2 dξ

|ξ|2s.

Squaring out the integral, it will suffice to show thatˆ ˆ ˆe i(x−y)·ξe−(t(x)+t(y))|ξ|2 dξ

|ξ|2sw(x)w(y) dµ(x)dµ(y) . En−2s(µ).

Page 91: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Thus, it remains to prove that, for 0 < s < n/2,∣∣∣∣ˆ e i(x−y)·ξe−(t(x)+t(y))|ξ|2 dξ

|ξ|2s

∣∣∣∣ . 1

|x − y |n−2s

uniformly for all choices of t(x), t(y) > 0. Recalling that

| · |−2s = Cn,s | · |2s−n, this would follow from

1

λn/2e−|·|

2/λ ∗ 1

| · |n−2s.

1

| · |n−2s.

uniformly in λ. By changing variables, this would follow from

e−|·|2 ∗ 1

| · |n−2s.

1

| · |n−2s,

which can be checked by direct calculation.

Page 92: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Thus, it remains to prove that, for 0 < s < n/2,∣∣∣∣ˆ e i(x−y)·ξe−(t(x)+t(y))|ξ|2 dξ

|ξ|2s

∣∣∣∣ . 1

|x − y |n−2s

uniformly for all choices of t(x), t(y) > 0. Recalling that

| · |−2s = Cn,s | · |2s−n, this would follow from

1

λn/2e−|·|

2/λ ∗ 1

| · |n−2s.

1

| · |n−2s.

uniformly in λ. By changing variables, this would follow from

e−|·|2 ∗ 1

| · |n−2s.

1

| · |n−2s,

which can be checked by direct calculation.

Page 93: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Thus, it remains to prove that, for 0 < s < n/2,∣∣∣∣ˆ e i(x−y)·ξe−(t(x)+t(y))|ξ|2 dξ

|ξ|2s

∣∣∣∣ . 1

|x − y |n−2s

uniformly for all choices of t(x), t(y) > 0. Recalling that

| · |−2s = Cn,s | · |2s−n, this would follow from

1

λn/2e−|·|

2/λ ∗ 1

| · |n−2s.

1

| · |n−2s.

uniformly in λ. By changing variables, this would follow from

e−|·|2 ∗ 1

| · |n−2s.

1

| · |n−2s,

which can be checked by direct calculation.

Page 94: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Thus, it remains to prove that, for 0 < s < n/2,∣∣∣∣ˆ e i(x−y)·ξe−(t(x)+t(y))|ξ|2 dξ

|ξ|2s

∣∣∣∣ . 1

|x − y |n−2s

uniformly for all choices of t(x), t(y) > 0. Recalling that

| · |−2s = Cn,s | · |2s−n, this would follow from

1

λn/2e−|·|

2/λ ∗ 1

| · |n−2s.

1

| · |n−2s.

uniformly in λ. By changing variables, this would follow from

e−|·|2 ∗ 1

| · |n−2s.

1

| · |n−2s,

which can be checked by direct calculation.

Page 95: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Corollary

Let 0 < s < n/2 and let u be a solution to the heat equation withinitial data u0 ∈ Hs . Then

dimx ∈ Rn lim

t→0u(t, x) 6= u0(x)

≤ n − 2s.

As we saw before, u0 ∈ Hs can be singular on a set of dimensionless than n − 2s and so this is optimal.

Page 96: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Corollary

Let 0 < s < n/2 and let u be a solution to the heat equation withinitial data u0 ∈ Hs . Then

dimx ∈ Rn lim

t→0u(t, x) 6= u0(x)

≤ n − 2s.

As we saw before, u0 ∈ Hs can be singular on a set of dimensionless than n − 2s and so this is optimal.

Page 97: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Part 3:

Convergence for theSchrodinger equation

Page 98: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Lebesgue a.e. convergence for Schrodinger

Studied by many authors:

Carleson (1979), Dahlberg–Kenig (1982), Cowling (1983),Carbery (1985), Sjolin (1987), Vega (1988), Bourgain (1992/95),Moyua–Vargas–Vega (1996/99), Tao–Vargas (2000), Tao (2003),Lee (2006), Bourgain (2013).

Best known sufficient condition for Lebesgue a.e. convergence:

I s ≥ 1/4 in dimension n = 1 (Carleson);

I s > 12 −

14n in dimension n ≥ 2 (Lee, Bourgain).

Best known necessary condition for Lebesgue a.e. convergence:

I s ≥ 1/4 in dimension n = 1 (Dahlberg–Kenig);

I s ≥ 12 −

1n+2 in dimension n ≥ 2 (Luca–R.).

Page 99: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Lebesgue a.e. convergence for Schrodinger

Studied by many authors:

Carleson (1979), Dahlberg–Kenig (1982), Cowling (1983),Carbery (1985), Sjolin (1987), Vega (1988), Bourgain (1992/95),Moyua–Vargas–Vega (1996/99), Tao–Vargas (2000), Tao (2003),Lee (2006), Bourgain (2013).

Best known sufficient condition for Lebesgue a.e. convergence:

I s ≥ 1/4 in dimension n = 1 (Carleson);

I s > 12 −

14n in dimension n ≥ 2 (Lee, Bourgain).

Best known necessary condition for Lebesgue a.e. convergence:

I s ≥ 1/4 in dimension n = 1 (Dahlberg–Kenig);

I s ≥ 12 −

1n+2 in dimension n ≥ 2 (Luca–R.).

Page 100: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Lebesgue a.e. convergence for Schrodinger

Studied by many authors:

Carleson (1979), Dahlberg–Kenig (1982), Cowling (1983),Carbery (1985), Sjolin (1987), Vega (1988), Bourgain (1992/95),Moyua–Vargas–Vega (1996/99), Tao–Vargas (2000), Tao (2003),Lee (2006), Bourgain (2013).

Best known sufficient condition for Lebesgue a.e. convergence:

I s ≥ 1/4 in dimension n = 1 (Carleson);

I s > 12 −

14n in dimension n ≥ 2 (Lee, Bourgain).

Best known necessary condition for Lebesgue a.e. convergence:

I s ≥ 1/4 in dimension n = 1 (Dahlberg–Kenig);

I s ≥ 12 −

1n+2 in dimension n ≥ 2 (Luca–R.).

Page 101: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Maximal estimate for the Schrodinger equation

Theorem (Barcelo–Bennett–Carbery–R.)

Let n/4 ≤ s < n/2 and α > n − 2s. Then, for all α-dimensional µ,∥∥∥ sup0<t<1

|e it∆f |∥∥∥L1(dµ)

≤ Cµ‖f ‖Hs .

Proof: By the same proof as for the heat equation, one finallyarrives to the inequality∣∣∣∣e−i |·|2 ∗ 1

| · |n−2s

∣∣∣∣ ≤ Cn−2s1

| · |n−2s,

This can also be shown to be true by more difficult directcalculation as long as n/4 ≤ s < n/2.

Page 102: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Maximal estimate for the Schrodinger equation

Theorem (Barcelo–Bennett–Carbery–R.)

Let n/4 ≤ s < n/2 and α > n − 2s. Then, for all α-dimensional µ,∥∥∥ sup0<t<1

|e it∆f |∥∥∥L1(dµ)

≤ Cµ‖f ‖Hs .

Proof: By the same proof as for the heat equation, one finallyarrives to the inequality∣∣∣∣e−i |·|2 ∗ 1

| · |n−2s

∣∣∣∣ ≤ Cn−2s1

| · |n−2s,

This can also be shown to be true by more difficult directcalculation as long as n/4 ≤ s < n/2.

Page 103: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Maximal estimate for the Schrodinger equation

Theorem (Barcelo–Bennett–Carbery–R.)

Let n/4 ≤ s < n/2 and α > n − 2s. Then, for all α-dimensional µ,∥∥∥ sup0<t<1

|e it∆f |∥∥∥L1(dµ)

≤ Cµ‖f ‖Hs .

Proof: By the same proof as for the heat equation, one finallyarrives to the inequality∣∣∣∣e−i |·|2 ∗ 1

| · |n−2s

∣∣∣∣ ≤ Cn−2s1

| · |n−2s,

This can also be shown to be true by more difficult directcalculation as long as n/4 ≤ s < n/2.

Page 104: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Maximal estimate for the Schrodinger equation

Theorem (Barcelo–Bennett–Carbery–R.)

Let n/4 ≤ s < n/2 and α > n − 2s. Then, for all α-dimensional µ,∥∥∥ sup0<t<1

|e it∆f |∥∥∥L1(dµ)

≤ Cµ‖f ‖Hs .

Proof: By the same proof as for the heat equation, one finallyarrives to the inequality∣∣∣∣e−i |·|2 ∗ 1

| · |n−2s

∣∣∣∣ ≤ Cn−2s1

| · |n−2s,

This can also be shown to be true by more difficult directcalculation as long as n/4 ≤ s < n/2.

Page 105: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Corollary

Let n/4 ≤ s < n/2 and let u be a solution to the Schrodingerequation with initial data u0 ∈ Hs . Then

dimx ∈ Rn lim

t→0u(t, x) 6= u0(x)

≤ n − 2s.

Again this is sharp in the range s ≥ n/4.

We cannot go below this regularity in one dimension due to thenecessary condition of Dahlberg–Kenig.

In the next section we will see that we cannot go below thisregularity in higher dimensions either via a fractal version of theLuca–R.-necessary condition.

Page 106: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Corollary

Let n/4 ≤ s < n/2 and let u be a solution to the Schrodingerequation with initial data u0 ∈ Hs . Then

dimx ∈ Rn lim

t→0u(t, x) 6= u0(x)

≤ n − 2s.

Again this is sharp in the range s ≥ n/4.

We cannot go below this regularity in one dimension due to thenecessary condition of Dahlberg–Kenig.

In the next section we will see that we cannot go below thisregularity in higher dimensions either via a fractal version of theLuca–R.-necessary condition.

Page 107: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Corollary

Let n/4 ≤ s < n/2 and let u be a solution to the Schrodingerequation with initial data u0 ∈ Hs . Then

dimx ∈ Rn lim

t→0u(t, x) 6= u0(x)

≤ n − 2s.

Again this is sharp in the range s ≥ n/4.

We cannot go below this regularity in one dimension due to thenecessary condition of Dahlberg–Kenig.

In the next section we will see that we cannot go below thisregularity in higher dimensions either via a fractal version of theLuca–R.-necessary condition.

Page 108: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Corollary

Let n/4 ≤ s < n/2 and let u be a solution to the Schrodingerequation with initial data u0 ∈ Hs . Then

dimx ∈ Rn lim

t→0u(t, x) 6= u0(x)

≤ n − 2s.

Again this is sharp in the range s ≥ n/4.

We cannot go below this regularity in one dimension due to thenecessary condition of Dahlberg–Kenig.

In the next section we will see that we cannot go below thisregularity in higher dimensions either via a fractal version of theLuca–R.-necessary condition.

Page 109: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

αn(s) := supu0∈Hs(Rn)

dimx ∈ Rn : lim

t→0u(x , t) 6= u0(x)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.2

0.4

0.6

0.8

1

1.2

1.4

s

α1(s

)

αn(s) = n − 2s, n/4 ≤ s ≤ n/2.

What about lower regularity (s < n/4) in higher dimensions?

Page 110: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

αn(s) := supu0∈Hs(Rn)

dimx ∈ Rn : lim

t→0u(x , t) 6= u0(x)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.2

0.4

0.6

0.8

1

1.2

1.4

s

α1(s

)

αn(s) = n − 2s, n/4 ≤ s ≤ n/2.

What about lower regularity (s < n/4) in higher dimensions?

Page 111: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

αn(s) := supu0∈Hs(Rn)

dimx ∈ Rn : lim

t→0u(x , t) 6= u0(x)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.2

0.4

0.6

0.8

1

1.2

1.4

s

α1(s

)

αn(s) = n − 2s, n/4 ≤ s ≤ n/2.

What about lower regularity (s < n/4) in higher dimensions?

Page 112: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

αn(s) := supu0∈Hs(Rn)

dimx ∈ Rn : lim

t→0u(x , t) 6= u0(x)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.2

0.4

0.6

0.8

1

1.2

1.4

s

α1(s

)

αn(s) = n − 2s, n/4 ≤ s ≤ n/2.

What about lower regularity (s < n/4) in higher dimensions?

Page 113: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Part 4:

Counterexample for theSchrodinger equation:

lower bounds for αn

Page 114: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

αn(s) := supu0∈Hs(Rn)

dim

x ∈ Rn : limt→0

e it∆u0(x) 6= u0(x)

Theorem (Luca–R.)

αn(s) ≥

n , when s ≤ n

2(n+2)

n + 1− 2(n+2)sn , when n

2(n+2) ≤ s ≤ n4

n − 2s , when n4 ≤ s ≤ n

20 , when n

2 ≤ s

Page 115: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

αn(s) := supu0∈Hs(Rn)

dim

x ∈ Rn : limt→0

e it∆u0(x) 6= u0(x)

Theorem (Luca–R.)

αn(s) ≥

n , when s ≤ n

2(n+2)

n + 1− 2(n+2)sn , when n

2(n+2) ≤ s ≤ n4

n − 2s , when n4 ≤ s ≤ n

20 , when n

2 ≤ s

Page 116: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

αn(s) ≥ n when s < n2(n+2)

This bound follows from:

Theorem (Luca–R.)

Suppose that

limt→0

e it∆u0(x) = u0(x), a.e. x ∈ Rn

for all u0 ∈ Hs(Rn). Then

s ≥ n

2(n + 2).

which improves Dahlberg–Kenig for n ≥ 3 (coinciding when n = 2).

Page 117: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

αn(s) ≥ n when s < n2(n+2)

This bound follows from:

Theorem (Luca–R.)

Suppose that

limt→0

e it∆u0(x) = u0(x), a.e. x ∈ Rn

for all u0 ∈ Hs(Rn). Then

s ≥ n

2(n + 2).

which improves Dahlberg–Kenig for n ≥ 3 (coinciding when n = 2).

Page 118: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

αn(s) ≥ n when s < n2(n+2)

This bound follows from:

Theorem (Luca–R.)

Suppose that

limt→0

e it∆u0(x) = u0(x), a.e. x ∈ Rn

for all u0 ∈ Hs(Rn). Then

s ≥ n

2(n + 2).

which improves Dahlberg–Kenig for n ≥ 3 (coinciding when n = 2).

Page 119: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Proof

Lemma (Nikisin–Stein maximal principle)

Modulo endpoints:

limt→0

e it∆u0(x) = u0(x), a.e. x ∈ Rn,

for all u0 ∈ Hs(Rn) if and only if there is a constant C such that∥∥∥∥ sup0<t<1

|e it∆u0|∥∥∥∥L2(B(0,1))

≤ C‖u0‖Hs(Rn).

So it suffices to show that s ≥ n2(n+2) is necessary for∥∥∥∥ sup

0<t<1|e it∆f |

∥∥∥∥L2(B(0,1))

. Rs‖f ‖2,

whenever supp f ⊂ ξ : |ξ| ≤ R.

Page 120: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Proof

Lemma (Nikisin–Stein maximal principle)

Modulo endpoints:

limt→0

e it∆u0(x) = u0(x), a.e. x ∈ Rn,

for all u0 ∈ Hs(Rn) if and only if there is a constant C such that∥∥∥∥ sup0<t<1

|e it∆u0|∥∥∥∥L2(B(0,1))

≤ C‖u0‖Hs(Rn).

So it suffices to show that s ≥ n2(n+2) is necessary for∥∥∥∥ sup

0<t<1|e it∆f |

∥∥∥∥L2(B(0,1))

. Rs‖f ‖2,

whenever supp f ⊂ ξ : |ξ| ≤ R.

Page 121: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat
Page 122: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Constructive interference

+

+

+...

=

Page 123: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

The initial data

We consider the frequencies

Ω :=ξ ∈ 2πR1−κZn : |ξ| ≤ R

+ B(0, 1

10 ),

for 0 < κ < 1n+2 ,

and initial data defined by

f =1√|Ω|

χΩ, so that ‖f ‖2 = 1.

This data was introduced in the context of Mattila’s question byBarcelo–Bennett–Carbery–Ruiz–Vilela (2007).

Note that|Ω| ' number of frequencies ' Rnκ.

Page 124: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

The initial data

We consider the frequencies

Ω :=ξ ∈ 2πR1−κZn : |ξ| ≤ R

+ B(0, 1

10 ),

for 0 < κ < 1n+2 ,

and initial data defined by

f =1√|Ω|

χΩ, so that ‖f ‖2 = 1.

This data was introduced in the context of Mattila’s question byBarcelo–Bennett–Carbery–Ruiz–Vilela (2007).

Note that|Ω| ' number of frequencies ' Rnκ.

Page 125: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

The initial data

We consider the frequencies

Ω :=ξ ∈ 2πR1−κZn : |ξ| ≤ R

+ B(0, 1

10 ),

for 0 < κ < 1n+2 ,

and initial data defined by

f =1√|Ω|

χΩ, so that ‖f ‖2 = 1.

This data was introduced in the context of Mattila’s question byBarcelo–Bennett–Carbery–Ruiz–Vilela (2007).

Note that|Ω| ' number of frequencies ' Rnκ.

Page 126: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

The initial data

We consider the frequencies

Ω :=ξ ∈ 2πR1−κZn : |ξ| ≤ R

+ B(0, 1

10 ),

for 0 < κ < 1n+2 ,

and initial data defined by

f =1√|Ω|

χΩ, so that ‖f ‖2 = 1.

This data was introduced in the context of Mattila’s question byBarcelo–Bennett–Carbery–Ruiz–Vilela (2007).

Note that|Ω| ' number of frequencies ' Rnκ.

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The initial data

We consider the frequencies

Ω :=ξ ∈ 2πR1−κZn : |ξ| ≤ R

+ B(0, 1

10 ),

for 0 < κ < 1n+2 ,

and initial data defined by

f =1√|Ω|

χΩ, so that ‖f ‖2 = 1.

This data was introduced in the context of Mattila’s question byBarcelo–Bennett–Carbery–Ruiz–Vilela (2007).

Note that|Ω| ' number of frequencies ' Rnκ.

Page 128: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Periodic constructive interference

The constructive interference reappears periodically in time:

|e it∆f (x)| &√|Ω| for all (x , t) ∈ X × T ,

whereX :=

x ∈ Rκ−1Zn : |x | ≤ 2

+ B(0,R−1),

and

T :=t ∈ 1

2πR2(κ−1)Z : 0 < t < R−1

.

Page 129: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Periodic constructive interference

The constructive interference reappears periodically in time:

|e it∆f (x)| &√|Ω| for all (x , t) ∈ X × T ,

whereX :=

x ∈ Rκ−1Zn : |x | ≤ 2

+ B(0,R−1),

and

T :=t ∈ 1

2πR2(κ−1)Z : 0 < t < R−1

.

Page 130: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Periodically coherent solutions

X is the dual-set of Ω:

x · ξ ∈(Rκ−1Zn

)·(2πR1−κZn

)= 2πZ.

T is the dual-set of Ω · Ω:

tξ · ξ ∈(

1

2πR2(κ−1)Z

)(2πR1−κZn

)·(2πR1−κZn

)= 2πZ.

So that there is no cancellation in the integral:

e it∆f (x) ' 1√|Ω|

ˆΩe ix ·ξ−it|ξ|

2dξ ' |Ω|√

|Ω|.

Page 131: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Periodically coherent solutions

X is the dual-set of Ω:

x · ξ ∈(Rκ−1Zn

)·(2πR1−κZn

)= 2πZ.

T is the dual-set of Ω · Ω:

tξ · ξ ∈(

1

2πR2(κ−1)Z

)(2πR1−κZn

)·(2πR1−κZn

)= 2πZ.

So that there is no cancellation in the integral:

e it∆f (x) ' 1√|Ω|

ˆΩe ix ·ξ−it|ξ|

2dξ ' |Ω|√

|Ω|.

Page 132: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Periodically coherent solutions

X is the dual-set of Ω:

x · ξ ∈(Rκ−1Zn

)·(2πR1−κZn

)= 2πZ.

T is the dual-set of Ω · Ω:

tξ · ξ ∈(

1

2πR2(κ−1)Z

)(2πR1−κZn

)·(2πR1−κZn

)= 2πZ.

So that there is no cancellation in the integral:

e it∆f (x) ' 1√|Ω|

ˆΩe ix ·ξ−it|ξ|

2dξ ' |Ω|√

|Ω|.

Page 133: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Periodically coherent solutions

X is the dual-set of Ω:

x · ξ ∈(Rκ−1Zn

)·(2πR1−κZn

)= 2πZ.

T is the dual-set of Ω · Ω:

tξ · ξ ∈(

1

2πR2(κ−1)Z

)(2πR1−κZn

)·(2πR1−κZn

)= 2πZ.

So that there is no cancellation in the integral:

e it∆f (x) ' 1√|Ω|

ˆΩe ix ·ξ−it|ξ|

2dξ ' |Ω|√

|Ω|.

Page 134: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Periodically coherent solutions

X is the dual-set of Ω:

x · ξ ∈(Rκ−1Zn

)·(2πR1−κZn

)= 2πZ.

T is the dual-set of Ω · Ω:

tξ · ξ ∈(

1

2πR2(κ−1)Z

)(2πR1−κZn

)·(2πR1−κZn

)= 2πZ.

So that there is no cancellation in the integral:

e it∆f (x) ' 1√|Ω|

ˆΩe ix ·ξ−it|ξ|

2dξ ' |Ω|√

|Ω|.

Page 135: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Periodically coherent solutions

X is the dual-set of Ω:

x · ξ ∈(Rκ−1Zn

)·(2πR1−κZn

)= 2πZ.

T is the dual-set of Ω · Ω:

tξ · ξ ∈(

1

2πR2(κ−1)Z

)(2πR1−κZn

)·(2πR1−κZn

)= 2πZ.

So that there is no cancellation in the integral:

e it∆f (x) ' 1√|Ω|

ˆΩe ix ·ξ−it|ξ|

2dξ ' |Ω|√

|Ω|.

Page 136: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Periodically coherent solutions

Thus

|e it∆f (x)| &√|Ω| for all (x , t) ∈ X × T ,

But the interference always reappears in the same places so

sup0<t<1

|e it∆f (x)| &√|Ω| only for x ∈ X .

Page 137: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Periodically coherent solutions

Thus

|e it∆f (x)| &√|Ω| for all (x , t) ∈ X × T ,

But the interference always reappears in the same places so

sup0<t<1

|e it∆f (x)| &√|Ω| only for x ∈ X .

Page 138: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Travelling periodically coherent solutions

Instead we take

fθ(x) = e i12θ·x f (x), where θ ∈ Sn−1

so that

|e it∆fθ(x)| = |e it∆f (x − tθ)|,

which yields

sup0<t<1

|e it∆fθ(x)| &√|Ω| for all x ∈

⋃t∈T

X + tθ.

Page 139: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Travelling periodically coherent solutions

Instead we take

fθ(x) = e i12θ·x f (x), where θ ∈ Sn−1

so that

|e it∆fθ(x)| = |e it∆f (x − tθ)|,

which yields

sup0<t<1

|e it∆fθ(x)| &√|Ω| for all x ∈

⋃t∈T

X + tθ.

Page 140: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Lemma

Let 0 < κ < 1n+2 . Then there exists θ ∈ Sn−1 such that

B(0, 1/2) ⊂⋃t∈T

X + tθ.

This is optimal in the sense that it is not true for κ ≥ 1n+2 .

After scaling and quotienting out Zn, this follows from quantitiveergodic theory on the torus Tn.

Lemma (Luca–R.)

There exists θ ∈ Sn−1 such that for all y ∈ Tn there is at ∈ RδZ ∩ (0,R) such that

|y − tθ| ≤ R−(1−δ)/n logR.

Page 141: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Lemma

Let 0 < κ < 1n+2 . Then there exists θ ∈ Sn−1 such that

B(0, 1/2) ⊂⋃t∈T

X + tθ.

This is optimal in the sense that it is not true for κ ≥ 1n+2 .

After scaling and quotienting out Zn, this follows from quantitiveergodic theory on the torus Tn.

Lemma (Luca–R.)

There exists θ ∈ Sn−1 such that for all y ∈ Tn there is at ∈ RδZ ∩ (0,R) such that

|y − tθ| ≤ R−(1−δ)/n logR.

Page 142: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Lemma

Let 0 < κ < 1n+2 . Then there exists θ ∈ Sn−1 such that

B(0, 1/2) ⊂⋃t∈T

X + tθ.

This is optimal in the sense that it is not true for κ ≥ 1n+2 .

After scaling and quotienting out Zn, this follows from quantitiveergodic theory on the torus Tn.

Lemma (Luca–R.)

There exists θ ∈ Sn−1 such that for all y ∈ Tn there is at ∈ RδZ ∩ (0,R) such that

|y − tθ| ≤ R−(1−δ)/n logR.

Page 143: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Conclusion of the proofPlugging into the maximal estimate,∥∥∥∥ sup

0<t<1|e it∆fθ|

∥∥∥∥L2(B(0,1))

. Rs‖fθ‖2,

recalling that

sup0<t<1

|e it∆fθ| ≥√|Ω| on B(0, 1/2),

we obtain

√|Ω| . Rs‖fθ‖2.

Then as |Ω| & Rnκ and ‖fθ‖2 = 1, this yields

⇒ s ≥ nκ

2and then we take κ→ 1

n + 2.

Page 144: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Conclusion of the proofPlugging into the maximal estimate,∥∥∥∥ sup

0<t<1|e it∆fθ|

∥∥∥∥L2(B(0,1))

. Rs‖fθ‖2,

recalling that

sup0<t<1

|e it∆fθ| ≥√|Ω| on B(0, 1/2),

we obtain

√|Ω| . Rs‖fθ‖2.

Then as |Ω| & Rnκ and ‖fθ‖2 = 1, this yields

⇒ s ≥ nκ

2and then we take κ→ 1

n + 2.

Page 145: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Conclusion of the proofPlugging into the maximal estimate,∥∥∥∥ sup

0<t<1|e it∆fθ|

∥∥∥∥L2(B(0,1))

. Rs‖fθ‖2,

recalling that

sup0<t<1

|e it∆fθ| ≥√|Ω| on B(0, 1/2),

we obtain

√|Ω| . Rs‖fθ‖2.

Then as |Ω| & Rnκ and ‖fθ‖2 = 1, this yields

⇒ s ≥ nκ

2and then we take κ→ 1

n + 2.

Page 146: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Conclusion of the proofPlugging into the maximal estimate,∥∥∥∥ sup

0<t<1|e it∆fθ|

∥∥∥∥L2(B(0,1))

. Rs‖fθ‖2,

recalling that

sup0<t<1

|e it∆fθ| ≥√|Ω| on B(0, 1/2),

we obtain

√|Ω| . Rs‖fθ‖2.

Then as |Ω| & Rnκ and ‖fθ‖2 = 1, this yields

⇒ s ≥ nκ

2and then we take κ→ 1

n + 2.

Page 147: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

αn(s) ≥ n + 1− 2(n+2)sn when n

2(n+2) ≤ s ≤ n4

This follows from:

Theorem (Luca–Rogers)

Let n/2 ≤ α ≤ n and suppose that, for all u0 ∈ Hs(Rn),

limt→0

e it∆u0(x) = u0(x)

for all x off an α-dimensional set. Then

s ≥ n

2(n + 2)

(n − α + 1

).

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αn(s) ≥ n + 1− 2(n+2)sn when n

2(n+2) ≤ s ≤ n4

This follows from:

Theorem (Luca–Rogers)

Let n/2 ≤ α ≤ n and suppose that, for all u0 ∈ Hs(Rn),

limt→0

e it∆u0(x) = u0(x)

for all x off an α-dimensional set. Then

s ≥ n

2(n + 2)

(n − α + 1

).

Page 149: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

αn(s) ≥ n + 1− 2(n+2)sn when n

2(n+2) ≤ s ≤ n4

This follows from:

Theorem (Luca–Rogers)

Let n/2 ≤ α ≤ n and suppose that, for all u0 ∈ Hs(Rn),

limt→0

e it∆u0(x) = u0(x)

for all x off an α-dimensional set. Then

s ≥ n

2(n + 2)

(n − α + 1

).

Page 150: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

ProofThe Nikisin–Stein maximal principle does not hold in this context,and so we first give a direct proof of the Lebesgue measure result.

We consider a sum of the previous initial data

f :=∑j>1

fθj , θj ∈ Sd−1,

where we take R = 2j and normalise in a different way, so that

fθj (x) := e i12θj ·x fj(x), fj = 2−j(nκ−ε)χΩj

,

Ωj :=ξ ∈ 2π2j(1−κ)Zn : |ξ| ≤ 2j

+ B(0, 1

10 ).

Note that |Ωj | ' 2jnκ, so that ‖fj‖Hs ' 2−jnκ2

+jε+js .

Then if s < nκ2 − ε we can sum so that f ∈ Hs .

To generalise to the fractal case we will take 1n+2 ≤ κ <

n−α+1n+2 .

Page 151: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

ProofThe Nikisin–Stein maximal principle does not hold in this context,and so we first give a direct proof of the Lebesgue measure result.

We consider a sum of the previous initial data

f :=∑j>1

fθj , θj ∈ Sd−1,

where we take R = 2j and normalise in a different way, so that

fθj (x) := e i12θj ·x fj(x), fj = 2−j(nκ−ε)χΩj

,

Ωj :=ξ ∈ 2π2j(1−κ)Zn : |ξ| ≤ 2j

+ B(0, 1

10 ).

Note that |Ωj | ' 2jnκ, so that ‖fj‖Hs ' 2−jnκ2

+jε+js .

Then if s < nκ2 − ε we can sum so that f ∈ Hs .

To generalise to the fractal case we will take 1n+2 ≤ κ <

n−α+1n+2 .

Page 152: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

ProofThe Nikisin–Stein maximal principle does not hold in this context,and so we first give a direct proof of the Lebesgue measure result.

We consider a sum of the previous initial data

f :=∑j>1

fθj , θj ∈ Sd−1,

where we take R = 2j and normalise in a different way, so that

fθj (x) := e i12θj ·x fj(x), fj = 2−j(nκ−ε)χΩj

,

Ωj :=ξ ∈ 2π2j(1−κ)Zn : |ξ| ≤ 2j

+ B(0, 1

10 ).

Note that |Ωj | ' 2jnκ, so that ‖fj‖Hs ' 2−jnκ2

+jε+js .

Then if s < nκ2 − ε we can sum so that f ∈ Hs .

To generalise to the fractal case we will take 1n+2 ≤ κ <

n−α+1n+2 .

Page 153: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

ProofThe Nikisin–Stein maximal principle does not hold in this context,and so we first give a direct proof of the Lebesgue measure result.

We consider a sum of the previous initial data

f :=∑j>1

fθj , θj ∈ Sd−1,

where we take R = 2j and normalise in a different way, so that

fθj (x) := e i12θj ·x fj(x), fj = 2−j(nκ−ε)χΩj

,

Ωj :=ξ ∈ 2π2j(1−κ)Zn : |ξ| ≤ 2j

+ B(0, 1

10 ).

Note that |Ωj | ' 2jnκ, so that ‖fj‖Hs ' 2−jnκ2

+jε+js .

Then if s < nκ2 − ε we can sum so that f ∈ Hs .

To generalise to the fractal case we will take 1n+2 ≤ κ <

n−α+1n+2 .

Page 154: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

ProofThe Nikisin–Stein maximal principle does not hold in this context,and so we first give a direct proof of the Lebesgue measure result.

We consider a sum of the previous initial data

f :=∑j>1

fθj , θj ∈ Sd−1,

where we take R = 2j and normalise in a different way, so that

fθj (x) := e i12θj ·x fj(x), fj = 2−j(nκ−ε)χΩj

,

Ωj :=ξ ∈ 2π2j(1−κ)Zn : |ξ| ≤ 2j

+ B(0, 1

10 ).

Note that |Ωj | ' 2jnκ, so that ‖fj‖Hs ' 2−jnκ2

+jε+js .

Then if s < nκ2 − ε we can sum so that f ∈ Hs .

To generalise to the fractal case we will take 1n+2 ≤ κ <

n−α+1n+2 .

Page 155: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

By the previous calculations, for all x ∈ Ej := ∪t∈TjXj + tθj , where

Xj :=x ∈ 2j(κ−1)Zn : |x | ≤ 2

+ B(0, 2−j),

Tj :=t ∈ 1

2π22j(κ−1)Z : 0 < t < 2−j

,

there is a tj(x) ∈ Tj such that |e itj (x)∆fθj (x)| & 2jε.

One can also show (essentially) that |e itj (x)∆∑

k 6=j fθk (x)| ≤ C .

If κ < 1n+2 , then B(0, 1/2) ⊂

⋂j>1 Ej , and we are done.

If κ ≥ 1n+2 , we consider the limit set

lim supj→∞

Ej :=⋂j>1

⋃k>j

Ek

and prove that this is α-dimensional.

For this we use that the limit is ‘α–Hausdorff dense’.

Page 156: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

By the previous calculations, for all x ∈ Ej := ∪t∈TjXj + tθj , where

Xj :=x ∈ 2j(κ−1)Zn : |x | ≤ 2

+ B(0, 2−j),

Tj :=t ∈ 1

2π22j(κ−1)Z : 0 < t < 2−j

,

there is a tj(x) ∈ Tj such that |e itj (x)∆fθj (x)| & 2jε.

One can also show (essentially) that |e itj (x)∆∑

k 6=j fθk (x)| ≤ C .

If κ < 1n+2 , then B(0, 1/2) ⊂

⋂j>1 Ej , and we are done.

If κ ≥ 1n+2 , we consider the limit set

lim supj→∞

Ej :=⋂j>1

⋃k>j

Ek

and prove that this is α-dimensional.

For this we use that the limit is ‘α–Hausdorff dense’.

Page 157: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

By the previous calculations, for all x ∈ Ej := ∪t∈TjXj + tθj , where

Xj :=x ∈ 2j(κ−1)Zn : |x | ≤ 2

+ B(0, 2−j),

Tj :=t ∈ 1

2π22j(κ−1)Z : 0 < t < 2−j

,

there is a tj(x) ∈ Tj such that |e itj (x)∆fθj (x)| & 2jε.

One can also show (essentially) that |e itj (x)∆∑

k 6=j fθk (x)| ≤ C .

If κ < 1n+2 , then B(0, 1/2) ⊂

⋂j>1 Ej , and we are done.

If κ ≥ 1n+2 , we consider the limit set

lim supj→∞

Ej :=⋂j>1

⋃k>j

Ek

and prove that this is α-dimensional.

For this we use that the limit is ‘α–Hausdorff dense’.

Page 158: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

By the previous calculations, for all x ∈ Ej := ∪t∈TjXj + tθj , where

Xj :=x ∈ 2j(κ−1)Zn : |x | ≤ 2

+ B(0, 2−j),

Tj :=t ∈ 1

2π22j(κ−1)Z : 0 < t < 2−j

,

there is a tj(x) ∈ Tj such that |e itj (x)∆fθj (x)| & 2jε.

One can also show (essentially) that |e itj (x)∆∑

k 6=j fθk (x)| ≤ C .

If κ < 1n+2 , then B(0, 1/2) ⊂

⋂j>1 Ej , and we are done.

If κ ≥ 1n+2 , we consider the limit set

lim supj→∞

Ej :=⋂j>1

⋃k>j

Ek

and prove that this is α-dimensional.

For this we use that the limit is ‘α–Hausdorff dense’.

Page 159: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

By the previous calculations, for all x ∈ Ej := ∪t∈TjXj + tθj , where

Xj :=x ∈ 2j(κ−1)Zn : |x | ≤ 2

+ B(0, 2−j),

Tj :=t ∈ 1

2π22j(κ−1)Z : 0 < t < 2−j

,

there is a tj(x) ∈ Tj such that |e itj (x)∆fθj (x)| & 2jε.

One can also show (essentially) that |e itj (x)∆∑

k 6=j fθk (x)| ≤ C .

If κ < 1n+2 , then B(0, 1/2) ⊂

⋂j>1 Ej , and we are done.

If κ ≥ 1n+2 , we consider the limit set

lim supj→∞

Ej :=⋂j>1

⋃k>j

Ek

and prove that this is α-dimensional.

For this we use that the limit is ‘α–Hausdorff dense’.

Page 160: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Falconer’s density theorem

Consider the Hausdorff content Hα∞ defined by

Hα∞(E ) := inf∑

i

δαi : E ⊂⋃i

B(xi , δi ).

Theorem (Falconer (1985))

Suppose that, for all balls Br ⊂ B(0, 1) of radius r ,

lim infj→∞

Hα∞(Ej ∩ B(x , r)) ≥ crα. (†)

Then dim(

lim supj→∞ Ej

)≥ α.

The proof is completed by checking the density condition (†) withEj =

⋃t∈Tj

Xj + tθj using a variant of the ergodic lemma.

Page 161: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Falconer’s density theorem

Consider the Hausdorff content Hα∞ defined by

Hα∞(E ) := inf∑

i

δαi : E ⊂⋃i

B(xi , δi ).

Theorem (Falconer (1985))

Suppose that, for all balls Br ⊂ B(0, 1) of radius r ,

lim infj→∞

Hα∞(Ej ∩ B(x , r)) ≥ crα. (†)

Then dim(

lim supj→∞ Ej

)≥ α.

The proof is completed by checking the density condition (†) withEj =

⋃t∈Tj

Xj + tθj using a variant of the ergodic lemma.

Page 162: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Falconer’s density theorem

Consider the Hausdorff content Hα∞ defined by

Hα∞(E ) := inf∑

i

δαi : E ⊂⋃i

B(xi , δi ).

Theorem (Falconer (1985))

Suppose that, for all balls Br ⊂ B(0, 1) of radius r ,

lim infj→∞

Hα∞(Ej ∩ B(x , r)) ≥ crα. (†)

Then dim(

lim supj→∞ Ej

)≥ α.

The proof is completed by checking the density condition (†) withEj =

⋃t∈Tj

Xj + tθj using a variant of the ergodic lemma.

Page 163: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Part 5:

Decay for the Fourier transformof fractal measures

Page 164: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

δxn=0

(R(ξ, ξn)

)= 1

(2π)n/2

´Rn−1 e

iRx ·ξ dx is independent of ξn.

Thus, the Fourier transform of certain (n − 1)-dimensionalmeasures do not decay in every direction.

But perhaps they decay on average......

Let βn(α) denote the supremum of the numbers β for which

‖µ(R · )‖2L2(Sn−1) . cα(µ)‖µ‖R−β

whenever R > 1 and µ is α-dimensional and supported in B(0, 1).

Question (Mattila (1987))

Who is βn(α) ?

Equivalently βn(α) is the supremum of the numbers β for which∥∥(gdσ)∨(R · )∥∥L1(dµ)

.√

cα(µ)‖µ‖R−β/2‖g‖L2(Sn−1).

Page 165: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

δxn=0

(R(ξ, ξn)

)= 1

(2π)n/2

´Rn−1 e

iRx ·ξ dx is independent of ξn.

Thus, the Fourier transform of certain (n − 1)-dimensionalmeasures do not decay in every direction.

But perhaps they decay on average......

Let βn(α) denote the supremum of the numbers β for which

‖µ(R · )‖2L2(Sn−1) . cα(µ)‖µ‖R−β

whenever R > 1 and µ is α-dimensional and supported in B(0, 1).

Question (Mattila (1987))

Who is βn(α) ?

Equivalently βn(α) is the supremum of the numbers β for which∥∥(gdσ)∨(R · )∥∥L1(dµ)

.√

cα(µ)‖µ‖R−β/2‖g‖L2(Sn−1).

Page 166: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

δxn=0

(R(ξ, ξn)

)= 1

(2π)n/2

´Rn−1 e

iRx ·ξ dx is independent of ξn.

Thus, the Fourier transform of certain (n − 1)-dimensionalmeasures do not decay in every direction.

But perhaps they decay on average......

Let βn(α) denote the supremum of the numbers β for which

‖µ(R · )‖2L2(Sn−1) . cα(µ)‖µ‖R−β

whenever R > 1 and µ is α-dimensional and supported in B(0, 1).

Question (Mattila (1987))

Who is βn(α) ?

Equivalently βn(α) is the supremum of the numbers β for which∥∥(gdσ)∨(R · )∥∥L1(dµ)

.√

cα(µ)‖µ‖R−β/2‖g‖L2(Sn−1).

Page 167: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

δxn=0

(R(ξ, ξn)

)= 1

(2π)n/2

´Rn−1 e

iRx ·ξ dx is independent of ξn.

Thus, the Fourier transform of certain (n − 1)-dimensionalmeasures do not decay in every direction.

But perhaps they decay on average......

Let βn(α) denote the supremum of the numbers β for which

‖µ(R · )‖2L2(Sn−1) . cα(µ)‖µ‖R−β

whenever R > 1 and µ is α-dimensional and supported in B(0, 1).

Question (Mattila (1987))

Who is βn(α) ?

Equivalently βn(α) is the supremum of the numbers β for which∥∥(gdσ)∨(R · )∥∥L1(dµ)

.√

cα(µ)‖µ‖R−β/2‖g‖L2(Sn−1).

Page 168: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

δxn=0

(R(ξ, ξn)

)= 1

(2π)n/2

´Rn−1 e

iRx ·ξ dx is independent of ξn.

Thus, the Fourier transform of certain (n − 1)-dimensionalmeasures do not decay in every direction.

But perhaps they decay on average......

Let βn(α) denote the supremum of the numbers β for which

‖µ(R · )‖2L2(Sn−1) . cα(µ)‖µ‖R−β

whenever R > 1 and µ is α-dimensional and supported in B(0, 1).

Question (Mattila (1987))

Who is βn(α) ?

Equivalently βn(α) is the supremum of the numbers β for which∥∥(gdσ)∨(R · )∥∥L1(dµ)

.√

cα(µ)‖µ‖R−β/2‖g‖L2(Sn−1).

Page 169: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

δxn=0

(R(ξ, ξn)

)= 1

(2π)n/2

´Rn−1 e

iRx ·ξ dx is independent of ξn.

Thus, the Fourier transform of certain (n − 1)-dimensionalmeasures do not decay in every direction.

But perhaps they decay on average......

Let βn(α) denote the supremum of the numbers β for which

‖µ(R · )‖2L2(Sn−1) . cα(µ)‖µ‖R−β

whenever R > 1 and µ is α-dimensional and supported in B(0, 1).

Question (Mattila (1987))

Who is βn(α) ?

Equivalently βn(α) is the supremum of the numbers β for which∥∥(gdσ)∨(R · )∥∥L1(dµ)

.√

cα(µ)‖µ‖R−β/2‖g‖L2(Sn−1).

Page 170: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Previous results

β2(α) =

α, α ∈ (0, 1/2],

Mattila (1987)1/2, α ∈ [1/2, 1],

α/2, α ∈ [1, 2], Wolff (1999).

βn(α) ≥

α, α ∈ (0, n−12 ],

Mattila (1987)n−1

2 , α ∈ [n−12 , n2 ],

α− 1 + n+2−2α4 , α ∈ [n2 ,

n+22 ], Erdogan (2005)

α− 1, α ∈ [n+22 , n], Sjolin (1993).

Page 171: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Previous results

β2(α) =

α, α ∈ (0, 1/2],

Mattila (1987)1/2, α ∈ [1/2, 1],

α/2, α ∈ [1, 2], Wolff (1999).

βn(α) ≥

α, α ∈ (0, n−12 ],

Mattila (1987)n−1

2 , α ∈ [n−12 , n2 ],

α− 1 + n+2−2α4 , α ∈ [n2 ,

n+22 ], Erdogan (2005)

α− 1, α ∈ [n+22 , n], Sjolin (1993).

Page 172: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Lemma (Bridging lemma)

Let u be a solution to ∂tu = i(−∆)m/2u with initial datau ∈ Hs(Rn) with 0 < s < n/2. Then if βn(α) > n − 2s, then

dim

x ∈ Rn : limt→0

u(x , t) 6= u0(x)≤ α.

Proof: It will suffice to prove, for all α-dimensional µ,∥∥∥ sup0<t<1

|e it(−∆)m/2f |∥∥∥L1(dµ)

. Cµ‖f ‖Hs .

Writing f = | · |−s g and using polar coordinates,

(2π)n/2|e it(−∆)m/2f (x)|

=

∣∣∣∣ˆRn

e−it|ξ|m |ξ|−s g(ξ)e ix ·ξ dξ

∣∣∣∣=

∣∣∣∣ˆ ∞0

e−itRmRn−1−s

Sn−1

g(Rω) e iRx ·ωdσ(ω) dR

∣∣∣∣ .

Page 173: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Lemma (Bridging lemma)

Let u be a solution to ∂tu = i(−∆)m/2u with initial datau ∈ Hs(Rn) with 0 < s < n/2. Then if βn(α) > n − 2s, then

dim

x ∈ Rn : limt→0

u(x , t) 6= u0(x)≤ α.

Proof: It will suffice to prove, for all α-dimensional µ,∥∥∥ sup0<t<1

|e it(−∆)m/2f |∥∥∥L1(dµ)

. Cµ‖f ‖Hs .

Writing f = | · |−s g and using polar coordinates,

(2π)n/2|e it(−∆)m/2f (x)|

=

∣∣∣∣ˆRn

e−it|ξ|m |ξ|−s g(ξ)e ix ·ξ dξ

∣∣∣∣=

∣∣∣∣ˆ ∞0

e−itRmRn−1−s

Sn−1

g(Rω) e iRx ·ωdσ(ω) dR

∣∣∣∣ .

Page 174: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Lemma (Bridging lemma)

Let u be a solution to ∂tu = i(−∆)m/2u with initial datau ∈ Hs(Rn) with 0 < s < n/2. Then if βn(α) > n − 2s, then

dim

x ∈ Rn : limt→0

u(x , t) 6= u0(x)≤ α.

Proof: It will suffice to prove, for all α-dimensional µ,∥∥∥ sup0<t<1

|e it(−∆)m/2f |∥∥∥L1(dµ)

. Cµ‖f ‖Hs .

Writing f = | · |−s g and using polar coordinates,

(2π)n/2|e it(−∆)m/2f (x)|

=

∣∣∣∣ˆRn

e−it|ξ|m |ξ|−s g(ξ)e ix ·ξ dξ

∣∣∣∣=

∣∣∣∣ˆ ∞0

e−itRmRn−1−s

Sn−1

g(Rω) e iRx ·ωdσ(ω) dR

∣∣∣∣ .

Page 175: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Lemma (Bridging lemma)

Let u be a solution to ∂tu = i(−∆)m/2u with initial datau ∈ Hs(Rn) with 0 < s < n/2. Then if βn(α) > n − 2s, then

dim

x ∈ Rn : limt→0

u(x , t) 6= u0(x)≤ α.

Proof: It will suffice to prove, for all α-dimensional µ,∥∥∥ sup0<t<1

|e it(−∆)m/2f |∥∥∥L1(dµ)

. Cµ‖f ‖Hs .

Writing f = | · |−s g and using polar coordinates,

(2π)n/2|e it(−∆)m/2f (x)|

=

∣∣∣∣ˆRn

e−it|ξ|m |ξ|−s g(ξ)e ix ·ξ dξ

∣∣∣∣=

∣∣∣∣ˆ ∞0

e−itRmRn−1−s

Sn−1

g(Rω) e iRx ·ωdσ(ω) dR

∣∣∣∣ .

Page 176: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Lemma (Bridging lemma)

Let u be a solution to ∂tu = i(−∆)m/2u with initial datau ∈ Hs(Rn) with 0 < s < n/2. Then if βn(α) > n − 2s, then

dim

x ∈ Rn : limt→0

u(x , t) 6= u0(x)≤ α.

Proof: It will suffice to prove, for all α-dimensional µ,∥∥∥ sup0<t<1

|e it(−∆)m/2f |∥∥∥L1(dµ)

. Cµ‖f ‖Hs .

Writing f = | · |−s g and using polar coordinates,

(2π)n/2|e it(−∆)m/2f (x)|

=

∣∣∣∣ˆRn

e−it|ξ|m |ξ|−s g(ξ)e ix ·ξ dξ

∣∣∣∣=

∣∣∣∣ˆ ∞0

e−itRmRn−1−s

Sn−1

g(Rω) e iRx ·ωdσ(ω) dR

∣∣∣∣ .

Page 177: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

|e it(−∆)m/2f (x)| .

ˆ ∞0

Rn−1−s∣∣∣∣ˆ

Sn−1

g(Rω) e iRx ·ωdσ(ω)

∣∣∣∣ dR,so that, by Fubini,∥∥∥ sup

t∈R|e it(−∆)m/2

f |∥∥∥L1(dµ)

.ˆ ∞

0Rn−1−s∥∥(g(R ·)dσ

)∨(R · )

∥∥L1(dµ)

dR.

By the dual version of the Mattila inequality,∥∥(g(R ·)dσ)∨

(R · )∥∥L1(dµ)

≤ Cµ (1 + R)−β/2‖g(R · )‖L2(Sn−1).

for all β < βn(α), so that∥∥∥ supt∈R|e it(−∆)m/2

f |∥∥∥L1(dµ)

≤ Cµ

ˆ ∞0

Rn−1−s

(1 + R)β/2‖g(R · )‖L2(Sn−1)dR.

Finally, by Cauchy–Schwarz,

≤ Cµ

(ˆ ∞0

Rn−1−2s

(1 + R)βdR

)1/2(ˆ ∞0‖g(R · )‖2

L2(Sn−1)Rn−1dR

)1/2

≤ Cµ ‖g‖L2(Rn),

where for the final inequality we must take β > n − 2s.

Page 178: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

|e it(−∆)m/2f (x)| .

ˆ ∞0

Rn−1−s∣∣∣∣ˆ

Sn−1

g(Rω) e iRx ·ωdσ(ω)

∣∣∣∣ dR,so that, by Fubini,∥∥∥ sup

t∈R|e it(−∆)m/2

f |∥∥∥L1(dµ)

.ˆ ∞

0Rn−1−s∥∥(g(R ·)dσ

)∨(R · )

∥∥L1(dµ)

dR.

By the dual version of the Mattila inequality,∥∥(g(R ·)dσ)∨

(R · )∥∥L1(dµ)

≤ Cµ (1 + R)−β/2‖g(R · )‖L2(Sn−1).

for all β < βn(α), so that∥∥∥ supt∈R|e it(−∆)m/2

f |∥∥∥L1(dµ)

≤ Cµ

ˆ ∞0

Rn−1−s

(1 + R)β/2‖g(R · )‖L2(Sn−1)dR.

Finally, by Cauchy–Schwarz,

≤ Cµ

(ˆ ∞0

Rn−1−2s

(1 + R)βdR

)1/2(ˆ ∞0‖g(R · )‖2

L2(Sn−1)Rn−1dR

)1/2

≤ Cµ ‖g‖L2(Rn),

where for the final inequality we must take β > n − 2s.

Page 179: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

|e it(−∆)m/2f (x)| .

ˆ ∞0

Rn−1−s∣∣∣∣ˆ

Sn−1

g(Rω) e iRx ·ωdσ(ω)

∣∣∣∣ dR,so that, by Fubini,∥∥∥ sup

t∈R|e it(−∆)m/2

f |∥∥∥L1(dµ)

.ˆ ∞

0Rn−1−s∥∥(g(R ·)dσ

)∨(R · )

∥∥L1(dµ)

dR.

By the dual version of the Mattila inequality,∥∥(g(R ·)dσ)∨

(R · )∥∥L1(dµ)

≤ Cµ (1 + R)−β/2‖g(R · )‖L2(Sn−1).

for all β < βn(α), so that∥∥∥ supt∈R|e it(−∆)m/2

f |∥∥∥L1(dµ)

≤ Cµ

ˆ ∞0

Rn−1−s

(1 + R)β/2‖g(R · )‖L2(Sn−1)dR.

Finally, by Cauchy–Schwarz,

≤ Cµ

(ˆ ∞0

Rn−1−2s

(1 + R)βdR

)1/2(ˆ ∞0‖g(R · )‖2

L2(Sn−1)Rn−1dR

)1/2

≤ Cµ ‖g‖L2(Rn),

where for the final inequality we must take β > n − 2s.

Page 180: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

|e it(−∆)m/2f (x)| .

ˆ ∞0

Rn−1−s∣∣∣∣ˆ

Sn−1

g(Rω) e iRx ·ωdσ(ω)

∣∣∣∣ dR,so that, by Fubini,∥∥∥ sup

t∈R|e it(−∆)m/2

f |∥∥∥L1(dµ)

.ˆ ∞

0Rn−1−s∥∥(g(R ·)dσ

)∨(R · )

∥∥L1(dµ)

dR.

By the dual version of the Mattila inequality,∥∥(g(R ·)dσ)∨

(R · )∥∥L1(dµ)

≤ Cµ (1 + R)−β/2‖g(R · )‖L2(Sn−1).

for all β < βn(α), so that∥∥∥ supt∈R|e it(−∆)m/2

f |∥∥∥L1(dµ)

≤ Cµ

ˆ ∞0

Rn−1−s

(1 + R)β/2‖g(R · )‖L2(Sn−1)dR.

Finally, by Cauchy–Schwarz,

≤ Cµ

(ˆ ∞0

Rn−1−2s

(1 + R)βdR

)1/2(ˆ ∞0‖g(R · )‖2

L2(Sn−1)Rn−1dR

)1/2

≤ Cµ ‖g‖L2(Rn),

where for the final inequality we must take β > n − 2s.

Page 181: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

|e it(−∆)m/2f (x)| .

ˆ ∞0

Rn−1−s∣∣∣∣ˆ

Sn−1

g(Rω) e iRx ·ωdσ(ω)

∣∣∣∣ dR,so that, by Fubini,∥∥∥ sup

t∈R|e it(−∆)m/2

f |∥∥∥L1(dµ)

.ˆ ∞

0Rn−1−s∥∥(g(R ·)dσ

)∨(R · )

∥∥L1(dµ)

dR.

By the dual version of the Mattila inequality,∥∥(g(R ·)dσ)∨

(R · )∥∥L1(dµ)

≤ Cµ (1 + R)−β/2‖g(R · )‖L2(Sn−1).

for all β < βn(α), so that∥∥∥ supt∈R|e it(−∆)m/2

f |∥∥∥L1(dµ)

≤ Cµ

ˆ ∞0

Rn−1−s

(1 + R)β/2‖g(R · )‖L2(Sn−1)dR.

Finally, by Cauchy–Schwarz,

≤ Cµ

(ˆ ∞0

Rn−1−2s

(1 + R)βdR

)1/2(ˆ ∞0‖g(R · )‖2

L2(Sn−1)Rn−1dR

)1/2

≤ Cµ ‖g‖L2(Rn),

where for the final inequality we must take β > n − 2s.

Page 182: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

|e it(−∆)m/2f (x)| .

ˆ ∞0

Rn−1−s∣∣∣∣ˆ

Sn−1

g(Rω) e iRx ·ωdσ(ω)

∣∣∣∣ dR,so that, by Fubini,∥∥∥ sup

t∈R|e it(−∆)m/2

f |∥∥∥L1(dµ)

.ˆ ∞

0Rn−1−s∥∥(g(R ·)dσ

)∨(R · )

∥∥L1(dµ)

dR.

By the dual version of the Mattila inequality,∥∥(g(R ·)dσ)∨

(R · )∥∥L1(dµ)

≤ Cµ (1 + R)−β/2‖g(R · )‖L2(Sn−1).

for all β < βn(α), so that∥∥∥ supt∈R|e it(−∆)m/2

f |∥∥∥L1(dµ)

≤ Cµ

ˆ ∞0

Rn−1−s

(1 + R)β/2‖g(R · )‖L2(Sn−1)dR.

Finally, by Cauchy–Schwarz,

≤ Cµ

(ˆ ∞0

Rn−1−2s

(1 + R)βdR

)1/2(ˆ ∞0‖g(R · )‖2

L2(Sn−1)Rn−1dR

)1/2

≤ Cµ ‖g‖L2(Rn),

where for the final inequality we must take β > n − 2s.

Page 183: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Part 6:

Convergence for the waveequation

Page 184: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Recall that, with initial data u(·, 0) = u0 and ∂tu(·, 0) = u1, thesolution to the wave equation satisfies

u(ξ) = cos(t|ξ|)u0(ξ) +sin(t|ξ|)|ξ|

u1(ξ)

=1

2(e it|ξ| + e−it|ξ|)u0(ξ) +

1

2

(e it|ξ| − e−it|ξ|)

i |ξ|u1(ξ)

= e it|ξ|1

2

(u0(ξ) +

u1(ξ)

i |ξ|

)+ e−it|ξ|

1

2

(u0(ξ)− u1(ξ)

i |ξ|

)=: e it|ξ|f+(ξ) + e−it|ξ|f−(ξ).

With this notation, we can write

u(·, t) = e it(−∆)1/2f+ + e−it(−∆)1/2

f−.

If the initial data is in Hs × Hs−1, both f+ and f− belong to Hs .

Thus convergence of e it(−∆)1/2f to f for all f ∈ Hs implies

convergence of u(·, t) to u0 for all (u0, u1) ∈ Hs × Hs−1.

Page 185: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Recall that, with initial data u(·, 0) = u0 and ∂tu(·, 0) = u1, thesolution to the wave equation satisfies

u(ξ) = cos(t|ξ|)u0(ξ) +sin(t|ξ|)|ξ|

u1(ξ)

=1

2(e it|ξ| + e−it|ξ|)u0(ξ) +

1

2

(e it|ξ| − e−it|ξ|)

i |ξ|u1(ξ)

= e it|ξ|1

2

(u0(ξ) +

u1(ξ)

i |ξ|

)+ e−it|ξ|

1

2

(u0(ξ)− u1(ξ)

i |ξ|

)=: e it|ξ|f+(ξ) + e−it|ξ|f−(ξ).

With this notation, we can write

u(·, t) = e it(−∆)1/2f+ + e−it(−∆)1/2

f−.

If the initial data is in Hs × Hs−1, both f+ and f− belong to Hs .

Thus convergence of e it(−∆)1/2f to f for all f ∈ Hs implies

convergence of u(·, t) to u0 for all (u0, u1) ∈ Hs × Hs−1.

Page 186: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Recall that, with initial data u(·, 0) = u0 and ∂tu(·, 0) = u1, thesolution to the wave equation satisfies

u(ξ) = cos(t|ξ|)u0(ξ) +sin(t|ξ|)|ξ|

u1(ξ)

=1

2(e it|ξ| + e−it|ξ|)u0(ξ) +

1

2

(e it|ξ| − e−it|ξ|)

i |ξ|u1(ξ)

= e it|ξ|1

2

(u0(ξ) +

u1(ξ)

i |ξ|

)+ e−it|ξ|

1

2

(u0(ξ)− u1(ξ)

i |ξ|

)=: e it|ξ|f+(ξ) + e−it|ξ|f−(ξ).

With this notation, we can write

u(·, t) = e it(−∆)1/2f+ + e−it(−∆)1/2

f−.

If the initial data is in Hs × Hs−1, both f+ and f− belong to Hs .

Thus convergence of e it(−∆)1/2f to f for all f ∈ Hs implies

convergence of u(·, t) to u0 for all (u0, u1) ∈ Hs × Hs−1.

Page 187: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Recall that, with initial data u(·, 0) = u0 and ∂tu(·, 0) = u1, thesolution to the wave equation satisfies

u(ξ) = cos(t|ξ|)u0(ξ) +sin(t|ξ|)|ξ|

u1(ξ)

=1

2(e it|ξ| + e−it|ξ|)u0(ξ) +

1

2

(e it|ξ| − e−it|ξ|)

i |ξ|u1(ξ)

= e it|ξ|1

2

(u0(ξ) +

u1(ξ)

i |ξ|

)+ e−it|ξ|

1

2

(u0(ξ)− u1(ξ)

i |ξ|

)=: e it|ξ|f+(ξ) + e−it|ξ|f−(ξ).

With this notation, we can write

u(·, t) = e it(−∆)1/2f+ + e−it(−∆)1/2

f−.

If the initial data is in Hs × Hs−1, both f+ and f− belong to Hs .

Thus convergence of e it(−∆)1/2f to f for all f ∈ Hs implies

convergence of u(·, t) to u0 for all (u0, u1) ∈ Hs × Hs−1.

Page 188: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Recall that, with initial data u(·, 0) = u0 and ∂tu(·, 0) = u1, thesolution to the wave equation satisfies

u(ξ) = cos(t|ξ|)u0(ξ) +sin(t|ξ|)|ξ|

u1(ξ)

=1

2(e it|ξ| + e−it|ξ|)u0(ξ) +

1

2

(e it|ξ| − e−it|ξ|)

i |ξ|u1(ξ)

= e it|ξ|1

2

(u0(ξ) +

u1(ξ)

i |ξ|

)+ e−it|ξ|

1

2

(u0(ξ)− u1(ξ)

i |ξ|

)=: e it|ξ|f+(ξ) + e−it|ξ|f−(ξ).

With this notation, we can write

u(·, t) = e it(−∆)1/2f+ + e−it(−∆)1/2

f−.

If the initial data is in Hs × Hs−1, both f+ and f− belong to Hs .

Thus convergence of e it(−∆)1/2f to f for all f ∈ Hs implies

convergence of u(·, t) to u0 for all (u0, u1) ∈ Hs × Hs−1.

Page 189: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Recall that, with initial data u(·, 0) = u0 and ∂tu(·, 0) = u1, thesolution to the wave equation satisfies

u(ξ) = cos(t|ξ|)u0(ξ) +sin(t|ξ|)|ξ|

u1(ξ)

=1

2(e it|ξ| + e−it|ξ|)u0(ξ) +

1

2

(e it|ξ| − e−it|ξ|)

i |ξ|u1(ξ)

= e it|ξ|1

2

(u0(ξ) +

u1(ξ)

i |ξ|

)+ e−it|ξ|

1

2

(u0(ξ)− u1(ξ)

i |ξ|

)=: e it|ξ|f+(ξ) + e−it|ξ|f−(ξ).

With this notation, we can write

u(·, t) = e it(−∆)1/2f+ + e−it(−∆)1/2

f−.

If the initial data is in Hs × Hs−1, both f+ and f− belong to Hs .

Thus convergence of e it(−∆)1/2f to f for all f ∈ Hs implies

convergence of u(·, t) to u0 for all (u0, u1) ∈ Hs × Hs−1.

Page 190: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Recall that, with initial data u(·, 0) = u0 and ∂tu(·, 0) = u1, thesolution to the wave equation satisfies

u(ξ) = cos(t|ξ|)u0(ξ) +sin(t|ξ|)|ξ|

u1(ξ)

=1

2(e it|ξ| + e−it|ξ|)u0(ξ) +

1

2

(e it|ξ| − e−it|ξ|)

i |ξ|u1(ξ)

= e it|ξ|1

2

(u0(ξ) +

u1(ξ)

i |ξ|

)+ e−it|ξ|

1

2

(u0(ξ)− u1(ξ)

i |ξ|

)=: e it|ξ|f+(ξ) + e−it|ξ|f−(ξ).

With this notation, we can write

u(·, t) = e it(−∆)1/2f+ + e−it(−∆)1/2

f−.

If the initial data is in Hs × Hs−1, both f+ and f− belong to Hs .

Thus convergence of e it(−∆)1/2f to f for all f ∈ Hs implies

convergence of u(·, t) to u0 for all (u0, u1) ∈ Hs × Hs−1.

Page 191: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Corollary (of bridging lemma and Sjolin’s estimate)

Let u be a solution to the Schrodinger equation with initial datain Hs or to the wave equation with initial data in Hs × Hs−1. Then

dimx ∈ Rn : lim

t→0u(x , t) 6= u0(x)

≤ n − 2s + 1.

Proof: By the result of Sjolin, β(α) ≥ α− 1 so that β(α) > n− 2sas long as α > n − 2s + 1. Thus, by the bridging lemma,

dim

x ∈ Rn : limt→0

u(x , t) 6= u0(x)≤ n − 2s + 1.

Corollary (of the corollary)

Let u be a solution to the Schrodinger equation with initial datain H1 or to the wave equation with initial data in H1 × L2. Then

dimx ∈ Rn : lim

t→0u(x , t) 6= u0(x)

≤ n − 1.

Page 192: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Corollary (of bridging lemma and Sjolin’s estimate)

Let u be a solution to the Schrodinger equation with initial datain Hs or to the wave equation with initial data in Hs × Hs−1. Then

dimx ∈ Rn : lim

t→0u(x , t) 6= u0(x)

≤ n − 2s + 1.

Proof: By the result of Sjolin, β(α) ≥ α− 1 so that β(α) > n− 2sas long as α > n − 2s + 1. Thus, by the bridging lemma,

dim

x ∈ Rn : limt→0

u(x , t) 6= u0(x)≤ n − 2s + 1.

Corollary (of the corollary)

Let u be a solution to the Schrodinger equation with initial datain H1 or to the wave equation with initial data in H1 × L2. Then

dimx ∈ Rn : lim

t→0u(x , t) 6= u0(x)

≤ n − 1.

Page 193: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Corollary (of bridging lemma and Sjolin’s estimate)

Let u be a solution to the Schrodinger equation with initial datain Hs or to the wave equation with initial data in Hs × Hs−1. Then

dimx ∈ Rn : lim

t→0u(x , t) 6= u0(x)

≤ n − 2s + 1.

Proof: By the result of Sjolin, β(α) ≥ α− 1 so that β(α) > n− 2sas long as α > n − 2s + 1. Thus, by the bridging lemma,

dim

x ∈ Rn : limt→0

u(x , t) 6= u0(x)≤ n − 2s + 1.

Corollary (of the corollary)

Let u be a solution to the Schrodinger equation with initial datain H1 or to the wave equation with initial data in H1 × L2. Then

dimx ∈ Rn : lim

t→0u(x , t) 6= u0(x)

≤ n − 1.

Page 194: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Theorem (Luca–R.)

Let n ≥ 3. Then

βn(α) ≥ α− 1 +(n − α)2

(n − 1)(2n − α− 1).

This is an improvement in the range n/2 + 1 ≤ α < n.

The proof takes advantage of:I ‘multilinear restriction’ estimates due to Bennett–Carbery–TaoI ‘decomposition’ of Bourgain–Guth.I ‘interpolation’ with the argument of Sjolin.

Corollary

Let u be a solution to the Schrodinger equation with initial datain H1 or to the wave equation with initial data in H1 × L2. Then

dimx ∈ Rn : lim

t→0u(x , t) 6= u0(x)

< n − 1.

Thus the solution cannot diverge on spheres.

Page 195: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Theorem (Luca–R.)

Let n ≥ 3. Then

βn(α) ≥ α− 1 +(n − α)2

(n − 1)(2n − α− 1).

This is an improvement in the range n/2 + 1 ≤ α < n.

The proof takes advantage of:I ‘multilinear restriction’ estimates due to Bennett–Carbery–TaoI ‘decomposition’ of Bourgain–Guth.I ‘interpolation’ with the argument of Sjolin.

Corollary

Let u be a solution to the Schrodinger equation with initial datain H1 or to the wave equation with initial data in H1 × L2. Then

dimx ∈ Rn : lim

t→0u(x , t) 6= u0(x)

< n − 1.

Thus the solution cannot diverge on spheres.

Page 196: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Theorem (Luca–R.)

Let n ≥ 3. Then

βn(α) ≥ α− 1 +(n − α)2

(n − 1)(2n − α− 1).

This is an improvement in the range n/2 + 1 ≤ α < n.

The proof takes advantage of:I ‘multilinear restriction’ estimates due to Bennett–Carbery–TaoI ‘decomposition’ of Bourgain–Guth.I ‘interpolation’ with the argument of Sjolin.

Corollary

Let u be a solution to the Schrodinger equation with initial datain H1 or to the wave equation with initial data in H1 × L2. Then

dimx ∈ Rn : lim

t→0u(x , t) 6= u0(x)

< n − 1.

Thus the solution cannot diverge on spheres.

Page 197: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Theorem (Luca–R.)

Let n ≥ 3. Then

βn(α) ≥ α− 1 +(n − α)2

(n − 1)(2n − α− 1).

This is an improvement in the range n/2 + 1 ≤ α < n.

The proof takes advantage of:I ‘multilinear restriction’ estimates due to Bennett–Carbery–TaoI ‘decomposition’ of Bourgain–Guth.I ‘interpolation’ with the argument of Sjolin.

Corollary

Let u be a solution to the Schrodinger equation with initial datain H1 or to the wave equation with initial data in H1 × L2. Then

dimx ∈ Rn : lim

t→0u(x , t) 6= u0(x)

< n − 1.

Thus the solution cannot diverge on spheres.

Page 198: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Theorem (Luca–R.)

Let n ≥ 3. Then

βn(α) ≥ α− 1 +(n − α)2

(n − 1)(2n − α− 1).

This is an improvement in the range n/2 + 1 ≤ α < n.

The proof takes advantage of:I ‘multilinear restriction’ estimates due to Bennett–Carbery–TaoI ‘decomposition’ of Bourgain–Guth.I ‘interpolation’ with the argument of Sjolin.

Corollary

Let u be a solution to the Schrodinger equation with initial datain H1 or to the wave equation with initial data in H1 × L2. Then

dimx ∈ Rn : lim

t→0u(x , t) 6= u0(x)

< n − 1.

Thus the solution cannot diverge on spheres.

Page 199: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Theorem (Luca–R.)

Let n ≥ 3. Then

βn(α) ≥ α− 1 +(n − α)2

(n − 1)(2n − α− 1).

This is an improvement in the range n/2 + 1 ≤ α < n.

The proof takes advantage of:I ‘multilinear restriction’ estimates due to Bennett–Carbery–TaoI ‘decomposition’ of Bourgain–Guth.I ‘interpolation’ with the argument of Sjolin.

Corollary

Let u be a solution to the Schrodinger equation with initial datain H1 or to the wave equation with initial data in H1 × L2. Then

dimx ∈ Rn : lim

t→0u(x , t) 6= u0(x)

< n − 1.

Thus the solution cannot diverge on spheres.

Page 200: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Theorem (Luca–R.)

Let n ≥ 3. Then

βn(α) ≥ α− 1 +(n − α)2

(n − 1)(2n − α− 1).

This is an improvement in the range n/2 + 1 ≤ α < n.

The proof takes advantage of:I ‘multilinear restriction’ estimates due to Bennett–Carbery–TaoI ‘decomposition’ of Bourgain–Guth.I ‘interpolation’ with the argument of Sjolin.

Corollary

Let u be a solution to the Schrodinger equation with initial datain H1 or to the wave equation with initial data in H1 × L2. Then

dimx ∈ Rn : lim

t→0u(x , t) 6= u0(x)

< n − 1.

Thus the solution cannot diverge on spheres.

Page 201: Pointwise convergence to initial datamath.shinshu-u.ac.jp/~tsutsui/HAAM2016summer/Keith.pdf · Summary IPart 1: Set-up and introduction to the PDEs. IPart 2: Convergence for the heat

Arigatou gozaimasu!