Almost everywhere convergence of Fourier Series … We shall review some historical results on the...
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Almost everywhere convergence of FourierSeries and Kalton’s lattice constants
Marıa J. Carro
University of Barcelona
Caceres, marzo 2016
Marıa J. Carro Fourier Series and Kalton’s lattice constants
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Abstract
We shall review some historical results on the problem of almosteverywhere convergence of Fourier series and present some re-cent developments on this topic using the so called Kalton latticeconstants.
-3 -2 -1 1 2 3
-1.5
-1.0
-0.5
0.5
1.0
1.5
Marıa J. Carro Fourier Series and Kalton’s lattice constants
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N. Kalton
Introduced his lattice constants in order to study when a spacecontains uniformly complemented `n2.
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Kalton’s constant
Quasi-Banach constantLet (X , ‖ · ‖X ) be a quasi-Banach function space. Then:
∥∥∥ n∑i=1
fi∥∥∥
X≤ cn(X )
n∑i=1
‖fi‖X .
Examples1 If 1 < p ≤ ∞, cn(Lp) = 1, for every n.
2 If p ≤ 1, cn(Lp) = n1p−1
3 If
L1,∞ ={
f : ‖f‖L1,∞ = supy>0
y |{x : |f (x)| > y | <∞},
then cn(L1,∞) = 1 + log n.
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Kalton’s constant
Quasi-Banach constantLet (X , ‖ · ‖X ) be a quasi-Banach function space. Then:
∥∥∥ n∑i=1
fi∥∥∥
X≤ cn(X )
n∑i=1
‖fi‖X .
Examples1 If 1 < p ≤ ∞, cn(Lp) = 1, for every n.
2 If p ≤ 1, cn(Lp) = n1p−1
3 If
L1,∞ ={
f : ‖f‖L1,∞ = supy>0
y |{x : |f (x)| > y | <∞},
then cn(L1,∞) = 1 + log n.
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Kalton’s constant
Lattice constantsIn 1993, N. Kalton defined en(X ) to be the least constant suchthat, ∥∥∥ n∑
i=1
fi∥∥∥
X≤ en(X ) max
1≤i≤n‖fi‖X ,
and dn(X ) the least constant so that
n∑i=1
∥∥∥fi∥∥∥
X≤ dn(X )
∥∥∥ n∑i=1
fi∥∥∥
X
for every collection of pairwise disjoint functions.
N.J. Kalton, Lattice Structures on Banach Spaces, Memoirs Amer. Math. Soc.Vol. 103, No 493, 1993.
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Properties of Kalton’s constant
Under certain conditions on X ,
1 dn(X ) = en(X ∗).2 dn(X ∗) = en(X ).3
limdn
nα= 0 =⇒ X is
11− α concave.
4 If lim dn√log n
= 0, then X does not contain uniformly
complemented copies of `2n.5 etc.
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Some similar constants
On many interesting examples, en(X ) coincide with the leastconstant satysfying
∥∥∥ n∑i=1
fi∥∥∥
X≤ en(X ),
for every collection of pairwise disjoint functions so that ‖fi‖X =1, and in fact, if we define fn(x) as the least constant satisfying
∥∥∥ n∑i=1
χAi
‖χAi‖X
∥∥∥X≤ fn(X ),
we also have that en(X ) = fn(X ).
Mastylo, Mieczysaw, Lattice structures on some Banach spaces. Proc. Amer.Math. Soc. 140 (2012), no. 4, 1413–1422.
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Embedding of spaces
A very easy and useful remark
By definition of fn(X ), given n, there exists {An1,A
n2, · · · ,An
n} acollection of pairwise disjoint sets so that
fn(X )
2≤∥∥∥ n∑
i=1
χAni
‖χAni‖X
∥∥∥X
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Embedding of spaces
TheoremLet Y be a Banach space. Then
Y ⊂ X =⇒ fn(X )
n. sup
i
‖χAni‖Y
‖χAni‖X, ∀n.
Proof
fn(X )
2≤
∥∥∥ n∑i=1
χAni
‖χAni‖X
∥∥∥X.∥∥∥ n∑
i=1
χAni
‖χAni‖X
∥∥∥Y
. n supi
‖χAni‖Y
‖χAni‖X.
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Connection with Fourier series
Jean Baptiste Joseph Fourier (1768–1830).
Fourier introduced this series for the purpose of solving theheat equation on a metal plate and his initial results werepublished in 1807.
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The almost everywhere convergence of the Fourierseries
Let T ≡ [0,1) and let f ∈ L1(T). Then, the Fourier coefficientsare defined as
f (k) =∫ 1
0f (x)e−2πikx dx , k ∈ Z,
and the Fourier series of f at x ∈ T is given by
S[f ](x) =∞∑
k=−∞f (k)e2πikx .
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The almost everywhere convergence of the Fourierseries
Let T ≡ [0,1) and let f ∈ L1(T). Then, the Fourier coefficientsare defined as
f (k) =∫ 1
0f (x)e−2πikx dx , k ∈ Z,
and the Fourier series of f at x ∈ T is given by
S[f ](x) =∞∑
k=−∞f (k)e2πikx .
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The problem of the almost everywhere convergence of the Fourierseries consists in studying when:
limn→∞
Snf (x) = f (x), a.e. x ∈ T
whereSnf (x) =
∑|k |≤n
f (k)e2πikx .
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Paul du Bois-Reymond (1831–1889).
A Negative Result
In 1873, du Bois-Reymond constructed a continuous functionwhose Fourier series is not convergent at a point.
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Andrey Kolmogorov (1903 –1987).
Another Negative Result
In 1922, Andrey Kolmogorov gave an example of a functionf ∈ L1(T) whose Fourier series diverges almost everywhere.
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Lennart Carleson (1928– ).
The Positive Result
In 1966, L. Carleson proved that if f ∈ L2(T), then
limn→∞
Snf (x) = f (x), a.e. x ∈ T.
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In fact:
Carleson-Hunt, 1967
If f ∈ Lp(T), for some p > 1, then
limn→∞
Snf (x) = f (x), a.e. x ∈ T.
Open question today:
To characterize the space of all integrable functions for whichthe almost everywhere convergence of the Fourier series holds.
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In fact:
Carleson-Hunt, 1967
If f ∈ Lp(T), for some p > 1, then
limn→∞
Snf (x) = f (x), a.e. x ∈ T.
Open question today:
To characterize the space of all integrable functions for whichthe almost everywhere convergence of the Fourier series holds.
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Carleson maximal operator
Let S be the Carleson maximal operator,
Sf (x) = supn∈N|Snf (x)|.
Essentially, to solve the problem of the almost everywhere con-vergence in a space X , we have to prove that
S : X −→ L0
is bounded, where L0 is the set of measurable functions with theconvergence in measure topology.
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Carleson-Hunt main estimate
For every measurable set E ⊂ T and every y > 0,
∣∣∣{x ∈ [0,1] : |SχE(x)| > y}∣∣∣1/p≤ C
y(p − 1)|E |1/p, p > 1,
Consecuencia:
For every p > 1,
S : Lp −→ Lp,C
p − 1.
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The situation up to now
For every 1 < p < 2 < q,
Lq(T) ⊂ L2(T) ⊂ Lp(T) ⊂ ?????? ⊂ L1(T)
All the above spaces are of the same form:
∫ 1
0|f (x)|D(|f (x)|)dx <∞.
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The situation up to now
For every 1 < p < 2 < q,
Lq(T) ⊂ L2(T) ⊂ Lp(T) ⊂ ?????? ⊂ L1(T)
All the above spaces are of the same form:
∫ 1
0|f (x)|D(|f (x)|)dx <∞.
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Open question:
Find the best D such that the almost everywhere convergenceof the Fourier series holds true for every f satisfying:
∫ 1
0|f (x)|D(|f (x)|)dx <∞.
D(s) = 1, D(s) = sε, ε > 0.
Notation:
log(1) y = log y , log(n) y = log(n−1) log y , y >> 1.
Observe thatlog(n) y ≤ log(n−1) y .
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Open question:
Find the best D such that the almost everywhere convergenceof the Fourier series holds true for every f satisfying:
∫ 1
0|f (x)|D(|f (x)|)dx <∞.
D(s) = 1, D(s) = sε, ε > 0.
Notation:
log(1) y = log y , log(n) y = log(n−1) log y , y >> 1.
Observe thatlog(n) y ≤ log(n−1) y .
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Open question:
Find the best D such that the almost everywhere convergenceof the Fourier series holds true for every f satisfying:
∫ 1
0|f (x)|D(|f (x)|)dx <∞.
D(s) = 1, D(s) = sε, ε > 0.
Notation:
log(1) y = log y , log(n) y = log(n−1) log y , y >> 1.
Observe thatlog(n) y ≤ log(n−1) y .
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Possible and non possible functions D
State of the art:
1 ≤√
log ylog(2) y
≤ log y ≤ log y log(3) y
≤ log y log(2) y ≤ (log y)2 ≤ yε
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Possible and non possible functions D
State of the art:
1 ≤√
log ylog(2) y
≤ log y ≤ log y log(3) y
≤ log y log(2) y ≤ (log y)2 ≤ yε
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Possible and non possible functions D
Credits go to...
S. Yano, 1951:
D(y) = (log y)2
P. Sjolin, 1969:
D(y) = log y log(2) y
Y. Antonov, 1996:
D(y) = log y log(3) y
S. V. Konyagin, 2004:
D(y) =
√log y
log(2) y.
Marıa J. Carro Fourier Series and Kalton’s lattice constants
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Possible and non possible functions D
Credits go to...
S. Yano, 1951:
D(y) = (log y)2
P. Sjolin, 1969:
D(y) = log y log(2) y
Y. Antonov, 1996:
D(y) = log y log(3) y
S. V. Konyagin, 2004:
D(y) =
√log y
log(2) y.
Marıa J. Carro Fourier Series and Kalton’s lattice constants
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Possible and non possible functions D
Credits go to...
S. Yano, 1951:
D(y) = (log y)2
P. Sjolin, 1969:
D(y) = log y log(2) y
Y. Antonov, 1996:
D(y) = log y log(3) y
S. V. Konyagin, 2004:
D(y) =
√log y
log(2) y.
Marıa J. Carro Fourier Series and Kalton’s lattice constants
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Possible and non possible functions D
Credits go to...
S. Yano, 1951:
D(y) = (log y)2
P. Sjolin, 1969:
D(y) = log y log(2) y
Y. Antonov, 1996:
D(y) = log y log(3) y
S. V. Konyagin, 2004:
D(y) =
√log y
log(2) y.
Marıa J. Carro Fourier Series and Kalton’s lattice constants
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Possible and non possible functions D
Credits go to...
S. Yano, 1951:
D(y) = (log y)2
P. Sjolin, 1969:
D(y) = log y log(2) y
Y. Antonov, 1996:
D(y) = log y log(3) y
S. V. Konyagin, 2004:
D(y) =
√log y
log(2) y.
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Yano’s contribution, 1951
L(log L)2
If T is a sublinear operator such that
T : Lp −→ Lp,C
(p − 1)m ,
thenT : L(log L)m −→ L1.
Corollary
S : L(log L)2 −→ L1.
Yano, Shigeki Notes on Fourier analysis. An extrapolation theorem. J. Math.Soc. Japan 3, (1951). 296–305.
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Antonov’s contribution
Carleson-Hunt’s estimate, 1967
For every measurable set E ⊂ T and every y > 0,
‖SχE‖L1,∞ . |E |(
1 + log1|E |
):= D(|E |)
1996
For every bounded function by 1 and every y > 0,
‖Sf‖L1,∞ . D(‖f‖1)
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Antonov’s contribution
For every measurable function f ,
f =∑n∈Z
22n+1fn(x)
where
fn(x) =f (x)χ{22n≤|f |<22n+1}(x)
22n+1
Corollary
S : L log L log3 L −→ L1,∞.
Antonov, N. Yu. Convergence of Fourier series. East J. Approx. 2 (1996), no.2, 187–196
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Arias de Reyna’s contribution, 2004
Let f ∈ L1(T) and let us write
f =∑
n
fn, fn ∈ L∞.
Then:|Sf | ≤
∑n
|Sfn| =∑
n
‖fn‖∞∣∣∣S( fn‖fn‖∞
)∣∣∣
Using Stein-Weiss Lemma
||Sf ||1,∞ =∑
n
(1 + log n)||fn||∞∥∥∥∥S(
fn||fn||∞
)∥∥∥∥1,∞
.∑
n
(1 + log n)||fn||∞D( ||fn||1||fn||∞
).
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Arias de Reyna’s contribution, 2004
Let f ∈ L1(T) and let us write
f =∑
n
fn, fn ∈ L∞.
Then:|Sf | ≤
∑n
|Sfn| =∑
n
‖fn‖∞∣∣∣S( fn‖fn‖∞
)∣∣∣Using Stein-Weiss Lemma
||Sf ||1,∞ =∑
n
(1 + log n)||fn||∞∥∥∥∥S(
fn||fn||∞
)∥∥∥∥1,∞
.∑
n
(1 + log n)||fn||∞D( ||fn||1||fn||∞
).
Marıa J. Carro Fourier Series and Kalton’s lattice constants
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Arias de Reyna’s space
Definition
A measurable function f ∈ L0 belongs to QA if there exists a se-quence (fn)∞n=1, with fn ∈ L∞, such that
f =∞∑
n=1
fn, a.e.
and∞∑
n=1
(1 + log n)‖fn‖∞D( ‖fn‖1‖fn‖∞
)<∞.
We endow this space with the quasi-norm
‖f‖QA = inf{ ∞∑
n=1
(1 + log n)‖fn‖∞D( ‖fn‖1‖fn‖∞
)}.
Marıa J. Carro Fourier Series and Kalton’s lattice constants
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The best result nowadays:
TheoremFor every function f ∈ QA,
limn→∞
Snf (x) = f (x),
almost everywhere.
Recovering Antonov’s result:
L log L log(3) L(T) ⊂ QA
Arias-de-Reyna, J. Pointwise convergence of Fourier series. J. London Math.Soc. (2) 65 (2002), no. 1, 139–153.
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The best result nowadays:
TheoremFor every function f ∈ QA,
limn→∞
Snf (x) = f (x),
almost everywhere.
Recovering Antonov’s result:
L log L log(3) L(T) ⊂ QA
Arias-de-Reyna, J. Pointwise convergence of Fourier series. J. London Math.Soc. (2) 65 (2002), no. 1, 139–153.
Marıa J. Carro Fourier Series and Kalton’s lattice constants
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Questions in the air:
First one:
L log L log(4) L(T) ⊂ QA?
Conjecture:
L log L log(3) L(T)
is the best Lorentz space embedded in QA.
General questionFind the best Lorentz space contained in QA.
Marıa J. Carro Fourier Series and Kalton’s lattice constants
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Questions in the air:
First one:
L log L log(4) L(T) ⊂ QA?
Conjecture:
L log L log(3) L(T)
is the best Lorentz space embedded in QA.
General questionFind the best Lorentz space contained in QA.
Marıa J. Carro Fourier Series and Kalton’s lattice constants
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Questions in the air:
First one:
L log L log(4) L(T) ⊂ QA?
Conjecture:
L log L log(3) L(T)
is the best Lorentz space embedded in QA.
General questionFind the best Lorentz space contained in QA.
Marıa J. Carro Fourier Series and Kalton’s lattice constants
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Questions in the air:
First one:
L log L log(4) L(T) ⊂ QA?
Conjecture:
L log L log(3) L(T)
is the best Lorentz space embedded in QA.
General questionFind the best Lorentz space contained in QA.
Marıa J. Carro Fourier Series and Kalton’s lattice constants
![Page 46: Almost everywhere convergence of Fourier Series … We shall review some historical results on the problem of almost everywhere convergence of Fourier series and present some re-cent](https://reader034.fdocuments.net/reader034/viewer/2022051601/5ae423ec7f8b9ad47c8f472c/html5/thumbnails/46.jpg)
Kalton’s constants
Functional properties of QA
QA is a quasi-Banach r.i. space.
QA ↪→ L log L(T) .
The Banach envelope of QA is L log L(T).
QA is logconvex in the sense of Kalton; that is:
‖x1 + ...+ xn‖ ≤ Cn∑
j=1
(1 + log j)‖xj‖ .
C., M.J.; Mastylo, M.; Rodrıguez-Piazza, L. Almost everywhere convergentFourier series. J. Fourier Anal. Appl. 18 (2012), no. 2, 266–286.
Marıa J. Carro Fourier Series and Kalton’s lattice constants
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Kalton’s constants
Functional properties of QA
QA is a quasi-Banach r.i. space.
QA ↪→ L log L(T) .
The Banach envelope of QA is L log L(T).
QA is logconvex in the sense of Kalton; that is:
‖x1 + ...+ xn‖ ≤ Cn∑
j=1
(1 + log j)‖xj‖ .
C., M.J.; Mastylo, M.; Rodrıguez-Piazza, L. Almost everywhere convergentFourier series. J. Fourier Anal. Appl. 18 (2012), no. 2, 266–286.
Marıa J. Carro Fourier Series and Kalton’s lattice constants
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Kalton’s constants
Functional properties of QA
QA is a quasi-Banach r.i. space.
QA ↪→ L log L(T) .
The Banach envelope of QA is L log L(T).
QA is logconvex in the sense of Kalton; that is:
‖x1 + ...+ xn‖ ≤ Cn∑
j=1
(1 + log j)‖xj‖ .
C., M.J.; Mastylo, M.; Rodrıguez-Piazza, L. Almost everywhere convergentFourier series. J. Fourier Anal. Appl. 18 (2012), no. 2, 266–286.
Marıa J. Carro Fourier Series and Kalton’s lattice constants
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Kalton’s constants
Functional properties of QA
QA is a quasi-Banach r.i. space.
QA ↪→ L log L(T) .
The Banach envelope of QA is L log L(T).
QA is logconvex in the sense of Kalton; that is:
‖x1 + ...+ xn‖ ≤ Cn∑
j=1
(1 + log j)‖xj‖ .
C., M.J.; Mastylo, M.; Rodrıguez-Piazza, L. Almost everywhere convergentFourier series. J. Fourier Anal. Appl. 18 (2012), no. 2, 266–286.
Marıa J. Carro Fourier Series and Kalton’s lattice constants
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Kalton’s constants
Functional properties of QA
QA is a quasi-Banach r.i. space.
QA ↪→ L log L(T) .
The Banach envelope of QA is L log L(T).
QA is logconvex in the sense of Kalton; that is:
‖x1 + ...+ xn‖ ≤ Cn∑
j=1
(1 + log j)‖xj‖ .
C., M.J.; Mastylo, M.; Rodrıguez-Piazza, L. Almost everywhere convergentFourier series. J. Fourier Anal. Appl. 18 (2012), no. 2, 266–286.
Marıa J. Carro Fourier Series and Kalton’s lattice constants
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Kalton’s constants
Functional properties of QA
QA is a quasi-Banach r.i. space.
QA ↪→ L log L(T) .
The Banach envelope of QA is L log L(T).
QA is logconvex in the sense of Kalton; that is:
‖x1 + ...+ xn‖ ≤ Cn∑
j=1
(1 + log j)‖xj‖ .
C., M.J.; Mastylo, M.; Rodrıguez-Piazza, L. Almost everywhere convergentFourier series. J. Fourier Anal. Appl. 18 (2012), no. 2, 266–286.
Marıa J. Carro Fourier Series and Kalton’s lattice constants
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Kalton’s constants
Theorem
For each positive integer n,
c n(1 + log n) ≤ en(QA) ≈ fn(QA) ≤ n(1 + log n) .
Moreover:
For each positive integer n,
c n(1 + log n) ≤∥∥∥ n∑
i=1
χAni
‖χAni‖X
∥∥∥QA
with Ani as follows:
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Kalton’s constantsBrief Article
The Author
January 29, 2016
1
· · ·
A1 A2 A3 · · · Am
1
where, if µj = |Aj |, then
µj+1 =(µj
e
)n3(1+log n), j = 1, · · · ,n.
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L log L log(4) L(T) ⊂ QA?
TheoremLet Y be a Banach space. Then
Y ⊂ X =⇒ fn(X )
n. sup
i
‖χAni‖Y
‖χAni‖X, ∀n.
Is it true that?
(1 + log n) . supi
‖χAni‖Y
‖χAni‖X, ∀n.
Marıa J. Carro Fourier Series and Kalton’s lattice constants
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Answers on the earth:
First one:
L log L log(4) L(T) * QA.
The conjecture is false:
L log L log(3) L(T)
is not the best Lorentz space embedded in QA.
In fact, more can be said:
Answer to the general question:
There is not the best Lorentz space contained in QA.
Marıa J. Carro Fourier Series and Kalton’s lattice constants
![Page 56: Almost everywhere convergence of Fourier Series … We shall review some historical results on the problem of almost everywhere convergence of Fourier series and present some re-cent](https://reader034.fdocuments.net/reader034/viewer/2022051601/5ae423ec7f8b9ad47c8f472c/html5/thumbnails/56.jpg)
Answers on the earth:
First one:
L log L log(4) L(T) * QA.
The conjecture is false:
L log L log(3) L(T)
is not the best Lorentz space embedded in QA.
In fact, more can be said:
Answer to the general question:
There is not the best Lorentz space contained in QA.
Marıa J. Carro Fourier Series and Kalton’s lattice constants
![Page 57: Almost everywhere convergence of Fourier Series … We shall review some historical results on the problem of almost everywhere convergence of Fourier series and present some re-cent](https://reader034.fdocuments.net/reader034/viewer/2022051601/5ae423ec7f8b9ad47c8f472c/html5/thumbnails/57.jpg)
Answers on the earth:
First one:
L log L log(4) L(T) * QA.
The conjecture is false:
L log L log(3) L(T)
is not the best Lorentz space embedded in QA.
In fact, more can be said:
Answer to the general question:
There is not the best Lorentz space contained in QA.
Marıa J. Carro Fourier Series and Kalton’s lattice constants
![Page 58: Almost everywhere convergence of Fourier Series … We shall review some historical results on the problem of almost everywhere convergence of Fourier series and present some re-cent](https://reader034.fdocuments.net/reader034/viewer/2022051601/5ae423ec7f8b9ad47c8f472c/html5/thumbnails/58.jpg)
Answers on the earth:
First one:
L log L log(4) L(T) * QA.
The conjecture is false:
L log L log(3) L(T)
is not the best Lorentz space embedded in QA.
In fact, more can be said:
Answer to the general question:
There is not the best Lorentz space contained in QA.
Marıa J. Carro Fourier Series and Kalton’s lattice constants
![Page 59: Almost everywhere convergence of Fourier Series … We shall review some historical results on the problem of almost everywhere convergence of Fourier series and present some re-cent](https://reader034.fdocuments.net/reader034/viewer/2022051601/5ae423ec7f8b9ad47c8f472c/html5/thumbnails/59.jpg)
Carleson maximal operator
Let S be the Carleson maximal operator,
Sf (x) = supn∈N|Snf (x)|.
Essentially, to solve the problem of the almost everywhere con-vergence in a space X , we have to prove that
S : X −→ L0
is bounded, where L0 is the set of measurable functions with theconvergence in measure topology.
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Carleson maximal operator
Let S be the Carleson maximal operator,
Sf (x) = supn∈N|Snf (x)|.
Essentially, to solve the problem of the almost everywhere con-vergence in a space X⊂ L1, we have to prove that
S : X −→ L1,∞
is bounded.
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A bigger space
LetS(L1,∞) = {f : Sf ∈ L1,∞}
with the norm‖f‖S(L1,∞) = ‖Sf‖L1,∞ .
Corollary
i) If f ∈ S(L1,∞),
limn
Snf (x) = f (x),a.e.x
ii)QA ⊂ S(L1,∞)
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A bigger space
LetS(L1,∞) = {f : Sf ∈ L1,∞}
with the norm‖f‖S(L1,∞) = ‖Sf‖L1,∞ .
Corollary
i) If f ∈ S(L1,∞),
limn
Snf (x) = f (x),a.e.x
ii)QA ⊂ S(L1,∞)
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Our next goal: to study the space S(L1,∞)
Properties
i) It is a quasi-Banach space.ii)
fn(S(L1,∞)) ≤ en(S(L1,∞)) . n(1 + log n)
Open questions
i) n(1 + log n) . fn(S(L1,∞))?
ii) How to construct Ani ?
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The end
THANKS FOR YOUR ATTENTION
Marıa J. Carro Fourier Series and Kalton’s lattice constants