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Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Selected Problems in Classical Function Theory
Dmitry [email protected]
http://shell.cas.usf.edu/ dkhavins/
University of South Florida
August 17, 2013
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Outline
1 Introduction: Spaces of Analytic Functions
2 Analytic Functions with Positive Boundary Values
3 Analytic Content: Approximating z̄ in Bergman and SmirnovNorms
4 Putnam’s Inequality for Toeplitz Operators in Bergman Spaces
5 Final Remarks
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Outline
1 Introduction: Spaces of Analytic Functions
2 Analytic Functions with Positive Boundary Values
3 Analytic Content: Approximating z̄ in Bergman and SmirnovNorms
4 Putnam’s Inequality for Toeplitz Operators in Bergman Spaces
5 Final Remarks
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Outline
1 Introduction: Spaces of Analytic Functions
2 Analytic Functions with Positive Boundary Values
3 Analytic Content: Approximating z̄ in Bergman and SmirnovNorms
4 Putnam’s Inequality for Toeplitz Operators in Bergman Spaces
5 Final Remarks
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Outline
1 Introduction: Spaces of Analytic Functions
2 Analytic Functions with Positive Boundary Values
3 Analytic Content: Approximating z̄ in Bergman and SmirnovNorms
4 Putnam’s Inequality for Toeplitz Operators in Bergman Spaces
5 Final Remarks
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Outline
1 Introduction: Spaces of Analytic Functions
2 Analytic Functions with Positive Boundary Values
3 Analytic Content: Approximating z̄ in Bergman and SmirnovNorms
4 Putnam’s Inequality for Toeplitz Operators in Bergman Spaces
5 Final Remarks
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Place of Action
open disk
not a disk!, a f. c. domain G
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Place of Action
open disk
not a disk!, a f. c. domain G
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Place of Action
open disk
not a disk!,
a f. c. domain G
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Place of Action
open disk
not a disk!, a f. c. domain G
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Bergman spaces
Definition
For 0 < p <∞, define the Bergman space as
Ap =
{f analytic in D :
(∫D|f (z)|pdA(z)
) 1p
=: ‖f ‖p <∞
},
where dA = 1πdxdy denotes normalized area measure in the unit
disk D.
Ap(G ) are defined accordingly
Stefan Bergman (b. 1895 d.1977)
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Bergman spaces
Definition
For 0 < p <∞, define the Bergman space as
Ap =
{f analytic in D :
(∫D|f (z)|pdA(z)
) 1p
=: ‖f ‖p <∞
},
where dA = 1πdxdy denotes normalized area measure in the unit
disk D. Ap(G ) are defined accordingly
Stefan Bergman (b. 1895 d.1977)
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Hardy spaces
Definition
For 0 < p <∞, define the Hardy space as
Hp :=
{f analytic in D : sup
0<r≤1
1
2π
∫ 2π
0|f (re it)|pdt =: ‖f ‖pHp <∞
}.
(H∞ = {f analytic in D : ‖f ‖∞ := sup {|f (z)|, z ∈ D} <∞}).
Godfrey Harold Hardy (b. 1877 d. 1947)
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Hardy classes in multiply connected domains
Definition
An analytic function f (z) in G belongs the the Hardy class Hp(G )for p : 0 < p <∞ if the subharmonic function |f |p has a harmonicmajorant in G .
Remark: Hardy classes (spaces) are conformally equivalent, i.e., ifϕ : K → G is the conformal mapping of an n-connected circulardomain K onto G , then f ∈ Hp(G ) if and only if f ◦ ϕ ∈ Hp(K ) .
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Hardy classes in multiply connected domains
Definition
An analytic function f (z) in G belongs the the Hardy class Hp(G )for p : 0 < p <∞ if the subharmonic function |f |p has a harmonicmajorant in G .
Remark: Hardy classes (spaces) are conformally equivalent, i.e., ifϕ : K → G is the conformal mapping of an n-connected circulardomain K onto G , then f ∈ Hp(G ) if and only if f ◦ ϕ ∈ Hp(K ) .
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
V. I. Smirnov classes
Let G be an n-connected domain in thecomplex plane bounded by Jordan rectifiable curves γ1,...,γn andlet Γ =
⋃ni=1 γi .
Definition
An analytic function f (z) in G is said to belong to the Smirnovclass Ep(G ) for p : 0 < p <∞ if there exists a sequence ofrectifiable curves {Γi} in G converging to Γ such that
lim supi→∞
∫Γi
|f (z)|p|dz | <∞.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
V. I. Smirnov (1887 - 1974)
Remark (i) The critical difference between Smirnov classes andHardy classes is that the former are NOT conformally equivalent.
(ii) For p ≥ 1 Hardy functions are represented by Poisson integralsof their boundary values, while Smirnov functions are representedby Cauchy integrals, i.e., in the former case the kernel is positive,
in the latter - complex.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
V. I. Smirnov (1887 - 1974)
Remark (i) The critical difference between Smirnov classes andHardy classes is that the former are NOT conformally equivalent.
(ii) For p ≥ 1 Hardy functions are represented by Poisson integralsof their boundary values, while Smirnov functions are representedby Cauchy integrals, i.e., in the former case the kernel is positive,
in the latter - complex.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
V. I. Smirnov (1887 - 1974)
Remark (i) The critical difference between Smirnov classes andHardy classes is that the former are NOT conformally equivalent.
(ii) For p ≥ 1 Hardy functions are represented by Poisson integralsof their boundary values, while Smirnov functions are representedby Cauchy integrals, i.e., in the former case the kernel is positive,
in the latter - complex.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
V. I. Smirnov (1887 - 1974)
Remark (i) The critical difference between Smirnov classes andHardy classes is that the former are NOT conformally equivalent.
(ii) For p ≥ 1 Hardy functions are represented by Poisson integralsof their boundary values, while Smirnov functions are representedby Cauchy integrals, i.e., in the former case the kernel is positive,
in the latter - complex.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
J. Neuwirth - D. J. Newman (1930 - 2007) Theorem, 1967
Theorem
Let f ∈ H1/2(D) be ≥ 0 a.e. on the unit circle T. Then f isconstant. (The function z
(1+z)2 implies that 1/2 is sharp.)
Remarks (i) If “a.e.” is meant wrt the harmonic measure thetheorem automatically extends to arbitrary domains.(ii)
Theorem
(L. De Castro - DK, 2012) Let G be a simply connected Smirnovdomain with rectifiable boundary Γ. Let p0 ≥ 1 be defined as thesmallest p ≥ 1 such that f ∈ Ep(G ) and f has real boundaryvalues a.e. on Γ imply that f is a constant. Then all f ∈ Ep0/2
such that f ≥ 0 a.e. on Γ are constants.
(iii) The theorem is false for non-Smirnov domains (DK -1982).
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
J. Neuwirth - D. J. Newman (1930 - 2007) Theorem, 1967
Theorem
Let f ∈ H1/2(D) be ≥ 0 a.e. on the unit circle T. Then f isconstant. (The function z
(1+z)2 implies that 1/2 is sharp.)
Remarks (i) If “a.e.” is meant wrt the harmonic measure thetheorem automatically extends to arbitrary domains.
(ii)
Theorem
(L. De Castro - DK, 2012) Let G be a simply connected Smirnovdomain with rectifiable boundary Γ. Let p0 ≥ 1 be defined as thesmallest p ≥ 1 such that f ∈ Ep(G ) and f has real boundaryvalues a.e. on Γ imply that f is a constant. Then all f ∈ Ep0/2
such that f ≥ 0 a.e. on Γ are constants.
(iii) The theorem is false for non-Smirnov domains (DK -1982).
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
J. Neuwirth - D. J. Newman (1930 - 2007) Theorem, 1967
Theorem
Let f ∈ H1/2(D) be ≥ 0 a.e. on the unit circle T. Then f isconstant. (The function z
(1+z)2 implies that 1/2 is sharp.)
Remarks (i) If “a.e.” is meant wrt the harmonic measure thetheorem automatically extends to arbitrary domains.(ii)
Theorem
(L. De Castro - DK, 2012) Let G be a simply connected Smirnovdomain with rectifiable boundary Γ. Let p0 ≥ 1 be defined as thesmallest p ≥ 1 such that f ∈ Ep(G ) and f has real boundaryvalues a.e. on Γ imply that f is a constant. Then all f ∈ Ep0/2
such that f ≥ 0 a.e. on Γ are constants.
(iii) The theorem is false for non-Smirnov domains (DK -1982).
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
J. Neuwirth - D. J. Newman (1930 - 2007) Theorem, 1967
Theorem
Let f ∈ H1/2(D) be ≥ 0 a.e. on the unit circle T. Then f isconstant. (The function z
(1+z)2 implies that 1/2 is sharp.)
Remarks (i) If “a.e.” is meant wrt the harmonic measure thetheorem automatically extends to arbitrary domains.(ii)
Theorem
(L. De Castro - DK, 2012) Let G be a simply connected Smirnovdomain with rectifiable boundary Γ. Let p0 ≥ 1 be defined as thesmallest p ≥ 1 such that f ∈ Ep(G ) and f has real boundaryvalues a.e. on Γ imply that f is a constant. Then all f ∈ Ep0/2
such that f ≥ 0 a.e. on Γ are constants.
(iii) The theorem is false for non-Smirnov domains (DK -1982).
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Neuwirth - Newman Theorem in Multiply ConnectedDomains
S.C. Smirnov Domains. f (z) = B(z)S(z)F 2(z), where B(z) is atransplanted Blaschke product, S(z) is a bounded singular innerfunction, and F (z) ∈ Ep0 is an outer function. But, on Γ,B(z)S(z)F 2(z) = |f (z)| = F (z)F (z) a.e., soF (z) = B(z)S(z)F (z) ∈ Ep0(G ) which implies thatF (z) + F (z) ∈ Ep0(G ) and is real-valued, hence a constant.Thus,f (z) = const · B(z)S(z) is a bounded function with non-negativeboundary values, hence a constant as well.Difficulty for M.C.D. f = QBSF 2, where B is the generalizedBlaschke product, S is a singular inner function, F 2 is an outerfactor, F ∈ Ep0 , Q is an invertible bounded analytic function, and|Q| is a local constant on Γ. The problem is that |B| and |S | arelocal constants a.e. on the boundary of G . Hence, f ≥ 0 a.e. on Γonly yields on Γ that f = QBSF 2 = |QBS |F F , i.e., F coincideswith different analytic functions on different boundary components.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Neuwirth - Newman Theorem in Multiply ConnectedDomains
S.C. Smirnov Domains.
f (z) = B(z)S(z)F 2(z), where B(z) is atransplanted Blaschke product, S(z) is a bounded singular innerfunction, and F (z) ∈ Ep0 is an outer function. But, on Γ,B(z)S(z)F 2(z) = |f (z)| = F (z)F (z) a.e., soF (z) = B(z)S(z)F (z) ∈ Ep0(G ) which implies thatF (z) + F (z) ∈ Ep0(G ) and is real-valued, hence a constant.Thus,f (z) = const · B(z)S(z) is a bounded function with non-negativeboundary values, hence a constant as well.Difficulty for M.C.D. f = QBSF 2, where B is the generalizedBlaschke product, S is a singular inner function, F 2 is an outerfactor, F ∈ Ep0 , Q is an invertible bounded analytic function, and|Q| is a local constant on Γ. The problem is that |B| and |S | arelocal constants a.e. on the boundary of G . Hence, f ≥ 0 a.e. on Γonly yields on Γ that f = QBSF 2 = |QBS |F F , i.e., F coincideswith different analytic functions on different boundary components.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Neuwirth - Newman Theorem in Multiply ConnectedDomains
S.C. Smirnov Domains. f (z) = B(z)S(z)F 2(z), where B(z) is atransplanted Blaschke product, S(z) is a bounded singular innerfunction, and F (z) ∈ Ep0 is an outer function.
But, on Γ,B(z)S(z)F 2(z) = |f (z)| = F (z)F (z) a.e., soF (z) = B(z)S(z)F (z) ∈ Ep0(G ) which implies thatF (z) + F (z) ∈ Ep0(G ) and is real-valued, hence a constant.Thus,f (z) = const · B(z)S(z) is a bounded function with non-negativeboundary values, hence a constant as well.Difficulty for M.C.D. f = QBSF 2, where B is the generalizedBlaschke product, S is a singular inner function, F 2 is an outerfactor, F ∈ Ep0 , Q is an invertible bounded analytic function, and|Q| is a local constant on Γ. The problem is that |B| and |S | arelocal constants a.e. on the boundary of G . Hence, f ≥ 0 a.e. on Γonly yields on Γ that f = QBSF 2 = |QBS |F F , i.e., F coincideswith different analytic functions on different boundary components.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Neuwirth - Newman Theorem in Multiply ConnectedDomains
S.C. Smirnov Domains. f (z) = B(z)S(z)F 2(z), where B(z) is atransplanted Blaschke product, S(z) is a bounded singular innerfunction, and F (z) ∈ Ep0 is an outer function. But, on Γ,B(z)S(z)F 2(z) = |f (z)| = F (z)F (z) a.e.,
soF (z) = B(z)S(z)F (z) ∈ Ep0(G ) which implies thatF (z) + F (z) ∈ Ep0(G ) and is real-valued, hence a constant.Thus,f (z) = const · B(z)S(z) is a bounded function with non-negativeboundary values, hence a constant as well.Difficulty for M.C.D. f = QBSF 2, where B is the generalizedBlaschke product, S is a singular inner function, F 2 is an outerfactor, F ∈ Ep0 , Q is an invertible bounded analytic function, and|Q| is a local constant on Γ. The problem is that |B| and |S | arelocal constants a.e. on the boundary of G . Hence, f ≥ 0 a.e. on Γonly yields on Γ that f = QBSF 2 = |QBS |F F , i.e., F coincideswith different analytic functions on different boundary components.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Neuwirth - Newman Theorem in Multiply ConnectedDomains
S.C. Smirnov Domains. f (z) = B(z)S(z)F 2(z), where B(z) is atransplanted Blaschke product, S(z) is a bounded singular innerfunction, and F (z) ∈ Ep0 is an outer function. But, on Γ,B(z)S(z)F 2(z) = |f (z)| = F (z)F (z) a.e., soF (z) = B(z)S(z)F (z) ∈ Ep0(G ) which implies thatF (z) + F (z) ∈ Ep0(G ) and is real-valued, hence a constant.
Thus,f (z) = const · B(z)S(z) is a bounded function with non-negativeboundary values, hence a constant as well.Difficulty for M.C.D. f = QBSF 2, where B is the generalizedBlaschke product, S is a singular inner function, F 2 is an outerfactor, F ∈ Ep0 , Q is an invertible bounded analytic function, and|Q| is a local constant on Γ. The problem is that |B| and |S | arelocal constants a.e. on the boundary of G . Hence, f ≥ 0 a.e. on Γonly yields on Γ that f = QBSF 2 = |QBS |F F , i.e., F coincideswith different analytic functions on different boundary components.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Neuwirth - Newman Theorem in Multiply ConnectedDomains
S.C. Smirnov Domains. f (z) = B(z)S(z)F 2(z), where B(z) is atransplanted Blaschke product, S(z) is a bounded singular innerfunction, and F (z) ∈ Ep0 is an outer function. But, on Γ,B(z)S(z)F 2(z) = |f (z)| = F (z)F (z) a.e., soF (z) = B(z)S(z)F (z) ∈ Ep0(G ) which implies thatF (z) + F (z) ∈ Ep0(G ) and is real-valued, hence a constant.Thus,f (z) = const · B(z)S(z) is a bounded function with non-negativeboundary values, hence a constant as well.
Difficulty for M.C.D. f = QBSF 2, where B is the generalizedBlaschke product, S is a singular inner function, F 2 is an outerfactor, F ∈ Ep0 , Q is an invertible bounded analytic function, and|Q| is a local constant on Γ. The problem is that |B| and |S | arelocal constants a.e. on the boundary of G . Hence, f ≥ 0 a.e. on Γonly yields on Γ that f = QBSF 2 = |QBS |F F , i.e., F coincideswith different analytic functions on different boundary components.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Neuwirth - Newman Theorem in Multiply ConnectedDomains
S.C. Smirnov Domains. f (z) = B(z)S(z)F 2(z), where B(z) is atransplanted Blaschke product, S(z) is a bounded singular innerfunction, and F (z) ∈ Ep0 is an outer function. But, on Γ,B(z)S(z)F 2(z) = |f (z)| = F (z)F (z) a.e., soF (z) = B(z)S(z)F (z) ∈ Ep0(G ) which implies thatF (z) + F (z) ∈ Ep0(G ) and is real-valued, hence a constant.Thus,f (z) = const · B(z)S(z) is a bounded function with non-negativeboundary values, hence a constant as well.Difficulty for M.C.D.
f = QBSF 2, where B is the generalizedBlaschke product, S is a singular inner function, F 2 is an outerfactor, F ∈ Ep0 , Q is an invertible bounded analytic function, and|Q| is a local constant on Γ. The problem is that |B| and |S | arelocal constants a.e. on the boundary of G . Hence, f ≥ 0 a.e. on Γonly yields on Γ that f = QBSF 2 = |QBS |F F , i.e., F coincideswith different analytic functions on different boundary components.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Neuwirth - Newman Theorem in Multiply ConnectedDomains
S.C. Smirnov Domains. f (z) = B(z)S(z)F 2(z), where B(z) is atransplanted Blaschke product, S(z) is a bounded singular innerfunction, and F (z) ∈ Ep0 is an outer function. But, on Γ,B(z)S(z)F 2(z) = |f (z)| = F (z)F (z) a.e., soF (z) = B(z)S(z)F (z) ∈ Ep0(G ) which implies thatF (z) + F (z) ∈ Ep0(G ) and is real-valued, hence a constant.Thus,f (z) = const · B(z)S(z) is a bounded function with non-negativeboundary values, hence a constant as well.Difficulty for M.C.D. f = QBSF 2, where B is the generalizedBlaschke product, S is a singular inner function, F 2 is an outerfactor, F ∈ Ep0 , Q is an invertible bounded analytic function, and|Q| is a local constant on Γ.
The problem is that |B| and |S | arelocal constants a.e. on the boundary of G . Hence, f ≥ 0 a.e. on Γonly yields on Γ that f = QBSF 2 = |QBS |F F , i.e., F coincideswith different analytic functions on different boundary components.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Neuwirth - Newman Theorem in Multiply ConnectedDomains
S.C. Smirnov Domains. f (z) = B(z)S(z)F 2(z), where B(z) is atransplanted Blaschke product, S(z) is a bounded singular innerfunction, and F (z) ∈ Ep0 is an outer function. But, on Γ,B(z)S(z)F 2(z) = |f (z)| = F (z)F (z) a.e., soF (z) = B(z)S(z)F (z) ∈ Ep0(G ) which implies thatF (z) + F (z) ∈ Ep0(G ) and is real-valued, hence a constant.Thus,f (z) = const · B(z)S(z) is a bounded function with non-negativeboundary values, hence a constant as well.Difficulty for M.C.D. f = QBSF 2, where B is the generalizedBlaschke product, S is a singular inner function, F 2 is an outerfactor, F ∈ Ep0 , Q is an invertible bounded analytic function, and|Q| is a local constant on Γ. The problem is that |B| and |S | arelocal constants a.e. on the boundary of G .
Hence, f ≥ 0 a.e. on Γonly yields on Γ that f = QBSF 2 = |QBS |F F , i.e., F coincideswith different analytic functions on different boundary components.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Neuwirth - Newman Theorem in Multiply ConnectedDomains
S.C. Smirnov Domains. f (z) = B(z)S(z)F 2(z), where B(z) is atransplanted Blaschke product, S(z) is a bounded singular innerfunction, and F (z) ∈ Ep0 is an outer function. But, on Γ,B(z)S(z)F 2(z) = |f (z)| = F (z)F (z) a.e., soF (z) = B(z)S(z)F (z) ∈ Ep0(G ) which implies thatF (z) + F (z) ∈ Ep0(G ) and is real-valued, hence a constant.Thus,f (z) = const · B(z)S(z) is a bounded function with non-negativeboundary values, hence a constant as well.Difficulty for M.C.D. f = QBSF 2, where B is the generalizedBlaschke product, S is a singular inner function, F 2 is an outerfactor, F ∈ Ep0 , Q is an invertible bounded analytic function, and|Q| is a local constant on Γ. The problem is that |B| and |S | arelocal constants a.e. on the boundary of G . Hence, f ≥ 0 a.e. on Γonly yields on Γ that f = QBSF 2 = |QBS |F F , i.e., F coincideswith different analytic functions on different boundary components.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Neuwirth - Newman Theorem in Multiply ConnectedDomains
Conjecture
(L. De Castro - DK, 2012) Let G be a f.c. Smirnov domain. Ifp0 ≥ 1 is the smallest index for which all functions in Ep0 with realboundary values are constants, then all Ep0/2 - functions withpositive boundary values are constants.
Example L. De Castro - DK, 2012.
In an n - connected domain G of “cardioid type”with m interiorcusps on ∂G all E 1-functions with positive boundary values areconstants. (For such domains p0 = 2, cf. LDC - DK, 2012)).
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Neuwirth - Newman Theorem in Multiply ConnectedDomains
Conjecture
(L. De Castro - DK, 2012) Let G be a f.c. Smirnov domain. Ifp0 ≥ 1 is the smallest index for which all functions in Ep0 with realboundary values are constants, then all Ep0/2 - functions withpositive boundary values are constants.
Example L. De Castro - DK, 2012.
In an n - connected domain G of “cardioid type”with m interiorcusps on ∂G all E 1-functions with positive boundary values areconstants. (For such domains p0 = 2, cf. LDC - DK, 2012)).
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Neuwirth - Newman Theorem in Multiply ConnectedDomains
Conjecture
(L. De Castro - DK, 2012) Let G be a f.c. Smirnov domain. Ifp0 ≥ 1 is the smallest index for which all functions in Ep0 with realboundary values are constants, then all Ep0/2 - functions withpositive boundary values are constants.
Example L. De Castro - DK, 2012.
In an n - connected domain G of “cardioid type”with m interiorcusps on ∂G all E 1-functions with positive boundary values areconstants. (For such domains p0 = 2, cf. LDC - DK, 2012)).
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Neuwirth - Newman Theorem in Multiply ConnectedDomains
Conjecture
(L. De Castro - DK, 2012) Let G be a f.c. Smirnov domain. Ifp0 ≥ 1 is the smallest index for which all functions in Ep0 with realboundary values are constants, then all Ep0/2 - functions withpositive boundary values are constants.
Example L. De Castro - DK, 2012.
In an n - connected domain G of “cardioid type”with m interiorcusps on ∂G all E 1-functions with positive boundary values areconstants. (For such domains p0 = 2, cf. LDC - DK, 2012)).
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Neuwirth - Newman Theorem in Multiply ConnectedDomains
Conjecture
(L. De Castro - DK, 2012) Let G be a f.c. Smirnov domain. Ifp0 ≥ 1 is the smallest index for which all functions in Ep0 with realboundary values are constants, then all Ep0/2 - functions withpositive boundary values are constants.
Example L. De Castro - DK, 2012.
In an n - connected domain G of “cardioid type”with m interiorcusps on ∂G all E 1-functions with positive boundary values areconstants.
(For such domains p0 = 2, cf. LDC - DK, 2012)).
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Neuwirth - Newman Theorem in Multiply ConnectedDomains
Conjecture
(L. De Castro - DK, 2012) Let G be a f.c. Smirnov domain. Ifp0 ≥ 1 is the smallest index for which all functions in Ep0 with realboundary values are constants, then all Ep0/2 - functions withpositive boundary values are constants.
Example L. De Castro - DK, 2012.
In an n - connected domain G of “cardioid type”with m interiorcusps on ∂G all E 1-functions with positive boundary values areconstants. (For such domains p0 = 2, cf. LDC - DK, 2012)).
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Analytic Content
Let G be a finitely connected region in C bounded by n simpleclosed analytic curves γj , j = 1, . . . , n.
Definition
The analytic content of a domain G is
λ(G ) := infφ∈H∞(G)
‖ z̄ − φ ‖H∞(G)
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Analytic Content
Let G be a finitely connected region in C bounded by n simpleclosed analytic curves γj , j = 1, . . . , n.
Definition
The analytic content of a domain G is
λ(G ) := infφ∈H∞(G)
‖ z̄ − φ ‖H∞(G)
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Theorem
(H. Alexander - ’73, DK -’82) Let A and P be the area andperimeter of G , respectively. Then
2A
P≤ λ(G ) ≤
√A
π,
so P2 ≥ 4πA. Moreover, λ(G ) =√
Aπ ⇔ G is a disk.
Question: For which G does the equality 2AP = λ(G ) hold?
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Theorem
(H. Alexander - ’73, DK -’82) Let A and P be the area andperimeter of G , respectively. Then
2A
P≤ λ(G ) ≤
√A
π,
so P2 ≥ 4πA.
Moreover, λ(G ) =√
Aπ ⇔ G is a disk.
Question: For which G does the equality 2AP = λ(G ) hold?
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Theorem
(H. Alexander - ’73, DK -’82) Let A and P be the area andperimeter of G , respectively. Then
2A
P≤ λ(G ) ≤
√A
π,
so P2 ≥ 4πA. Moreover, λ(G ) =√
Aπ ⇔ G is a disk.
Question: For which G does the equality 2AP = λ(G ) hold?
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Theorem
(H. Alexander - ’73, DK -’82) Let A and P be the area andperimeter of G , respectively. Then
2A
P≤ λ(G ) ≤
√A
π,
so P2 ≥ 4πA. Moreover, λ(G ) =√
Aπ ⇔ G is a disk.
Question: For which G does the equality 2AP = λ(G ) hold?
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Equivalent form of achievement of lower bound
Theorem
TFAE:
(i) λ = 2AP ;
(ii)There is φ ∈ H∞(G ) such that z̄(s)− iλ ˙̄z(s) = φ(z(s)) on Γ,where s is the arc-length parameter;(iii) 1
A
∫G fdA = 1
P
∫Γ fds for all f ∈ H∞(G ).
Remarks:
(iii) holds for annuli G = {r < |z | < R}.(ii) implies that if Γ contains a circular arc, G is a disk or anannulus.
(DK-’82) Are these the only two possibilities? YES!(A.Abanov, CB, DK, R. Teodorescu (2012 - 2014))
In the simply connected case the extremal domain is a disk(DK -’86).
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Equivalent form of achievement of lower bound
Theorem
TFAE:(i) λ = 2A
P ;
(ii)There is φ ∈ H∞(G ) such that z̄(s)− iλ ˙̄z(s) = φ(z(s)) on Γ,where s is the arc-length parameter;(iii) 1
A
∫G fdA = 1
P
∫Γ fds for all f ∈ H∞(G ).
Remarks:
(iii) holds for annuli G = {r < |z | < R}.(ii) implies that if Γ contains a circular arc, G is a disk or anannulus.
(DK-’82) Are these the only two possibilities? YES!(A.Abanov, CB, DK, R. Teodorescu (2012 - 2014))
In the simply connected case the extremal domain is a disk(DK -’86).
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Equivalent form of achievement of lower bound
Theorem
TFAE:(i) λ = 2A
P ;(ii)There is φ ∈ H∞(G ) such that z̄(s)− iλ ˙̄z(s) = φ(z(s)) on Γ,where s is the arc-length parameter;
(iii) 1A
∫G fdA = 1
P
∫Γ fds for all f ∈ H∞(G ).
Remarks:
(iii) holds for annuli G = {r < |z | < R}.(ii) implies that if Γ contains a circular arc, G is a disk or anannulus.
(DK-’82) Are these the only two possibilities? YES!(A.Abanov, CB, DK, R. Teodorescu (2012 - 2014))
In the simply connected case the extremal domain is a disk(DK -’86).
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Equivalent form of achievement of lower bound
Theorem
TFAE:(i) λ = 2A
P ;(ii)There is φ ∈ H∞(G ) such that z̄(s)− iλ ˙̄z(s) = φ(z(s)) on Γ,where s is the arc-length parameter;(iii) 1
A
∫G fdA = 1
P
∫Γ fds for all f ∈ H∞(G ).
Remarks:
(iii) holds for annuli G = {r < |z | < R}.(ii) implies that if Γ contains a circular arc, G is a disk or anannulus.
(DK-’82) Are these the only two possibilities? YES!(A.Abanov, CB, DK, R. Teodorescu (2012 - 2014))
In the simply connected case the extremal domain is a disk(DK -’86).
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Equivalent form of achievement of lower bound
Theorem
TFAE:(i) λ = 2A
P ;(ii)There is φ ∈ H∞(G ) such that z̄(s)− iλ ˙̄z(s) = φ(z(s)) on Γ,where s is the arc-length parameter;(iii) 1
A
∫G fdA = 1
P
∫Γ fds for all f ∈ H∞(G ).
Remarks:
(iii) holds for annuli G = {r < |z | < R}.
(ii) implies that if Γ contains a circular arc, G is a disk or anannulus.
(DK-’82) Are these the only two possibilities? YES!(A.Abanov, CB, DK, R. Teodorescu (2012 - 2014))
In the simply connected case the extremal domain is a disk(DK -’86).
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Equivalent form of achievement of lower bound
Theorem
TFAE:(i) λ = 2A
P ;(ii)There is φ ∈ H∞(G ) such that z̄(s)− iλ ˙̄z(s) = φ(z(s)) on Γ,where s is the arc-length parameter;(iii) 1
A
∫G fdA = 1
P
∫Γ fds for all f ∈ H∞(G ).
Remarks:
(iii) holds for annuli G = {r < |z | < R}.(ii) implies that if Γ contains a circular arc, G is a disk or anannulus.
(DK-’82) Are these the only two possibilities? YES!(A.Abanov, CB, DK, R. Teodorescu (2012 - 2014))
In the simply connected case the extremal domain is a disk(DK -’86).
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Equivalent form of achievement of lower bound
Theorem
TFAE:(i) λ = 2A
P ;(ii)There is φ ∈ H∞(G ) such that z̄(s)− iλ ˙̄z(s) = φ(z(s)) on Γ,where s is the arc-length parameter;(iii) 1
A
∫G fdA = 1
P
∫Γ fds for all f ∈ H∞(G ).
Remarks:
(iii) holds for annuli G = {r < |z | < R}.(ii) implies that if Γ contains a circular arc, G is a disk or anannulus.
(DK-’82) Are these the only two possibilities?
YES!(A.Abanov, CB, DK, R. Teodorescu (2012 - 2014))
In the simply connected case the extremal domain is a disk(DK -’86).
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Equivalent form of achievement of lower bound
Theorem
TFAE:(i) λ = 2A
P ;(ii)There is φ ∈ H∞(G ) such that z̄(s)− iλ ˙̄z(s) = φ(z(s)) on Γ,where s is the arc-length parameter;(iii) 1
A
∫G fdA = 1
P
∫Γ fds for all f ∈ H∞(G ).
Remarks:
(iii) holds for annuli G = {r < |z | < R}.(ii) implies that if Γ contains a circular arc, G is a disk or anannulus.
(DK-’82) Are these the only two possibilities? YES!(A.Abanov, CB, DK, R. Teodorescu (2012 - 2014))
In the simply connected case the extremal domain is a disk(DK -’86).
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Equivalent form of achievement of lower bound
Theorem
TFAE:(i) λ = 2A
P ;(ii)There is φ ∈ H∞(G ) such that z̄(s)− iλ ˙̄z(s) = φ(z(s)) on Γ,where s is the arc-length parameter;(iii) 1
A
∫G fdA = 1
P
∫Γ fds for all f ∈ H∞(G ).
Remarks:
(iii) holds for annuli G = {r < |z | < R}.(ii) implies that if Γ contains a circular arc, G is a disk or anannulus.
(DK-’82) Are these the only two possibilities? YES!(A.Abanov, CB, DK, R. Teodorescu (2012 - 2014))
In the simply connected case the extremal domain is a disk(DK -’86).
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
E p Analytic Content
Definition
Forp ≥ 1, λEp := inf
φ∈Ep(G)‖ z̄ − φ ‖Ep(G) .
Theorem
(Z. Guadarrama - DK, 2007) Let A,P denote the area andperimeter of the f.c. domain G. For p ≥ 1, q = p
p−1 , we have
2Aq√
P≤ λEp ≤
√A
πP
1p .
Questions. (i) Are disks and annuli the only extremal domains forall λEp ,p≥1, as well? (ii) Do the extremal functions for λEp
characterize the domain G ?
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
E p Analytic Content
Definition
Forp ≥ 1, λEp := inf
φ∈Ep(G)‖ z̄ − φ ‖Ep(G) .
Theorem
(Z. Guadarrama - DK, 2007) Let A,P denote the area andperimeter of the f.c. domain G. For p ≥ 1, q = p
p−1 , we have
2Aq√
P≤ λEp ≤
√A
πP
1p .
Questions. (i) Are disks and annuli the only extremal domains forall λEp ,p≥1, as well? (ii) Do the extremal functions for λEp
characterize the domain G ?
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
E p Analytic Content
Definition
Forp ≥ 1, λEp := inf
φ∈Ep(G)‖ z̄ − φ ‖Ep(G) .
Theorem
(Z. Guadarrama - DK, 2007) Let A,P denote the area andperimeter of the f.c. domain G. For p ≥ 1, q = p
p−1 , we have
2Aq√
P≤ λEp ≤
√A
πP
1p .
Questions. (i) Are disks and annuli the only extremal domains forall λEp ,p≥1, as well? (ii) Do the extremal functions for λEp
characterize the domain G ?
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
E p Analytic Content
Definition
Forp ≥ 1, λEp := inf
φ∈Ep(G)‖ z̄ − φ ‖Ep(G) .
Theorem
(Z. Guadarrama - DK, 2007) Let A,P denote the area andperimeter of the f.c. domain G. For p ≥ 1, q = p
p−1 , we have
2Aq√
P≤ λEp ≤
√A
πP
1p .
Questions. (i) Are disks and annuli the only extremal domains forall λEp ,p≥1, as well?
(ii) Do the extremal functions for λEp
characterize the domain G ?
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
E p Analytic Content
Definition
Forp ≥ 1, λEp := inf
φ∈Ep(G)‖ z̄ − φ ‖Ep(G) .
Theorem
(Z. Guadarrama - DK, 2007) Let A,P denote the area andperimeter of the f.c. domain G. For p ≥ 1, q = p
p−1 , we have
2Aq√
P≤ λEp ≤
√A
πP
1p .
Questions. (i) Are disks and annuli the only extremal domains forall λEp ,p≥1, as well? (ii) Do the extremal functions for λEp
characterize the domain G ?
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Constant Best Approximations Characterize Disks
Theorem
(Z. Guadarrama - DK, 2007) Let Γ := ∂G be real analytic andp ≥ 1. If the best approximation (BA) to z̄ in Ep is a constant,then G is a disk.
Starting Point: WLOG BA is zero. Duality yieldsf ∗z̄dz = const|z |pds on Γ, where f ∗ is the extremal for the dual
problem. Dividing by z , f ∗(z)z dz = const|z |p−2ds. For p = 1, the
argument principle ⇒ f ∗ is a unimodular constant, and hence G isa disk. p > 1 is more complicated, p ∈ N has to be treatedseparately. If G is s.c., the regularity hypothesis can be relaxedsignificantly to assume merely that G is a Smirnov domain.Problem. Extend the above theorem to f.c. Smirnov domains.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Constant Best Approximations Characterize Disks
Theorem
(Z. Guadarrama - DK, 2007) Let Γ := ∂G be real analytic andp ≥ 1. If the best approximation (BA) to z̄ in Ep is a constant,then G is a disk.
Starting Point: WLOG BA is zero. Duality yieldsf ∗z̄dz = const|z |pds on Γ, where f ∗ is the extremal for the dual
problem. Dividing by z , f ∗(z)z dz = const|z |p−2ds. For p = 1, the
argument principle ⇒ f ∗ is a unimodular constant, and hence G isa disk. p > 1 is more complicated, p ∈ N has to be treatedseparately. If G is s.c., the regularity hypothesis can be relaxedsignificantly to assume merely that G is a Smirnov domain.Problem. Extend the above theorem to f.c. Smirnov domains.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Constant Best Approximations Characterize Disks
Theorem
(Z. Guadarrama - DK, 2007) Let Γ := ∂G be real analytic andp ≥ 1. If the best approximation (BA) to z̄ in Ep is a constant,then G is a disk.
Starting Point:
WLOG BA is zero. Duality yieldsf ∗z̄dz = const|z |pds on Γ, where f ∗ is the extremal for the dual
problem. Dividing by z , f ∗(z)z dz = const|z |p−2ds. For p = 1, the
argument principle ⇒ f ∗ is a unimodular constant, and hence G isa disk. p > 1 is more complicated, p ∈ N has to be treatedseparately. If G is s.c., the regularity hypothesis can be relaxedsignificantly to assume merely that G is a Smirnov domain.Problem. Extend the above theorem to f.c. Smirnov domains.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Constant Best Approximations Characterize Disks
Theorem
(Z. Guadarrama - DK, 2007) Let Γ := ∂G be real analytic andp ≥ 1. If the best approximation (BA) to z̄ in Ep is a constant,then G is a disk.
Starting Point: WLOG BA is zero.
Duality yieldsf ∗z̄dz = const|z |pds on Γ, where f ∗ is the extremal for the dual
problem. Dividing by z , f ∗(z)z dz = const|z |p−2ds. For p = 1, the
argument principle ⇒ f ∗ is a unimodular constant, and hence G isa disk. p > 1 is more complicated, p ∈ N has to be treatedseparately. If G is s.c., the regularity hypothesis can be relaxedsignificantly to assume merely that G is a Smirnov domain.Problem. Extend the above theorem to f.c. Smirnov domains.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Constant Best Approximations Characterize Disks
Theorem
(Z. Guadarrama - DK, 2007) Let Γ := ∂G be real analytic andp ≥ 1. If the best approximation (BA) to z̄ in Ep is a constant,then G is a disk.
Starting Point: WLOG BA is zero. Duality yieldsf ∗z̄dz = const|z |pds on Γ, where f ∗ is the extremal for the dual
problem.
Dividing by z , f ∗(z)z dz = const|z |p−2ds. For p = 1, the
argument principle ⇒ f ∗ is a unimodular constant, and hence G isa disk. p > 1 is more complicated, p ∈ N has to be treatedseparately. If G is s.c., the regularity hypothesis can be relaxedsignificantly to assume merely that G is a Smirnov domain.Problem. Extend the above theorem to f.c. Smirnov domains.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Constant Best Approximations Characterize Disks
Theorem
(Z. Guadarrama - DK, 2007) Let Γ := ∂G be real analytic andp ≥ 1. If the best approximation (BA) to z̄ in Ep is a constant,then G is a disk.
Starting Point: WLOG BA is zero. Duality yieldsf ∗z̄dz = const|z |pds on Γ, where f ∗ is the extremal for the dual
problem. Dividing by z ,
f ∗(z)z dz = const|z |p−2ds. For p = 1, the
argument principle ⇒ f ∗ is a unimodular constant, and hence G isa disk. p > 1 is more complicated, p ∈ N has to be treatedseparately. If G is s.c., the regularity hypothesis can be relaxedsignificantly to assume merely that G is a Smirnov domain.Problem. Extend the above theorem to f.c. Smirnov domains.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Constant Best Approximations Characterize Disks
Theorem
(Z. Guadarrama - DK, 2007) Let Γ := ∂G be real analytic andp ≥ 1. If the best approximation (BA) to z̄ in Ep is a constant,then G is a disk.
Starting Point: WLOG BA is zero. Duality yieldsf ∗z̄dz = const|z |pds on Γ, where f ∗ is the extremal for the dual
problem. Dividing by z , f ∗(z)z dz = const|z |p−2ds.
For p = 1, theargument principle ⇒ f ∗ is a unimodular constant, and hence G isa disk. p > 1 is more complicated, p ∈ N has to be treatedseparately. If G is s.c., the regularity hypothesis can be relaxedsignificantly to assume merely that G is a Smirnov domain.Problem. Extend the above theorem to f.c. Smirnov domains.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Constant Best Approximations Characterize Disks
Theorem
(Z. Guadarrama - DK, 2007) Let Γ := ∂G be real analytic andp ≥ 1. If the best approximation (BA) to z̄ in Ep is a constant,then G is a disk.
Starting Point: WLOG BA is zero. Duality yieldsf ∗z̄dz = const|z |pds on Γ, where f ∗ is the extremal for the dual
problem. Dividing by z , f ∗(z)z dz = const|z |p−2ds. For p = 1, the
argument principle ⇒ f ∗ is a unimodular constant
, and hence G isa disk. p > 1 is more complicated, p ∈ N has to be treatedseparately. If G is s.c., the regularity hypothesis can be relaxedsignificantly to assume merely that G is a Smirnov domain.Problem. Extend the above theorem to f.c. Smirnov domains.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Constant Best Approximations Characterize Disks
Theorem
(Z. Guadarrama - DK, 2007) Let Γ := ∂G be real analytic andp ≥ 1. If the best approximation (BA) to z̄ in Ep is a constant,then G is a disk.
Starting Point: WLOG BA is zero. Duality yieldsf ∗z̄dz = const|z |pds on Γ, where f ∗ is the extremal for the dual
problem. Dividing by z , f ∗(z)z dz = const|z |p−2ds. For p = 1, the
argument principle ⇒ f ∗ is a unimodular constant, and hence G isa disk.
p > 1 is more complicated, p ∈ N has to be treatedseparately. If G is s.c., the regularity hypothesis can be relaxedsignificantly to assume merely that G is a Smirnov domain.Problem. Extend the above theorem to f.c. Smirnov domains.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Constant Best Approximations Characterize Disks
Theorem
(Z. Guadarrama - DK, 2007) Let Γ := ∂G be real analytic andp ≥ 1. If the best approximation (BA) to z̄ in Ep is a constant,then G is a disk.
Starting Point: WLOG BA is zero. Duality yieldsf ∗z̄dz = const|z |pds on Γ, where f ∗ is the extremal for the dual
problem. Dividing by z , f ∗(z)z dz = const|z |p−2ds. For p = 1, the
argument principle ⇒ f ∗ is a unimodular constant, and hence G isa disk. p > 1 is more complicated,
p ∈ N has to be treatedseparately. If G is s.c., the regularity hypothesis can be relaxedsignificantly to assume merely that G is a Smirnov domain.Problem. Extend the above theorem to f.c. Smirnov domains.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Constant Best Approximations Characterize Disks
Theorem
(Z. Guadarrama - DK, 2007) Let Γ := ∂G be real analytic andp ≥ 1. If the best approximation (BA) to z̄ in Ep is a constant,then G is a disk.
Starting Point: WLOG BA is zero. Duality yieldsf ∗z̄dz = const|z |pds on Γ, where f ∗ is the extremal for the dual
problem. Dividing by z , f ∗(z)z dz = const|z |p−2ds. For p = 1, the
argument principle ⇒ f ∗ is a unimodular constant, and hence G isa disk. p > 1 is more complicated, p ∈ N has to be treatedseparately.
If G is s.c., the regularity hypothesis can be relaxedsignificantly to assume merely that G is a Smirnov domain.Problem. Extend the above theorem to f.c. Smirnov domains.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Constant Best Approximations Characterize Disks
Theorem
(Z. Guadarrama - DK, 2007) Let Γ := ∂G be real analytic andp ≥ 1. If the best approximation (BA) to z̄ in Ep is a constant,then G is a disk.
Starting Point: WLOG BA is zero. Duality yieldsf ∗z̄dz = const|z |pds on Γ, where f ∗ is the extremal for the dual
problem. Dividing by z , f ∗(z)z dz = const|z |p−2ds. For p = 1, the
argument principle ⇒ f ∗ is a unimodular constant, and hence G isa disk. p > 1 is more complicated, p ∈ N has to be treatedseparately. If G is s.c., the regularity hypothesis can be relaxedsignificantly to assume merely that G is a Smirnov domain.
Problem. Extend the above theorem to f.c. Smirnov domains.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Constant Best Approximations Characterize Disks
Theorem
(Z. Guadarrama - DK, 2007) Let Γ := ∂G be real analytic andp ≥ 1. If the best approximation (BA) to z̄ in Ep is a constant,then G is a disk.
Starting Point: WLOG BA is zero. Duality yieldsf ∗z̄dz = const|z |pds on Γ, where f ∗ is the extremal for the dual
problem. Dividing by z , f ∗(z)z dz = const|z |p−2ds. For p = 1, the
argument principle ⇒ f ∗ is a unimodular constant, and hence G isa disk. p > 1 is more complicated, p ∈ N has to be treatedseparately. If G is s.c., the regularity hypothesis can be relaxedsignificantly to assume merely that G is a Smirnov domain.Problem. Extend the above theorem to f.c. Smirnov domains.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Do BA to z̄ Characterize Annuli
Theorem
(ZG - DK, 2007) Let Γ := ∂G be real analytic and p = 1(!). If thebest approximation (BA) to z̄ in E 1 is a rational functiong(z) = c
z−a , then G is an annulus centered at a.
Conjecture
The theorem holds for all p > 1 and all f.c. Smirnov domains.
Unknown Territory: Study domains where BA to z̄ in Ep are,say, rational functions. It is easy to see that it leads to a largerclass than (by now celebrated) quadrature domains.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Do BA to z̄ Characterize Annuli
Theorem
(ZG - DK, 2007) Let Γ := ∂G be real analytic and p = 1(!). If thebest approximation (BA) to z̄ in E 1 is a rational functiong(z) = c
z−a , then G is an annulus centered at a.
Conjecture
The theorem holds for all p > 1 and all f.c. Smirnov domains.
Unknown Territory: Study domains where BA to z̄ in Ep are,say, rational functions. It is easy to see that it leads to a largerclass than (by now celebrated) quadrature domains.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Do BA to z̄ Characterize Annuli
Theorem
(ZG - DK, 2007) Let Γ := ∂G be real analytic and p = 1(!). If thebest approximation (BA) to z̄ in E 1 is a rational functiong(z) = c
z−a , then G is an annulus centered at a.
Conjecture
The theorem holds for all p > 1 and all f.c. Smirnov domains.
Unknown Territory: Study domains where BA to z̄ in Ep are,say, rational functions. It is easy to see that it leads to a largerclass than (by now celebrated) quadrature domains.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Do BA to z̄ Characterize Annuli
Theorem
(ZG - DK, 2007) Let Γ := ∂G be real analytic and p = 1(!). If thebest approximation (BA) to z̄ in E 1 is a rational functiong(z) = c
z−a , then G is an annulus centered at a.
Conjecture
The theorem holds for all p > 1 and all f.c. Smirnov domains.
Unknown Territory: Study domains where BA to z̄ in Ep are,say, rational functions. It is easy to see that it leads to a largerclass than (by now celebrated) quadrature domains.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Do BA to z̄ Characterize Annuli
Theorem
(ZG - DK, 2007) Let Γ := ∂G be real analytic and p = 1(!). If thebest approximation (BA) to z̄ in E 1 is a rational functiong(z) = c
z−a , then G is an annulus centered at a.
Conjecture
The theorem holds for all p > 1 and all f.c. Smirnov domains.
Unknown Territory: Study domains where BA to z̄ in Ep are,say, rational functions. It is easy to see that it leads to a largerclass than (by now celebrated) quadrature domains.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Do BA to z̄ Characterize Annuli
Theorem
(ZG - DK, 2007) Let Γ := ∂G be real analytic and p = 1(!). If thebest approximation (BA) to z̄ in E 1 is a rational functiong(z) = c
z−a , then G is an annulus centered at a.
Conjecture
The theorem holds for all p > 1 and all f.c. Smirnov domains.
Unknown Territory: Study domains where BA to z̄ in Ep are,say, rational functions.
It is easy to see that it leads to a largerclass than (by now celebrated) quadrature domains.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Do BA to z̄ Characterize Annuli
Theorem
(ZG - DK, 2007) Let Γ := ∂G be real analytic and p = 1(!). If thebest approximation (BA) to z̄ in E 1 is a rational functiong(z) = c
z−a , then G is an annulus centered at a.
Conjecture
The theorem holds for all p > 1 and all f.c. Smirnov domains.
Unknown Territory: Study domains where BA to z̄ in Ep are,say, rational functions. It is easy to see that it leads to a largerclass than (by now celebrated) quadrature domains.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Ap Analytic Content
Definition
Forp ≥ 1, λAp := inf
φ∈Ap(G)‖ z̄ − φ ‖Ap(G) .
Theorem
(ZG - DK, 2007) (i) Let G be a Smirnov domain and p ≥ 1. If BAto z̄ in Ap is a constant, then G is a disk.
(ii)If BA to z̄ in Ap is g(z) = cz−a , then G is an annulus centered
at a.
The Bergman spaces are easier!
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Ap Analytic Content
Definition
Forp ≥ 1, λAp := inf
φ∈Ap(G)‖ z̄ − φ ‖Ap(G) .
Theorem
(ZG - DK, 2007) (i) Let G be a Smirnov domain and p ≥ 1. If BAto z̄ in Ap is a constant, then G is a disk.
(ii)If BA to z̄ in Ap is g(z) = cz−a , then G is an annulus centered
at a.
The Bergman spaces are easier!
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Ap Analytic Content
Definition
Forp ≥ 1, λAp := inf
φ∈Ap(G)‖ z̄ − φ ‖Ap(G) .
Theorem
(ZG - DK, 2007) (i) Let G be a Smirnov domain and p ≥ 1. If BAto z̄ in Ap is a constant, then G is a disk.
(ii)If BA to z̄ in Ap is g(z) = cz−a , then G is an annulus centered
at a.
The Bergman spaces are easier!
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Ap Analytic Content
Definition
Forp ≥ 1, λAp := inf
φ∈Ap(G)‖ z̄ − φ ‖Ap(G) .
Theorem
(ZG - DK, 2007) (i) Let G be a Smirnov domain and p ≥ 1. If BAto z̄ in Ap is a constant, then G is a disk.
(ii)If BA to z̄ in Ap is g(z) = cz−a , then G is an annulus centered
at a.
The Bergman spaces are easier!
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Ap Analytic Content
Definition
Forp ≥ 1, λAp := inf
φ∈Ap(G)‖ z̄ − φ ‖Ap(G) .
Theorem
(ZG - DK, 2007) (i) Let G be a Smirnov domain and p ≥ 1. If BAto z̄ in Ap is a constant, then G is a disk.
(ii)If BA to z̄ in Ap is g(z) = cz−a , then G is an annulus centered
at a.
The Bergman spaces are easier!
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Ap Analytic Content
Definition
Forp ≥ 1, λAp := inf
φ∈Ap(G)‖ z̄ − φ ‖Ap(G) .
Theorem
(ZG - DK, 2007) (i) Let G be a Smirnov domain and p ≥ 1. If BAto z̄ in Ap is a constant, then G is a disk.
(ii)If BA to z̄ in Ap is g(z) = cz−a , then G is an annulus centered
at a.
The Bergman spaces are easier!
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Here is why
((i), p > 1, BA= 0):The duality yields
uz̄ = const|z |p
z̄, u ∈W 1,q
0 is a solution of the dual problem.
Integrating wrt z̄ gives (in G !):
u = const|z |p + h, where h ∈ H∞(G ).
Hence h|Γ is real-valued, so |z | = const on Γ.
What are the sharp bounds for λAp? Nothing is known aboutthe domains with other rational BA (in Ap).
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Here is why ((i), p > 1, BA= 0):
The duality yields
uz̄ = const|z |p
z̄, u ∈W 1,q
0 is a solution of the dual problem.
Integrating wrt z̄ gives (in G !):
u = const|z |p + h, where h ∈ H∞(G ).
Hence h|Γ is real-valued, so |z | = const on Γ.
What are the sharp bounds for λAp? Nothing is known aboutthe domains with other rational BA (in Ap).
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Here is why ((i), p > 1, BA= 0):The duality yields
uz̄ = const|z |p
z̄, u ∈W 1,q
0 is a solution of the dual problem.
Integrating wrt z̄ gives (in G !):
u = const|z |p + h, where h ∈ H∞(G ).
Hence h|Γ is real-valued, so |z | = const on Γ.
What are the sharp bounds for λAp? Nothing is known aboutthe domains with other rational BA (in Ap).
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Here is why ((i), p > 1, BA= 0):The duality yields
uz̄ = const|z |p
z̄, u ∈W 1,q
0 is a solution of the dual problem.
Integrating wrt z̄ gives (in G !):
u = const|z |p + h, where h ∈ H∞(G ).
Hence h|Γ is real-valued, so |z | = const on Γ.
What are the sharp bounds for λAp? Nothing is known aboutthe domains with other rational BA (in Ap).
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Here is why ((i), p > 1, BA= 0):The duality yields
uz̄ = const|z |p
z̄, u ∈W 1,q
0 is a solution of the dual problem.
Integrating wrt z̄ gives (in G !):
u = const|z |p + h, where h ∈ H∞(G ).
Hence h|Γ is real-valued, so |z | = const on Γ.
What are the sharp bounds for λAp? Nothing is known aboutthe domains with other rational BA (in Ap).
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Here is why ((i), p > 1, BA= 0):The duality yields
uz̄ = const|z |p
z̄, u ∈W 1,q
0 is a solution of the dual problem.
Integrating wrt z̄ gives (in G !):
u = const|z |p + h, where h ∈ H∞(G ).
Hence h|Γ is real-valued, so |z | = const on Γ.
What are the sharp bounds for λAp? Nothing is known aboutthe domains with other rational BA (in Ap).
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Putnam’s Inequality (1970) on E 2 and the IsoperimetricInequality
Let φ be analytic in a neighborhood of the f.c. domainG , ∂G := Γ,A := A(G ) = area of G ,P := P(Γ) = perimeter of G .Let T := Tφ : E 2 → E 2,Tf = φf , [T ∗,T ] = T ∗T − TT ∗.Putnam’s inequality then states
A(φ(G ))
π≥ ‖[T ∗,T ]‖ ≥ 4(A(φ(G )))2
‖φ′‖2E2(G)
· P(DK - 1985).
φ = z =⇒ P2 ≥ 4πA, the isoperimetric inequality. Thus, Putnam’sinequality in E 2 context is sharp.Question: What are the bounds in the Bergman space context?
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Putnam’s Inequality (1970) on E 2 and the IsoperimetricInequality
Let φ be analytic in a neighborhood of the f.c. domainG , ∂G := Γ,A := A(G ) = area of G ,P := P(Γ) = perimeter of G .
Let T := Tφ : E 2 → E 2,Tf = φf , [T ∗,T ] = T ∗T − TT ∗.Putnam’s inequality then states
A(φ(G ))
π≥ ‖[T ∗,T ]‖ ≥ 4(A(φ(G )))2
‖φ′‖2E2(G)
· P(DK - 1985).
φ = z =⇒ P2 ≥ 4πA, the isoperimetric inequality. Thus, Putnam’sinequality in E 2 context is sharp.Question: What are the bounds in the Bergman space context?
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Putnam’s Inequality (1970) on E 2 and the IsoperimetricInequality
Let φ be analytic in a neighborhood of the f.c. domainG , ∂G := Γ,A := A(G ) = area of G ,P := P(Γ) = perimeter of G .Let T := Tφ : E 2 → E 2,Tf = φf , [T ∗,T ] = T ∗T − TT ∗.
Putnam’s inequality then states
A(φ(G ))
π≥ ‖[T ∗,T ]‖ ≥ 4(A(φ(G )))2
‖φ′‖2E2(G)
· P(DK - 1985).
φ = z =⇒ P2 ≥ 4πA, the isoperimetric inequality. Thus, Putnam’sinequality in E 2 context is sharp.Question: What are the bounds in the Bergman space context?
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Putnam’s Inequality (1970) on E 2 and the IsoperimetricInequality
Let φ be analytic in a neighborhood of the f.c. domainG , ∂G := Γ,A := A(G ) = area of G ,P := P(Γ) = perimeter of G .Let T := Tφ : E 2 → E 2,Tf = φf , [T ∗,T ] = T ∗T − TT ∗.Putnam’s inequality then states
A(φ(G ))
π≥ ‖[T ∗,T ]‖ ≥ 4(A(φ(G )))2
‖φ′‖2E2(G)
· P(DK - 1985).
φ = z =⇒ P2 ≥ 4πA, the isoperimetric inequality. Thus, Putnam’sinequality in E 2 context is sharp.Question: What are the bounds in the Bergman space context?
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Putnam’s Inequality (1970) on E 2 and the IsoperimetricInequality
Let φ be analytic in a neighborhood of the f.c. domainG , ∂G := Γ,A := A(G ) = area of G ,P := P(Γ) = perimeter of G .Let T := Tφ : E 2 → E 2,Tf = φf , [T ∗,T ] = T ∗T − TT ∗.Putnam’s inequality then states
A(φ(G ))
π≥ ‖[T ∗,T ]‖
≥ 4(A(φ(G )))2
‖φ′‖2E2(G)
· P(DK - 1985).
φ = z =⇒ P2 ≥ 4πA, the isoperimetric inequality. Thus, Putnam’sinequality in E 2 context is sharp.Question: What are the bounds in the Bergman space context?
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Putnam’s Inequality (1970) on E 2 and the IsoperimetricInequality
Let φ be analytic in a neighborhood of the f.c. domainG , ∂G := Γ,A := A(G ) = area of G ,P := P(Γ) = perimeter of G .Let T := Tφ : E 2 → E 2,Tf = φf , [T ∗,T ] = T ∗T − TT ∗.Putnam’s inequality then states
A(φ(G ))
π≥ ‖[T ∗,T ]‖ ≥ 4(A(φ(G )))2
‖φ′‖2E2(G)
· P(DK - 1985).
φ = z =⇒ P2 ≥ 4πA, the isoperimetric inequality. Thus, Putnam’sinequality in E 2 context is sharp.Question: What are the bounds in the Bergman space context?
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Putnam’s Inequality (1970) on E 2 and the IsoperimetricInequality
Let φ be analytic in a neighborhood of the f.c. domainG , ∂G := Γ,A := A(G ) = area of G ,P := P(Γ) = perimeter of G .Let T := Tφ : E 2 → E 2,Tf = φf , [T ∗,T ] = T ∗T − TT ∗.Putnam’s inequality then states
A(φ(G ))
π≥ ‖[T ∗,T ]‖ ≥ 4(A(φ(G )))2
‖φ′‖2E2(G)
· P(DK - 1985).
φ = z =⇒ P2 ≥ 4πA, the isoperimetric inequality.
Thus, Putnam’sinequality in E 2 context is sharp.Question: What are the bounds in the Bergman space context?
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Putnam’s Inequality (1970) on E 2 and the IsoperimetricInequality
Let φ be analytic in a neighborhood of the f.c. domainG , ∂G := Γ,A := A(G ) = area of G ,P := P(Γ) = perimeter of G .Let T := Tφ : E 2 → E 2,Tf = φf , [T ∗,T ] = T ∗T − TT ∗.Putnam’s inequality then states
A(φ(G ))
π≥ ‖[T ∗,T ]‖ ≥ 4(A(φ(G )))2
‖φ′‖2E2(G)
· P(DK - 1985).
φ = z =⇒ P2 ≥ 4πA, the isoperimetric inequality. Thus, Putnam’sinequality in E 2 context is sharp.
Question: What are the bounds in the Bergman space context?
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Putnam’s Inequality (1970) on E 2 and the IsoperimetricInequality
Let φ be analytic in a neighborhood of the f.c. domainG , ∂G := Γ,A := A(G ) = area of G ,P := P(Γ) = perimeter of G .Let T := Tφ : E 2 → E 2,Tf = φf , [T ∗,T ] = T ∗T − TT ∗.Putnam’s inequality then states
A(φ(G ))
π≥ ‖[T ∗,T ]‖ ≥ 4(A(φ(G )))2
‖φ′‖2E2(G)
· P(DK - 1985).
φ = z =⇒ P2 ≥ 4πA, the isoperimetric inequality. Thus, Putnam’sinequality in E 2 context is sharp.Question: What are the bounds in the Bergman space context?
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Putnam’s inequality in A2
Observation: (M. Fleeman -DK (2012), folowing S. Axler - J.Shapiro (1985)). For a subnormal operator T the equality inPutnam’s inequality occurs only if the spectrum sp(T ) is a diskand the spectral measure “sits” on the circumference.The crux isthe equality case in the upper bound for the analytic content:
λ(G ) ≤√
Aπ . Indeed,
‖[T ∗z ,Tz ]‖ = sup‖g‖2=1
{ inff analytic
‖z̄g−f ‖2} ≤ infhanalytic
‖z̄−h‖∞ = λ.
A calculation (straightforward but tedious!) reveals that inA2(D), ‖[T ∗z ,Tz ]‖ = 1/2, i.e., the upper bound is two timessmaller than in the general Putnam inequality.Question: Should Putnam’s inequality in the context of theBergman space be corrected by a factor 1/2?
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Putnam’s inequality in A2
Observation: (M. Fleeman -DK (2012), folowing S. Axler - J.Shapiro (1985)).
For a subnormal operator T the equality inPutnam’s inequality occurs only if the spectrum sp(T ) is a diskand the spectral measure “sits” on the circumference.The crux isthe equality case in the upper bound for the analytic content:
λ(G ) ≤√
Aπ . Indeed,
‖[T ∗z ,Tz ]‖ = sup‖g‖2=1
{ inff analytic
‖z̄g−f ‖2} ≤ infhanalytic
‖z̄−h‖∞ = λ.
A calculation (straightforward but tedious!) reveals that inA2(D), ‖[T ∗z ,Tz ]‖ = 1/2, i.e., the upper bound is two timessmaller than in the general Putnam inequality.Question: Should Putnam’s inequality in the context of theBergman space be corrected by a factor 1/2?
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Putnam’s inequality in A2
Observation: (M. Fleeman -DK (2012), folowing S. Axler - J.Shapiro (1985)). For a subnormal operator T the equality inPutnam’s inequality occurs only if the spectrum sp(T ) is a diskand the spectral measure “sits” on the circumference.
The crux isthe equality case in the upper bound for the analytic content:
λ(G ) ≤√
Aπ . Indeed,
‖[T ∗z ,Tz ]‖ = sup‖g‖2=1
{ inff analytic
‖z̄g−f ‖2} ≤ infhanalytic
‖z̄−h‖∞ = λ.
A calculation (straightforward but tedious!) reveals that inA2(D), ‖[T ∗z ,Tz ]‖ = 1/2, i.e., the upper bound is two timessmaller than in the general Putnam inequality.Question: Should Putnam’s inequality in the context of theBergman space be corrected by a factor 1/2?
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Putnam’s inequality in A2
Observation: (M. Fleeman -DK (2012), folowing S. Axler - J.Shapiro (1985)). For a subnormal operator T the equality inPutnam’s inequality occurs only if the spectrum sp(T ) is a diskand the spectral measure “sits” on the circumference.The crux isthe equality case in the upper bound for the analytic content:
λ(G ) ≤√
Aπ . Indeed,
‖[T ∗z ,Tz ]‖ = sup‖g‖2=1
{ inff analytic
‖z̄g−f ‖2} ≤ infhanalytic
‖z̄−h‖∞ = λ.
A calculation (straightforward but tedious!) reveals that inA2(D), ‖[T ∗z ,Tz ]‖ = 1/2, i.e., the upper bound is two timessmaller than in the general Putnam inequality.Question: Should Putnam’s inequality in the context of theBergman space be corrected by a factor 1/2?
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Putnam’s inequality in A2
Observation: (M. Fleeman -DK (2012), folowing S. Axler - J.Shapiro (1985)). For a subnormal operator T the equality inPutnam’s inequality occurs only if the spectrum sp(T ) is a diskand the spectral measure “sits” on the circumference.The crux isthe equality case in the upper bound for the analytic content:
λ(G ) ≤√
Aπ .
Indeed,
‖[T ∗z ,Tz ]‖ = sup‖g‖2=1
{ inff analytic
‖z̄g−f ‖2} ≤ infhanalytic
‖z̄−h‖∞ = λ.
A calculation (straightforward but tedious!) reveals that inA2(D), ‖[T ∗z ,Tz ]‖ = 1/2, i.e., the upper bound is two timessmaller than in the general Putnam inequality.Question: Should Putnam’s inequality in the context of theBergman space be corrected by a factor 1/2?
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Putnam’s inequality in A2
Observation: (M. Fleeman -DK (2012), folowing S. Axler - J.Shapiro (1985)). For a subnormal operator T the equality inPutnam’s inequality occurs only if the spectrum sp(T ) is a diskand the spectral measure “sits” on the circumference.The crux isthe equality case in the upper bound for the analytic content:
λ(G ) ≤√
Aπ . Indeed,
‖[T ∗z ,Tz ]‖ = sup‖g‖2=1
{ inff analytic
‖z̄g−f ‖2}
≤ infhanalytic
‖z̄−h‖∞ = λ.
A calculation (straightforward but tedious!) reveals that inA2(D), ‖[T ∗z ,Tz ]‖ = 1/2, i.e., the upper bound is two timessmaller than in the general Putnam inequality.Question: Should Putnam’s inequality in the context of theBergman space be corrected by a factor 1/2?
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Putnam’s inequality in A2
Observation: (M. Fleeman -DK (2012), folowing S. Axler - J.Shapiro (1985)). For a subnormal operator T the equality inPutnam’s inequality occurs only if the spectrum sp(T ) is a diskand the spectral measure “sits” on the circumference.The crux isthe equality case in the upper bound for the analytic content:
λ(G ) ≤√
Aπ . Indeed,
‖[T ∗z ,Tz ]‖ = sup‖g‖2=1
{ inff analytic
‖z̄g−f ‖2} ≤ infhanalytic
‖z̄−h‖∞ = λ.
A calculation (straightforward but tedious!) reveals that inA2(D), ‖[T ∗z ,Tz ]‖ = 1/2, i.e., the upper bound is two timessmaller than in the general Putnam inequality.Question: Should Putnam’s inequality in the context of theBergman space be corrected by a factor 1/2?
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Putnam’s inequality in A2
Observation: (M. Fleeman -DK (2012), folowing S. Axler - J.Shapiro (1985)). For a subnormal operator T the equality inPutnam’s inequality occurs only if the spectrum sp(T ) is a diskand the spectral measure “sits” on the circumference.The crux isthe equality case in the upper bound for the analytic content:
λ(G ) ≤√
Aπ . Indeed,
‖[T ∗z ,Tz ]‖ = sup‖g‖2=1
{ inff analytic
‖z̄g−f ‖2} ≤ infhanalytic
‖z̄−h‖∞ = λ.
A calculation (straightforward but tedious!) reveals that inA2(D), ‖[T ∗z ,Tz ]‖ = 1/2, i.e., the upper bound is two timessmaller than in the general Putnam inequality.
Question: Should Putnam’s inequality in the context of theBergman space be corrected by a factor 1/2?
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Putnam’s inequality in A2
Observation: (M. Fleeman -DK (2012), folowing S. Axler - J.Shapiro (1985)). For a subnormal operator T the equality inPutnam’s inequality occurs only if the spectrum sp(T ) is a diskand the spectral measure “sits” on the circumference.The crux isthe equality case in the upper bound for the analytic content:
λ(G ) ≤√
Aπ . Indeed,
‖[T ∗z ,Tz ]‖ = sup‖g‖2=1
{ inff analytic
‖z̄g−f ‖2} ≤ infhanalytic
‖z̄−h‖∞ = λ.
A calculation (straightforward but tedious!) reveals that inA2(D), ‖[T ∗z ,Tz ]‖ = 1/2, i.e., the upper bound is two timessmaller than in the general Putnam inequality.Question: Should Putnam’s inequality in the context of theBergman space be corrected by a factor 1/2?
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Toeplitz Operators on the Bergman Space andSaint-Venant Inequality
The torsional rigidity ρ of a domain G measures the resilience ofthe beam of cross section G to twisting. In terms of the Rayleightype quotient :
ρ := supψ∈C∞0
(2‖ψ‖1
‖∇ψ‖2
)2
.
Theorem
(S. Bell - T. Ferguson - E. Lundberg, 2012)
ρ
Area(φ(G ))≤ ‖[T ∗φ ,Tφ]‖ (Putnam’s) ≤ A(φ(G ))
π.
Hence, taking φ = z, we obtain from the “isoperimetric sandwich”
ρ ≤ (Area(G ))2
π.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Toeplitz Operators on the Bergman Space andSaint-Venant Inequality
The torsional rigidity ρ of a domain G measures the resilience ofthe beam of cross section G to twisting.
In terms of the Rayleightype quotient :
ρ := supψ∈C∞0
(2‖ψ‖1
‖∇ψ‖2
)2
.
Theorem
(S. Bell - T. Ferguson - E. Lundberg, 2012)
ρ
Area(φ(G ))≤ ‖[T ∗φ ,Tφ]‖ (Putnam’s) ≤ A(φ(G ))
π.
Hence, taking φ = z, we obtain from the “isoperimetric sandwich”
ρ ≤ (Area(G ))2
π.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Toeplitz Operators on the Bergman Space andSaint-Venant Inequality
The torsional rigidity ρ of a domain G measures the resilience ofthe beam of cross section G to twisting. In terms of the Rayleightype quotient :
ρ := supψ∈C∞0
(2‖ψ‖1
‖∇ψ‖2
)2
.
Theorem
(S. Bell - T. Ferguson - E. Lundberg, 2012)
ρ
Area(φ(G ))≤ ‖[T ∗φ ,Tφ]‖ (Putnam’s) ≤ A(φ(G ))
π.
Hence, taking φ = z, we obtain from the “isoperimetric sandwich”
ρ ≤ (Area(G ))2
π.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Toeplitz Operators on the Bergman Space andSaint-Venant Inequality
The torsional rigidity ρ of a domain G measures the resilience ofthe beam of cross section G to twisting. In terms of the Rayleightype quotient :
ρ := supψ∈C∞0
(2‖ψ‖1
‖∇ψ‖2
)2
.
Theorem
(S. Bell - T. Ferguson - E. Lundberg, 2012)
ρ
Area(φ(G ))≤ ‖[T ∗φ ,Tφ]‖ (Putnam’s) ≤ A(φ(G ))
π.
Hence, taking φ = z, we obtain from the “isoperimetric sandwich”
ρ ≤ (Area(G ))2
π.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Toeplitz Operators on the Bergman Space andSaint-Venant Inequality
The torsional rigidity ρ of a domain G measures the resilience ofthe beam of cross section G to twisting. In terms of the Rayleightype quotient :
ρ := supψ∈C∞0
(2‖ψ‖1
‖∇ψ‖2
)2
.
Theorem
(S. Bell - T. Ferguson - E. Lundberg, 2012)
ρ
Area(φ(G ))≤ ‖[T ∗φ ,Tφ]‖
(Putnam’s) ≤ A(φ(G ))
π.
Hence, taking φ = z, we obtain from the “isoperimetric sandwich”
ρ ≤ (Area(G ))2
π.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Toeplitz Operators on the Bergman Space andSaint-Venant Inequality
The torsional rigidity ρ of a domain G measures the resilience ofthe beam of cross section G to twisting. In terms of the Rayleightype quotient :
ρ := supψ∈C∞0
(2‖ψ‖1
‖∇ψ‖2
)2
.
Theorem
(S. Bell - T. Ferguson - E. Lundberg, 2012)
ρ
Area(φ(G ))≤ ‖[T ∗φ ,Tφ]‖ (Putnam’s)
≤ A(φ(G ))
π.
Hence, taking φ = z, we obtain from the “isoperimetric sandwich”
ρ ≤ (Area(G ))2
π.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Toeplitz Operators on the Bergman Space andSaint-Venant Inequality
The torsional rigidity ρ of a domain G measures the resilience ofthe beam of cross section G to twisting. In terms of the Rayleightype quotient :
ρ := supψ∈C∞0
(2‖ψ‖1
‖∇ψ‖2
)2
.
Theorem
(S. Bell - T. Ferguson - E. Lundberg, 2012)
ρ
Area(φ(G ))≤ ‖[T ∗φ ,Tφ]‖ (Putnam’s) ≤ A(φ(G ))
π.
Hence, taking φ = z, we obtain from the “isoperimetric sandwich”
ρ ≤ (Area(G ))2
π.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Toeplitz Operators on the Bergman Space andSaint-Venant Inequality
The torsional rigidity ρ of a domain G measures the resilience ofthe beam of cross section G to twisting. In terms of the Rayleightype quotient :
ρ := supψ∈C∞0
(2‖ψ‖1
‖∇ψ‖2
)2
.
Theorem
(S. Bell - T. Ferguson - E. Lundberg, 2012)
ρ
Area(φ(G ))≤ ‖[T ∗φ ,Tφ]‖ (Putnam’s) ≤ A(φ(G ))
π.
Hence, taking φ = z,
we obtain from the “isoperimetric sandwich”
ρ ≤ (Area(G ))2
π.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Toeplitz Operators on the Bergman Space andSaint-Venant Inequality
The torsional rigidity ρ of a domain G measures the resilience ofthe beam of cross section G to twisting. In terms of the Rayleightype quotient :
ρ := supψ∈C∞0
(2‖ψ‖1
‖∇ψ‖2
)2
.
Theorem
(S. Bell - T. Ferguson - E. Lundberg, 2012)
ρ
Area(φ(G ))≤ ‖[T ∗φ ,Tφ]‖ (Putnam’s) ≤ A(φ(G ))
π.
Hence, taking φ = z, we obtain from the “isoperimetric sandwich”
ρ ≤ (Area(G ))2
π.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Toeplitz Operators on the Bergman Space andSaint-Venant Inequality
The torsional rigidity ρ of a domain G measures the resilience ofthe beam of cross section G to twisting. In terms of the Rayleightype quotient :
ρ := supψ∈C∞0
(2‖ψ‖1
‖∇ψ‖2
)2
.
Theorem
(S. Bell - T. Ferguson - E. Lundberg, 2012)
ρ
Area(φ(G ))≤ ‖[T ∗φ ,Tφ]‖ (Putnam’s) ≤ A(φ(G ))
π.
Hence, taking φ = z, we obtain from the “isoperimetric sandwich”
ρ ≤ (Area(G ))2
π.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
J.-F. Olsen - M.C. Reguera Theorem, 2013
Bell - Ferguson - Lundberg estimate ρ ≤ (Area(G))2
π
was missing bya factor of 2 the celebrated Saint-Venant inequality conjectured in1856. It was first proved by G. Polya in 1948.
Conjecture
(Bell - Ferguson - Lundberg, 2012) For the Bergman space
‖[T ∗z ,Tz ]‖ ≤ Area(G)2π .
For simply connected domains G it is now a theorem! (J.-F. Olsen- M.C. Reguera, May 2013). Hence,
Saint-Venant’s Inequality, a new proof: ρ ≤ (Area(G))2
2π .The proof is tour de force calculation with power series. This iswhy the statement is restricted to s.c. domains.M. Fleeman -DK noted that refining Olsen - Reguera’s proofimplies that the equality for the self-commutator upper bound ins.c. domains holds only for disks. This yields an alternative proofthat St.-Venant’s inequality becomes equality only for disks.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
J.-F. Olsen - M.C. Reguera Theorem, 2013
Bell - Ferguson - Lundberg estimate ρ ≤ (Area(G))2
π was missing bya factor of 2 the celebrated Saint-Venant inequality conjectured in1856.
It was first proved by G. Polya in 1948.
Conjecture
(Bell - Ferguson - Lundberg, 2012) For the Bergman space
‖[T ∗z ,Tz ]‖ ≤ Area(G)2π .
For simply connected domains G it is now a theorem! (J.-F. Olsen- M.C. Reguera, May 2013). Hence,
Saint-Venant’s Inequality, a new proof: ρ ≤ (Area(G))2
2π .The proof is tour de force calculation with power series. This iswhy the statement is restricted to s.c. domains.M. Fleeman -DK noted that refining Olsen - Reguera’s proofimplies that the equality for the self-commutator upper bound ins.c. domains holds only for disks. This yields an alternative proofthat St.-Venant’s inequality becomes equality only for disks.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
J.-F. Olsen - M.C. Reguera Theorem, 2013
Bell - Ferguson - Lundberg estimate ρ ≤ (Area(G))2
π was missing bya factor of 2 the celebrated Saint-Venant inequality conjectured in1856. It was first proved by G. Polya in 1948.
Conjecture
(Bell - Ferguson - Lundberg, 2012) For the Bergman space
‖[T ∗z ,Tz ]‖ ≤ Area(G)2π .
For simply connected domains G it is now a theorem! (J.-F. Olsen- M.C. Reguera, May 2013). Hence,
Saint-Venant’s Inequality, a new proof: ρ ≤ (Area(G))2
2π .The proof is tour de force calculation with power series. This iswhy the statement is restricted to s.c. domains.M. Fleeman -DK noted that refining Olsen - Reguera’s proofimplies that the equality for the self-commutator upper bound ins.c. domains holds only for disks. This yields an alternative proofthat St.-Venant’s inequality becomes equality only for disks.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
J.-F. Olsen - M.C. Reguera Theorem, 2013
Bell - Ferguson - Lundberg estimate ρ ≤ (Area(G))2
π was missing bya factor of 2 the celebrated Saint-Venant inequality conjectured in1856. It was first proved by G. Polya in 1948.
Conjecture
(Bell - Ferguson - Lundberg, 2012) For the Bergman space
‖[T ∗z ,Tz ]‖ ≤ Area(G)2π .
For simply connected domains G it is now a theorem! (J.-F. Olsen- M.C. Reguera, May 2013). Hence,
Saint-Venant’s Inequality, a new proof: ρ ≤ (Area(G))2
2π .The proof is tour de force calculation with power series. This iswhy the statement is restricted to s.c. domains.M. Fleeman -DK noted that refining Olsen - Reguera’s proofimplies that the equality for the self-commutator upper bound ins.c. domains holds only for disks. This yields an alternative proofthat St.-Venant’s inequality becomes equality only for disks.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
J.-F. Olsen - M.C. Reguera Theorem, 2013
Bell - Ferguson - Lundberg estimate ρ ≤ (Area(G))2
π was missing bya factor of 2 the celebrated Saint-Venant inequality conjectured in1856. It was first proved by G. Polya in 1948.
Conjecture
(Bell - Ferguson - Lundberg, 2012) For the Bergman space
‖[T ∗z ,Tz ]‖ ≤ Area(G)2π .
For simply connected domains G it is now a theorem! (J.-F. Olsen- M.C. Reguera, May 2013).
Hence,
Saint-Venant’s Inequality, a new proof: ρ ≤ (Area(G))2
2π .The proof is tour de force calculation with power series. This iswhy the statement is restricted to s.c. domains.M. Fleeman -DK noted that refining Olsen - Reguera’s proofimplies that the equality for the self-commutator upper bound ins.c. domains holds only for disks. This yields an alternative proofthat St.-Venant’s inequality becomes equality only for disks.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
J.-F. Olsen - M.C. Reguera Theorem, 2013
Bell - Ferguson - Lundberg estimate ρ ≤ (Area(G))2
π was missing bya factor of 2 the celebrated Saint-Venant inequality conjectured in1856. It was first proved by G. Polya in 1948.
Conjecture
(Bell - Ferguson - Lundberg, 2012) For the Bergman space
‖[T ∗z ,Tz ]‖ ≤ Area(G)2π .
For simply connected domains G it is now a theorem! (J.-F. Olsen- M.C. Reguera, May 2013). Hence,
Saint-Venant’s Inequality, a new proof: ρ ≤ (Area(G))2
2π .
The proof is tour de force calculation with power series. This iswhy the statement is restricted to s.c. domains.M. Fleeman -DK noted that refining Olsen - Reguera’s proofimplies that the equality for the self-commutator upper bound ins.c. domains holds only for disks. This yields an alternative proofthat St.-Venant’s inequality becomes equality only for disks.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
J.-F. Olsen - M.C. Reguera Theorem, 2013
Bell - Ferguson - Lundberg estimate ρ ≤ (Area(G))2
π was missing bya factor of 2 the celebrated Saint-Venant inequality conjectured in1856. It was first proved by G. Polya in 1948.
Conjecture
(Bell - Ferguson - Lundberg, 2012) For the Bergman space
‖[T ∗z ,Tz ]‖ ≤ Area(G)2π .
For simply connected domains G it is now a theorem! (J.-F. Olsen- M.C. Reguera, May 2013). Hence,
Saint-Venant’s Inequality, a new proof: ρ ≤ (Area(G))2
2π .The proof is tour de force calculation with power series.
This iswhy the statement is restricted to s.c. domains.M. Fleeman -DK noted that refining Olsen - Reguera’s proofimplies that the equality for the self-commutator upper bound ins.c. domains holds only for disks. This yields an alternative proofthat St.-Venant’s inequality becomes equality only for disks.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
J.-F. Olsen - M.C. Reguera Theorem, 2013
Bell - Ferguson - Lundberg estimate ρ ≤ (Area(G))2
π was missing bya factor of 2 the celebrated Saint-Venant inequality conjectured in1856. It was first proved by G. Polya in 1948.
Conjecture
(Bell - Ferguson - Lundberg, 2012) For the Bergman space
‖[T ∗z ,Tz ]‖ ≤ Area(G)2π .
For simply connected domains G it is now a theorem! (J.-F. Olsen- M.C. Reguera, May 2013). Hence,
Saint-Venant’s Inequality, a new proof: ρ ≤ (Area(G))2
2π .The proof is tour de force calculation with power series. This iswhy the statement is restricted to s.c. domains.
M. Fleeman -DK noted that refining Olsen - Reguera’s proofimplies that the equality for the self-commutator upper bound ins.c. domains holds only for disks. This yields an alternative proofthat St.-Venant’s inequality becomes equality only for disks.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
J.-F. Olsen - M.C. Reguera Theorem, 2013
Bell - Ferguson - Lundberg estimate ρ ≤ (Area(G))2
π was missing bya factor of 2 the celebrated Saint-Venant inequality conjectured in1856. It was first proved by G. Polya in 1948.
Conjecture
(Bell - Ferguson - Lundberg, 2012) For the Bergman space
‖[T ∗z ,Tz ]‖ ≤ Area(G)2π .
For simply connected domains G it is now a theorem! (J.-F. Olsen- M.C. Reguera, May 2013). Hence,
Saint-Venant’s Inequality, a new proof: ρ ≤ (Area(G))2
2π .The proof is tour de force calculation with power series. This iswhy the statement is restricted to s.c. domains.M. Fleeman -DK noted that refining Olsen - Reguera’s proofimplies that the equality for the self-commutator upper bound ins.c. domains holds only for disks.
This yields an alternative proofthat St.-Venant’s inequality becomes equality only for disks.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
J.-F. Olsen - M.C. Reguera Theorem, 2013
Bell - Ferguson - Lundberg estimate ρ ≤ (Area(G))2
π was missing bya factor of 2 the celebrated Saint-Venant inequality conjectured in1856. It was first proved by G. Polya in 1948.
Conjecture
(Bell - Ferguson - Lundberg, 2012) For the Bergman space
‖[T ∗z ,Tz ]‖ ≤ Area(G)2π .
For simply connected domains G it is now a theorem! (J.-F. Olsen- M.C. Reguera, May 2013). Hence,
Saint-Venant’s Inequality, a new proof: ρ ≤ (Area(G))2
2π .The proof is tour de force calculation with power series. This iswhy the statement is restricted to s.c. domains.M. Fleeman -DK noted that refining Olsen - Reguera’s proofimplies that the equality for the self-commutator upper bound ins.c. domains holds only for disks. This yields an alternative proofthat St.-Venant’s inequality becomes equality only for disks.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Further Research
Find the “book” proof of Olsen - Reguera theorem, freeing itfrom the power series calculation and extending the result toarbitrary domains.
Is the sharp upper bound for the A2-content equal Area(G)2π ,
similarly to the situation with the analytic content?
What is the sharp lower bound for the A2 - content expressedin terms of “simple” geometric characteristics (e.g., area,perimeter, principal frequency) of the domain?
Refine the “isoperimetric sandwiches” to include theconnectivity of the domain. This is virtually unknownterritory. For the celebrated Carleman’s inequality boundingA2(G ) norm in terms of E 1(G ) norm, some initial steps weremade in S. Jacobs’ thesis in 1973 (unpublished). For theanalytic content, there is a result of DK - D. Luecking from1984. Nothing else has been done since.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Further Research
Find the “book” proof of Olsen - Reguera theorem, freeing itfrom the power series calculation and extending the result toarbitrary domains.
Is the sharp upper bound for the A2-content equal Area(G)2π ,
similarly to the situation with the analytic content?
What is the sharp lower bound for the A2 - content expressedin terms of “simple” geometric characteristics (e.g., area,perimeter, principal frequency) of the domain?
Refine the “isoperimetric sandwiches” to include theconnectivity of the domain. This is virtually unknownterritory. For the celebrated Carleman’s inequality boundingA2(G ) norm in terms of E 1(G ) norm, some initial steps weremade in S. Jacobs’ thesis in 1973 (unpublished). For theanalytic content, there is a result of DK - D. Luecking from1984. Nothing else has been done since.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Further Research
Find the “book” proof of Olsen - Reguera theorem, freeing itfrom the power series calculation and extending the result toarbitrary domains.
Is the sharp upper bound for the A2-content equal Area(G)2π ,
similarly to the situation with the analytic content?
What is the sharp lower bound for the A2 - content expressedin terms of “simple” geometric characteristics (e.g., area,perimeter, principal frequency) of the domain?
Refine the “isoperimetric sandwiches” to include theconnectivity of the domain. This is virtually unknownterritory. For the celebrated Carleman’s inequality boundingA2(G ) norm in terms of E 1(G ) norm, some initial steps weremade in S. Jacobs’ thesis in 1973 (unpublished). For theanalytic content, there is a result of DK - D. Luecking from1984. Nothing else has been done since.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Further Research
Find the “book” proof of Olsen - Reguera theorem, freeing itfrom the power series calculation and extending the result toarbitrary domains.
Is the sharp upper bound for the A2-content equal Area(G)2π ,
similarly to the situation with the analytic content?
What is the sharp lower bound for the A2 - content expressedin terms of “simple” geometric characteristics (e.g., area,perimeter, principal frequency) of the domain?
Refine the “isoperimetric sandwiches” to include theconnectivity of the domain. This is virtually unknownterritory. For the celebrated Carleman’s inequality boundingA2(G ) norm in terms of E 1(G ) norm, some initial steps weremade in S. Jacobs’ thesis in 1973 (unpublished). For theanalytic content, there is a result of DK - D. Luecking from1984. Nothing else has been done since.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Further Research
Find the “book” proof of Olsen - Reguera theorem, freeing itfrom the power series calculation and extending the result toarbitrary domains.
Is the sharp upper bound for the A2-content equal Area(G)2π ,
similarly to the situation with the analytic content?
What is the sharp lower bound for the A2 - content expressedin terms of “simple” geometric characteristics (e.g., area,perimeter, principal frequency) of the domain?
Refine the “isoperimetric sandwiches” to include theconnectivity of the domain.
This is virtually unknownterritory. For the celebrated Carleman’s inequality boundingA2(G ) norm in terms of E 1(G ) norm, some initial steps weremade in S. Jacobs’ thesis in 1973 (unpublished). For theanalytic content, there is a result of DK - D. Luecking from1984. Nothing else has been done since.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Further Research
Find the “book” proof of Olsen - Reguera theorem, freeing itfrom the power series calculation and extending the result toarbitrary domains.
Is the sharp upper bound for the A2-content equal Area(G)2π ,
similarly to the situation with the analytic content?
What is the sharp lower bound for the A2 - content expressedin terms of “simple” geometric characteristics (e.g., area,perimeter, principal frequency) of the domain?
Refine the “isoperimetric sandwiches” to include theconnectivity of the domain. This is virtually unknownterritory.
For the celebrated Carleman’s inequality boundingA2(G ) norm in terms of E 1(G ) norm, some initial steps weremade in S. Jacobs’ thesis in 1973 (unpublished). For theanalytic content, there is a result of DK - D. Luecking from1984. Nothing else has been done since.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Further Research
Find the “book” proof of Olsen - Reguera theorem, freeing itfrom the power series calculation and extending the result toarbitrary domains.
Is the sharp upper bound for the A2-content equal Area(G)2π ,
similarly to the situation with the analytic content?
What is the sharp lower bound for the A2 - content expressedin terms of “simple” geometric characteristics (e.g., area,perimeter, principal frequency) of the domain?
Refine the “isoperimetric sandwiches” to include theconnectivity of the domain. This is virtually unknownterritory. For the celebrated Carleman’s inequality boundingA2(G ) norm in terms of E 1(G ) norm, some initial steps weremade in S. Jacobs’ thesis in 1973 (unpublished).
For theanalytic content, there is a result of DK - D. Luecking from1984. Nothing else has been done since.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Further Research
Find the “book” proof of Olsen - Reguera theorem, freeing itfrom the power series calculation and extending the result toarbitrary domains.
Is the sharp upper bound for the A2-content equal Area(G)2π ,
similarly to the situation with the analytic content?
What is the sharp lower bound for the A2 - content expressedin terms of “simple” geometric characteristics (e.g., area,perimeter, principal frequency) of the domain?
Refine the “isoperimetric sandwiches” to include theconnectivity of the domain. This is virtually unknownterritory. For the celebrated Carleman’s inequality boundingA2(G ) norm in terms of E 1(G ) norm, some initial steps weremade in S. Jacobs’ thesis in 1973 (unpublished). For theanalytic content, there is a result of DK - D. Luecking from1984.
Nothing else has been done since.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
Further Research
Find the “book” proof of Olsen - Reguera theorem, freeing itfrom the power series calculation and extending the result toarbitrary domains.
Is the sharp upper bound for the A2-content equal Area(G)2π ,
similarly to the situation with the analytic content?
What is the sharp lower bound for the A2 - content expressedin terms of “simple” geometric characteristics (e.g., area,perimeter, principal frequency) of the domain?
Refine the “isoperimetric sandwiches” to include theconnectivity of the domain. This is virtually unknownterritory. For the celebrated Carleman’s inequality boundingA2(G ) norm in terms of E 1(G ) norm, some initial steps weremade in S. Jacobs’ thesis in 1973 (unpublished). For theanalytic content, there is a result of DK - D. Luecking from1984. Nothing else has been done since.
Introduction: Spaces of Analytic Functions Analytic Functions with Positive Boundary Values Analytic Content: Approximating z̄ in Bergman and Smirnov Norms Putnam’s Inequality for Toeplitz Operators in Bergman Spaces Final Remarks
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