Selected problems in classical function theory

Bénéteau, Catherine; Khavinson, Dmitry

doi:10.1090/conm/638/12813

Cited by 4 publications

(2 citation statements)

References 13 publications

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“…The purpose of this paper is to investigate isoperimetric inequalities for λ Ap (Ω) (1 ≤ p < ∞) in all dimensions, and to examine the cases of equality with the upper and lower bounds (cf. Problem 3.4 of [5]). We denote by q the dual exponent of p, whence 1/p + 1/q = 1 (or q = ∞ if p = 1), and note that the dual space L * p can be identified with L q .…”

Section: Introductionmentioning

confidence: 99%

Isoperimetric inequalities for Bergman analytic content

Gardiner¹,

Ghergu²,

Sjödin³

2019

Preprint

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The Bergman p-analytic content (1 ≤ p < ∞) of a planar domain Ω measures the L p (Ω)-distance between z and the Bergman space A p (Ω) of holomorphic functions. It has a natural analogue in all dimensions which is formulated in terms of harmonic vector fields. This paper investigates isoperimetric inequalities for Bergman p-analytic content in terms of the St Venant functional for torsional rigidity, and addresses the cases of equality with the upper and lower bounds.

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Section: Introductionmentioning

confidence: 99%

Isoperimetric inequalities for Bergman analytic content

Gardiner¹,

Ghergu²,

Sjödin³

2019

Preprint

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“…The lower bounds of the area of sp(T f ) were obtained in [9] (see [2], [1] [13] and [14] for generalizations to uniform algebras and further discussions). Together with Putnam's inequality such lower bounds were used to prove the isoperimetric inequality (see [4], [5] and the references there). Recently, there has been revived interest in the topic in the context of analytic Topelitz operators on the Bergman space (cf.…”

Section: Introductionmentioning

confidence: 99%

A note on the spectral area of Toeplitz operators

Chu

Khavinson

2015

Proc. Amer. Math. Soc.

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Bergman–Hölder Functions, Area Integral Means and Extremal Problems

Ferguson

2017

Integr. Equ. Oper. Theory

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We study certain weighted area integral means of analytic functions in the unit disc. We relate the growth of these means to the property of being mean Hölder continuous with respect to the Bergman space norm. In contrast with earlier work, we use the second iterated difference quotient instead of the first. We then give applications to Bergman space extremal problems.This paper deals with mean Hölder type smoothness conditions for functions in Bergman spaces on the unit disc D, and the relation of these conditions to extremal problems in Bergman spaces.The first main topic is area integral means and smoothness conditions. It is well known (due to Hardy and Littlewood) that f is analytic in the unit disc and |f ′ (re iθ )| ≤ C(1−r) −1+β for 0 < β ≤ 1 if and only if f is continuous in the closed unit disc and |f (e iθ+it ) − f (e iθ )| ≤ C ′ |t| β . This result can be thought of as dealing with the H ∞ norm of the boundary function. There is a similar result for 1 ≤ p < ∞, also due to Hardy and Littlewood, that states that for an analytic function f , the integral means M p (r, f ′ ) ≤ C(1 − r) −1+β if and only if f ∈ H p and f (·) − f (e it ·) H p ≤ C ′ |t| β . (See chapter 5 of [2]). Zygmund [18] obtained similar results for the second iterated difference. For example, he proved that for an analytic function f , one has that f is continuous in |z| ≤ 1 and |f (e it z) + f (e −it z) − 2f (z)| ≤ C|t| if and only if |f ′′ (z)| ≤ C(1 − r) −1 (see [2]). Similar results hold for powers of |t| greater than 0 and at most 2, and for integral means. Analogous properties hold for harmonic functions in higher dimensions (see e.g. Chapter V of [14]).In [8], the authors give results relating growth of area integral means of analytic functions to mean Hölder regularity of these functions. In this article, we prove similar results, but instead use the second iterated difference, like in the result of Zygmund. Also, we work on the standard weighted Bergman spaces instead of just the unweighted case. For example, we prove that if 0 < β ≤ 2 and −1 < α < ∞, and if A p α denotes the standard weighted Bergman space, then f (e it ·)+f (e −it ·)− Date: September 25, 2018.

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Selected problems in classical function theory

Cited by 4 publications

References 13 publications

Isoperimetric inequalities for Bergman analytic content

Isoperimetric inequalities for Bergman analytic content

A note on the spectral area of Toeplitz operators

Bergman–Hölder Functions, Area Integral Means and Extremal Problems

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