Extremal problems in Bergman spaces and an extension of Ryabykh's Hardy space regularity theorem for 1 &lt; p &lt; infinity

Ferguson, Timothy

doi:10.1512/iumj.2017.66.5949

“…For detailed treatments of extremal problems on the closely related classical Hardy spaces, we refer the reader to [3,Chapter 8], [8, Chapter IV] or [11]. To list only a few references, extremal problems in Bergman spaces have been studied in [5], [6], [7], [10], [12], [13], [14], [15] for linear functionals and in [16] for norm-attaining operators. The related history of contractive Date: 25 July, 2021.…”

Section: Introductionmentioning

confidence: 99%

Extremal problems for vanishing functions in Bergman spaces

Llinares

¹

,

Vukotić

²

2022

Proc. Amer. Math. Soc.

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“…For detailed treatments of extremal problems on the closely related classical Hardy spaces, we refer the reader to [3,Chapter 8], [8, Chapter IV] or [11]. To list only a few references, extremal problems in Bergman spaces have been studied in [5], [6], [7], [10], [12], [13], [14], [15] for linear functionals and in [16] for norm-attaining operators. The related history of contractive Date: 08 December, 2020.…”

Section: Introductionmentioning

confidence: 99%

Extremal problems for vanishing functions in Bergman spaces

Llinares,

Vukotić

2020

Preprint

0

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Bergman–Hölder Functions, Area Integral Means and Extremal Problems

Ferguson

¹

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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Extremal problems in Bergman spaces and an extension of Ryabykh's Hardy space regularity theorem for 1 < p < infinity

Cited by 3 publications

References 6 publications

Extremal problems for vanishing functions in Bergman spaces

Extremal problems for vanishing functions in Bergman spaces

Extremal problems for vanishing functions in Bergman spaces

Bergman–Hölder Functions, Area Integral Means and Extremal Problems

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