Bergman–Hölder Functions, Area Integral Means and Extremal Problems

Abstract

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.

Uniform Approximation of Extremal Functions in Weighted Bergman Spaces

“…We now discuss how to use the results in the previous section to bound the distance from a given function to F in the supremum norm. We will use the following theorem found in [8,Corollary 4.3]. The proof of this theorem shows that the same results hold if F is replaced by F n .…”

“…We now discuss extremal problems restricted to the space of polynomials of degree n. Let F n denote the extremal polynomial of degree n, for the extremal problem of maximizing Re φ(f ) where f ranges over all polynomials of degree at most n with norm 1. We will need the following theorem from [8].…”

“…See [2,3] for more information on extremal problems in spaces of analytic functions. See also [8,10,12,14] for more information on regularity questions related the extremal problems we discuss.…”

“…We refer to functions in the Λ * classes as being (mean) Bergman-Hölder continuous (see [8]). We discuss several estimates that relate the mean Bergman-Hölder continuity of A p α functions to the minimum error in approximating these functions with polynomials of fixed degree.…”

“…the L ∞ norm). To do so, we use results from [8] to obtain bounds on the C β norm of the extremal functions and the functions approximating them for certain β, as long as the integral kernels are sufficiently regular. We also use Theorem 4.2, which allows us to conclude that two functions that are each not too large in the C β norm and that are close in the A p α norm must actually be close in the uniform norm.…”