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On the local Hölder boundary smoothness of an analytic function in the unit ball compared with the smoothness of its modulus

Ioann Vasilyev

Abstract

Local boundary smoothness of an analytic function f on the unit ball of C^n is compared to the smoothness of its modulus. We prove that different conditions imposed on the zeros of f imply different drops of the smoothness. We also show that some of the drops are the best possible.

On the local Hölder boundary smoothness of an analytic function in the unit ball compared with the smoothness of its modulus

Abstract

Local boundary smoothness of an analytic function f on the unit ball of C^n is compared to the smoothness of its modulus. We prove that different conditions imposed on the zeros of f imply different drops of the smoothness. We also show that some of the drops are the best possible.

Paper Structure

This paper contains 4 sections, 9 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['grglthm2']}
  3. Proof of Theorem \ref{['grglthm1']}
  4. Proof of Theorem \ref{['grglthm5']}

Key Result

Theorem A

(Carleson--Jacobs--Havin--Shamoyan) Let $\alpha\in (0,1).$ Suppose that $f: \mathbb D \rightarrow \mathbb C$ is an outer function which has an $\alpha$-Hölder modulus on the boundary circle $\mathbb T$ of the open unit disc $\mathbb D$. Then the function $f$ itself is $\alpha/2$-Hölder on $\mathbb T

Theorems & Definitions (19)

  • Theorem A
  • Theorem B
  • Theorem C
  • Definition 1
  • Remark 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Remark 2
  • Theorem 1
  • ...and 9 more