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Norm inequalities for weighted Dirichlet spaces with applications to conformal maps

Fernando Pérez-González, Jouni Rättyä, Toni Vesikko

Abstract

A variety of norm inequalities related to Bergman and Dirichlet spaces induced by radial weights are considered. Some of the results obtained can be considered as generalizations of certain known special cases while most of the estimates discovered are completely new. The second objective of the paper is to apply the obtained norm inequalities to relate the growth of the maximum modulus of a conformal map $f$, measured in terms of a weighted integrability condition, to a geometric quantity involving the area of image under $f$ of a disc centered at the origin. Our findings in this direction yield new geometric characterizations of conformal maps in certain weighted Dirichlet and Besov spaces.

Norm inequalities for weighted Dirichlet spaces with applications to conformal maps

Abstract

A variety of norm inequalities related to Bergman and Dirichlet spaces induced by radial weights are considered. Some of the results obtained can be considered as generalizations of certain known special cases while most of the estimates discovered are completely new. The second objective of the paper is to apply the obtained norm inequalities to relate the growth of the maximum modulus of a conformal map , measured in terms of a weighted integrability condition, to a geometric quantity involving the area of image under of a disc centered at the origin. Our findings in this direction yield new geometric characterizations of conformal maps in certain weighted Dirichlet and Besov spaces.
Paper Structure (8 sections, 22 theorems, 179 equations)

This paper contains 8 sections, 22 theorems, 179 equations.

Table of Contents

  1. Introduction and main results
  2. Proof of Theorem \ref{['theorem:norm']}
  3. Proof of Theorem \ref{['thm-iff']}
  4. Proof of Theorem \ref{['thm:A2-weight-M']}
  5. Proof of Theorem \ref{['thm:Hp-Ap-S']}
  6. Proofs of Theorems \ref{['Thm1']} and \ref{['Thm2']}
  7. Proof of Theorem \ref{['thm:coefficients']}
  8. Declarations

Key Result

Theorem 1

Let $\omega$ be a radial weight and $0<p,q<\infty$. Then there exists a constant $C=C(p,q,\omega)>0$ such that if and only if either $\widehat{\omega}_{[q-2]}\in\widehat{\mathcal{D}}$ or $\widehat{\omega}_{[q-2]}\not\in L^1$.

Theorems & Definitions (35)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Corollary 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Lemma A
  • proof
  • ...and 25 more