Nonlinear potential theory on the ball, with applications to exceptional and boundary interpolation sets
type: Item
instance of: scholarly article
title: Nonlinear potential theory on the ball, with applications to exceptional and boundary interpolation sets
zbMATH Open document ID: 0833.31007
author: Igor E. Verbitsky
author: William S. Cohn
published in: Michigan Mathematical Journal
publication date: 1996-03-12T00:00:00Z
review text: Let \(S\) be the unit sphere in \(\mathbb{C}^n\), equipped with the usual rotation-invariant measure \(d\sigma\), and \(I_\beta \mu(\eta)= \int_S {{d\mu} \over {|1-\langle \eta, \zeta\rangle |^{n- \beta}}}\) for \(\eta\in S\), \(0< \beta< n\) and any finite measure \(\mu\in M(S)\). The energy of \(\mu\) is defined as \(E_p^\beta (\mu)= |I_\beta \mu|^{p'}_{L^{p'} (d\sigma)}\), where \(p'= p/ (p-1)\) for \(1<p\). If \(d\mu= f\cdot d\sigma\), \(f\in L^1 (d\sigma)\), set \(I_\beta \mu= I_\beta f\). The authors consider two holomorphic modifications of the Hedberg-Wolff potential. Let \(\mu\in M^+ (S)\), \(1< p< \infty\), \(0< \beta\leq n/p\), and \(B(\zeta, r)\) be the anisotropic ball of radius \(r\), centered at \(\zeta\in S\). For any \(\lambda >0\) and \(z\), \(|z|<1\), they set \[ U_\lambda^{\beta p} \mu (z)= \int_0^1 \int_S \Bigl[{{\mu(B (\zeta, 1-r))} \over {(1- r)^{n- \beta p}}} \Bigr ]^{p'-1} {{(1- r)^{\lambda- n}} \over {(1- r\langle z, \zeta\rangle )^\lambda}} d\sigma (\zeta) {{dr} \over {1-r}}. \] If \(0< \lambda< 1\), they set \[ V_\lambda^{\beta p} \mu (z)= \int_0^1 \Bigl [\int_S {{(1- r)^{\lambda+ \beta p-n}} \over {(1- r\langle z, \zeta\rangle )^\lambda}} d\mu (\zeta)\Bigr]^{p'-1} {{dr} \over {1- r}}. \] The main result is the following statement: Theorem 1. Let \(\mu\in M^+\), \(p>1\), \(0< \beta\leq n/p\) and \(1> \lambda> n- \beta p\). \[ (i) \quad \text{If }p\leq 2, \text{ then } \bigl|U_\lambda^{\beta p} \mu \bigr|^p_{H^p_\beta} \leq CE^\beta_p (\mu), \qquad (ii) \quad \text{If }p\geq 2, \text{ then } \bigl|V_\lambda^{\beta p} \mu \bigr|^p_{H^p_\beta}\leq CE^\beta_p (\mu). \] Here \(H^p_\beta\) is the Hardy-Sobolev space of functions holomorphic in the unit ball. Making use of these estimates, the authors investigate exceptional sets of functions defined on the unit ball, i.e., the subsets of the sphere where the function fails to have a finite admissible limit. In particular, they prove the following conjecture from the article of \textit{P. Ahern}…
reviewed by: Alexander I. Kheyfits
Mathematics Subject Classification ID: 31C45
Mathematics Subject Classification ID: 32A40
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 784583
zbMATH Keywords: nonlinear potentials
zbMATH Keywords: exceptional boundary sets
zbMATH Keywords: holomorphic modifications of the Hedberg-Wolff potential
zbMATH Keywords: Hardy-Sobolev space
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029005154
OpenAlex ID: W2027510863
DOI: 10.1307/MMJ/1029005154
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description: scientific article; zbMATH DE number 784583