Nonlinear potential theory on the ball, with applications to exceptional and boundary interpolation sets
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} and the first author [Indiana Univ. Math. J. 38, 417-453 (1989; Zbl 0691.46022)]: If \(p>1\), \(0\leq n- \beta p<1\), and \(K\) is a compact subset of \(S\) such that \(\text{Cap} (K, L^p_\beta) =0\), then \(K\) is exceptional for the space \(H^p_\beta\).
holomorphic modifications of the Hedberg-Wolff potential
scientific article; zbMATH DE number 784583
scientific article