Carleson-Sobolev measures for weighted Bloch spaces
Let \(B_n\) denote the unit ball in \( \mathbb C^n.\) The class of all homomorphic functions on \(B_n\) will be denoted by \(H(B_n).\) Let \(f \in H(B_n)\) have the homogeneous expansion \(f(z)=\sum_{k=1}^\infty f_k(z).\) For each non-negative integer \(j\), we define \({\mathcal R}^j f(z)=\sum_{k=1}^\infty k^jf_k(z).\) For each real parameter \(\alpha\), the weighted Bloch space \({\mathcal B}^\alpha(B_n)\) is defined to consist of all function \(f \in H(B_n)\) such that \( \sup_{z \in B_n} |{\mathcal R}^j f(z)|(1-|z|)^{j+\alpha} <\infty,\) where \(j\) is the least positive integer such that \(j+\alpha>0\). Let \({\mathcal A}^{-\log}(B_n)\) denote the set of all \(f \in H(B_n)\) such that \[ \sup_{z \in B_n} \frac{|f(z)|}{\log(e/(1-|z|))}<\infty. \] Let \(X \subset H(B_n)\) and \(0<q<\infty.\) A positive Borel measure \(\mu\) is called \(q\)-Carleson for \(X\) provided that the identity operator \(I :X \to L^q(\mu)\) is bounded. The author proves the following theorems: Theorem 1. Assume that \(\mu\) is a positive Borel measure on \(B_n,\) \(0<q<\infty,\) and \(\alpha>0.\) Then the following properties are equivalent: (a) \(\mu\) is a \(q\)-Carleson measure for \({\mathcal B}^\alpha(B_n);\) (b) \(\int_{B_n}(1-|z|)^{-\alpha q} d\mu(z) <\infty;\) (c) \(I :{\mathcal B}^\alpha(B_n) \to L^q(\mu)\) is a compact operator. Theorem 2. Assume that \(\mu\) is a positive Borel measure on \(B_n\) and \(0<q<\infty.\) Then the following properties are equivalent: (a) \(\mu\) is a \(q\)-Carleson measure for \({\mathcal A}^{-\log}(B_n);\) (b) \(\int_{B_n} (\log\frac{e}{1-|z|})^{q} d\mu(z) <\infty;\) (c) \(I : {\mathcal A}^{-\log}(B_n) \to L^q(\mu)\) is a compact operator. A generalization of the above theorems to the case of bounded, circular and strictly convex domains with a \(C^2\)-boundary are given.
scientific article; zbMATH DE number 5688816
scientific article