Carleson measures for variable exponent Bergman spaces
Let \(A^p(\mathbb{D})\), \(0< p <\infty\), denote the Bergman space of analytic functions in the unit disk \(\mathbb{D}\) in the complex plane, i.e., the space of holomorphic functions in the unit disk for which the integral \[ \int_{\mathbb{D}}|f(z)|^p dxdy \] is finite. A positive Borel measure \(\mu\) on \(\mathbb{D}\) is said to be a \textit{Carleson measure} for the Bergman space if there exists a constant \(C\) depending only on \(\mu\) such that for every function \(f\in A^p\) we have \[ \int_{\mathbb{D}}|f(z)|^p d\mu (z)\leq C\|f\|^p_{A^p(\mathbb{D})}. \] This notion was first introduced by Lenart Carleson in his proof of the Corona theorem for \(H^\infty(\mathbb{D})\). Around 1960, Carleson provided a complete characterisation of Carleson measures for the Hardy space \(H^p(\mathbb{D})\) where \(1\leq p<\infty\). He proved that a necessary and sufficient condition for \(\mu\) to be a Carleson measure is that \(\mu (S)\leq Ch\) for some constant \(C\) and all \textit{Carleson squares} \[ S=\{ re^{i\theta}: 1-h\leq r<1, |\theta -\theta_0|\leq h\} \] of side-length \(h\) at the boundary of the disk. According to a result of W. W. Hastings, the Carleson measures for the Bergman spaces have the analogous description \(\mu (S)\leq Ch^2\). There is also a description of Carleson measures in terms of the pseudo-hyperbolic metric by D. H. Luecking. In the paper under review, the authors provide a similar characterization for the variable-exponent Bergman spaces. A variable-exponent is a measurable function \(p:\mathbb{D} \to [1, \infty)\) such that its essential supremum on the unit disk is finite. The variable-exponent Bergman space \(A^{p(\cdot)}\) consists of the holomorphic functions in the unit disk for which the integral \[ \int_{\mathbb{D}}|f(z)|^{p(z)} dxdy \] is finite. By definition, a positive Borel measure \(\mu\) is said to be a Carleson measure for \(A^{p(\cdot)}(\mathbb{D})\) if there is a constant \(C\) such that \[ \| f\|_{L^{p(\cdot)}(\mathbb{D},\mu)}\leq C \| f\|_{A^{p(\cdot)}(\mathbb{D})}. \] It is well known that \(\mu\) is a Carleson measure for the ordinary Bergman space \(A^p(\mathbb{D})\) (here \(p\) is constant) if and only if there exist \(p\in (0,1)\) and a positive constant \(C\) such that \[ \mu(D_r(a))\leq C|D_r(a)|, \quad \forall a\in \mathbb{D}, \] where \(D_r(a)\) is the pseudo-hyperbolic disk with center \(a\) and radius \(r\), and \(|\cdot|\) stands for the usual area of subsets of the unit disk. The main result of the paper under review (Theorem 4.6) states that, if the variable-exponent \(p(\cdot)\) is log-Hölder continuous, then the same result holds true, that is, \(\mu\) is a Carleson measure for the variable-exponent Bergman space if and only if there exist constants \(0<r<1\), and \(C>0\) such that for every \(a\in\mathbb{D}\) we have \(\mu(D_r(a))\leq C|D_r(a)|\).
scientific article; zbMATH DE number 6814632
scientific article