Weighted norm inequalities for derivatives on Bergman spaces (Q6567990)

Let \(\mathbb{D}\) denote the open unit disk in the complex plane, and let \(\mathcal{H}(\mathbb{D})\) denote the algebra of analytic functions on the unit disk. A \textit{weight} \(w\) is a non-negative integrable function on the unit disk. The weight \(w\) is said to be \textit{radial} if \(w(z)=w(|z|)\) for \(z\in\mathbb{D}\), and if the integral \(\int_0^1 w(s)ds\) is finite. Given a weight function \(w\), and a real number \(0<p<\infty\), the weighted Bergman space \(A^p_w\) consists of analytic functions \(f\in \mathcal{H}(\mathbb{D})\) for which \N\[\N\Vert f\Vert^p_{A^p_w}=\int_\mathbb{D} |f(z)|^p w(z)dA(z)\N\]\Nis finite; here \(dA(z)=\pi^{-1}dxdy\) denotes the normalized area measure in the unit disk. The standard weighted Bergman space consists of analytic functions in the unit disk with the property that \N\[\N\Vert f\Vert_{A^p_w}^p=(\alpha+1)\int_\mathbb{D} |f(z)|^p(1-|z|^2)^\alpha dA(z)<\infty,\N\]\Nwhere \(\alpha\) is a real parameter in the interval \((-1,\infty)\). Indeed, \(w(z)=(\alpha+1)(1-|z|^2)^\alpha\) is called the standard radial weight for the Bergman space, and is usually denoted by \(dA_\alpha(z)\). \par It is well known that for each positive integer \(k\), the norm of a given function in the standard weighted Bergman space is equivalent to the following expression: \N\[\N\Vert f\Vert_{A^p_w}^p\asymp\int_\mathbb{D} |f^{(k)}(z)|^p(1-|z|^2)^{kp} w(z)dA(z)+\sum_{j=0}^{k-1}|f^{(j)}(0)|^p,\tag{1}\N\]\Nwhich means that \(f\in A^p_w\) if and only if \((1-|z|^2)^k f^{(k)}\in L^p(\mathbb{D},dA_\alpha)\) provided that \(w\) is the standard radial weight. The formula (1) has a number of generalizations for various classes of radial weights, see for instance [\textit{A. Aleman} and \textit{A. G. Siskakis}, Indiana Univ. Math. J. 46, No. 2, 337--356 (1997; Zbl 0951.47039); \textit{M. Pavlović} and the first author, Math. Nachr. 281, No. 11, 1612--1623 (2008; Zbl 1155.30018); the authors, Adv. Math. 391, Article ID 107950, 70 p. (2021; Zbl 1480.30039); \textit{A. G. Siskakis}, Acta Sci. Math. 66, No. 3--4, 651--664 (2000; Zbl 0994.46008)]. In particular, the current authors proved that formula (1) holds true if and only of the weight function \(w\) belongs to the class \(\mathcal{D}= \hat{\mathcal{D}}\cap \check{\mathcal{D}}\): a radial weight \(\nu\) belongs to \(\hat{\mathcal{D}}\) if there exists a constant \(C=C(\nu)>1\) such that \(\hat{\nu}(z):=\int_{|z|}^1\nu(s)ds\) satisfies \N\[\N\hat{\nu}(r)\le C\hat{\nu}\left(\frac{1+r}{2}\right),\quad 0\le r<1.\N\]\NIn the way, \(\nu\) belongs to \(\check{\mathcal{D}}\) provided that there are constants \(K=K(\nu)>1\) and \(C=C(\nu)>1\) such that \N\[\N\hat{\nu}(r)\ge C\hat{\nu}\left(1-\frac{1-r}{K}\right),\quad 0\le r<1.\N\]\NThe main objective of the paper under review is to obtain formula (1) for certain classes of \textit{non-radial} weights. The paper is quite technical with a handful of notations and symbols. To achieve their goal, the authors succeed to characterize the \(q\)-Carleson measures for the weighted Bergman space \(A^p_w\). As a application of their results, the authors describe the resolvent set of the integral operator \N\[\NT_g(f)(z)=\int_0^z g^\prime (\zeta) f(\zeta)d\zeta\N\]\Nacting on the Bergman space \(A^p_w\).