On the closures of Dirichlet type spaces in the Bloch space
Let $\mathbb{D}$ be the open unit disk in the complex plane, and let $\mathrm{Hol}(\mathbb{D})$ denote the space of holomorphic functions on the unit disk. A function $f\in\mathrm{Hol}(\mathbb{D})$ is said to belong to the Bloch space $\mathcal{B}$ if \[ \|f\|_{\mathcal{B}}=|f(0)|+\sup_{z\in\mathbf{D}}(1-|z|^2)|f^\prime (z)|<\infty. \] Similarly, for $s>-1$, the space $\mathcal{D}_s$ consists of $f\in \mathrm{Hol}(\mathbf{D})$ for which \[ \|f\|_{\mathcal{D}_s}=|f(0)|+\left (\int_{\mathbb{D}}(1-|z|^2)^s|f^\prime (z)|^2dA(z)\right )^{1/2}<\infty, \] where $dA(z)$ denotes the normalized Lebesgue area measure in the unit disk. For $s=0$, the space $\mathcal{D}_s$ reduces to the classical Dirichlet space, and for $s=1$, it will become the well-known Hardy space $H^2$ on the unit disk. \par It follows from the maximum modulus principle, or the Schwarz-Pick lemma that the Bloch space contains $H^\infty$, the space of bounded analytic functions on the unit disk. It is still an open problem to describe the closure of $H^\infty$ in the Bloch space $\mathcal{B}$; this question was raised by \textit{J. M. Anderson} et al. [J. Reine Angew. Math. 270, 12--37 (1974; Zbl 0292.30030)]. However, in 2010 a description of the closure of $H^p\cap \mathcal{B}$, for $1< p< \infty$, in the Bloch norm was obtained by \textit{N. M. Galán} and \textit{A. Nicolau} [St. Petersbg. Math. J. 22, No. 1, 55--59 (2011; Zbl 1211.30064)]. Later on in 2015, this result was improved by \textit{P. Galanopoulos} et al. [J. Math. Anal. Appl. 429, No. 2, 1214--1221 (2015; Zbl 1327.32008)] to incorporate the case $0< p< \infty$ in the unit ball of $\mathbb{C}^n$. \par In the paper under review, the authors provide a description of the closure in the Bloch norm of $\mathcal{D}_s\cap \mathcal{B}$. The main result of this paper, Theorem 2.1, states that for $-1<s\le 1$, a positive integer $n$, and a function $f\in\mathcal{B}$, the following holds: $f$ belongs to the closure in the Bloch norm of $\mathcal{D}_s\cap \mathcal{B}$ if and only if, for each $\epsilon >0$, the integral \[ \int_{\Omega_{n, \epsilon}(f)}(1-|z|^2)^{s-2}dA(z) \] is finite, where \[ \Omega_{n, \epsilon (f)}=\{ z\in \mathbb{D}: (1-|z|^2)^n|f^{(n)}(z)|\ge \epsilon \}. \] Similar descriptions are provided too.
scientific article; zbMATH DE number 7032869
scientific article