Some remarks on distances in spaces of analytic functions in bounded domains with $C^2$ boundary and admissible domains

The so-called Bergman representation formula (reproducing formula) and various estimates for Bergman projection with positive reproducing kernel and sharp Forelli–Rudin type estimates of Bergman kernel are playing a crucial role in certain new extremal problems related to so-called distance function in analytic function spaces in various domains in $C^{n}$.
In this paper based on some recent embedding theorems for analytic spaces in bounded domains with $C^{2}$ boundary and admissible domains new results for Bergman-type analytic function spaces related with this extremal problem will be provided. Some(not sharp) assertions for BMOA and Nevanlinna spaces, for analytic Besov spaces in any $D$ bounded domain with $C^2$ boundary or admissible domains in $C^{n}$ will be also provided.
We remark for readers in addition the problems related to regularity of Bergman projection which we use always in proof in various types of domains with various types of boundaries (or properties of boundaries) are currently and in the past already are under intensive attention. Many estimates for reproducing operators and there kernels and $L^p$ boundedness of Bergman projections have been also the object of considerable interest for more than 40 years. These tools serve as the core of all our proofs. When the boundary of the domain $D$ is sufficiently smooth decisive results were obtained in various settings. Our intention in this paper is the same as a previous our papers on this topic, namely we collect some facts from earlier investigation concerning Bergman projection and use them for our purposes in estimates of $dist_Y(f,\mathcal{X})$ function (distance function).
Based on our previous work and recent results on embeddings in classical analytic spaces in domains of various type in $C^{n}$ we provide several new general assertions for distance function in products of strictly pseudoconvex domains with smooth boundary, general bounded domains with $C^{2}$ boundary and in admissible domains in various spaces of analytic functions of several complex variables. These are first results of this type for bounded domains with $C^{2}$ boundary and admissible domains. In addition to our results we add some new sharp results for special kind of domains so called products of strictly pseudoconex domains with smooth boundary in $C^n$ extendind our sharp results in strictly pseudoconex domains.
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