Sharp restrictions of analytic function spaces of several variables
R. F. Shamoyan, N. M. Makhina
TL;DR
This expository work surveys sharp trace theorems for analytic and harmonic function spaces in several complex variables, focusing on tubular domains over symmetric cones and bounded strongly pseudoconvex domains. It develops and collects trace results for mixed-norm Bergman spaces, Herz-type spaces, Bloch and BMOA-type spaces, using Bergman-type projections and related integral operators (e.g., Martinelli–Bochner kernels and $T_{\beta}$ maps) to relate traces on product domains to traces on lower-dimensional domains. The results provide explicit index relations and embedding theorems that connect high- and low-dimensional spaces, with extensive treatment of tubular domains, pseudoconvex domains, and products, and include numerous sharp inclusions and representations. The work also outlines open problems and future directions, emphasizing the role of Bergman projections and lattice structures in establishing trace phenomena across a range of domains and function spaces, thereby guiding further developments in several complex variables and harmonic analysis.
Abstract
In this expository paper we collect many recent advances in analytic function spaces of several complex variables related with trace problem in tubular domains over symmetric cones and bounded strongly pseudoconvex domains with smooth boundary. We consider various function space of analytic functions of several variables in various domains in $C^n$ and provide or complete descriptions of traces or estimates of traces of various analytic function spaces in various domains obtained in recent years, by various authors. The problem to find sharp estimates of traces of Hardy analytic function spaces in the unit polydisk first was posed by W. Rudin in 1969. Since then many papers appeared in literature. We collect in this expository paper not only already many known results on traces of various analytic function spaces in product domains but also discuss various new interesting results related with this problem. Related to trace problem various results were provided previously by G. Henkin, E. Amar, H. Alexander and various other authors. Finnaly, note that our trace theorems are closely related with the Bergman type projections acting between function spaces with different dimensions. This expository paper contains mainly new results concerning traces in tubular and bounded strongly pseudoconvex domains, proofs of these theorems are based in particular also on various properties of Bergman type projection, in this expository paper we will also shortly discuss some new results obtained by first author on Bergman type projections in these complicated domains in $C^n$. This is the second part of our notes related with trace problem. In the first part we provided a large list of recent sharp results on traces in the polydisk and polyball and was published in [1].