Composition operators and embedding theorems for some function spaces of Dirichlet series

TL;DR

This paper studies the boundedness of composition operators on Dirichlet space-related function spaces, establishing new embedding theorems and analyzing the impact of polynomial symbols on these operators.

Contribution

It introduces new boundedness results for composition operators on Dirichlet and Hardy spaces, including optimal embedding theorems and analysis of polynomial symbols.

Findings

Characterized when composition operators are bounded between Dirichlet-type spaces.

Proved a new embedding theorem for Hardy spaces into Bergman spaces in the half-plane.

Obtained first non-trivial results for composition operators on Hardy spaces with polynomial symbols.

Abstract

We observe that local embedding problems for certain Hardy and Bergman spaces of Dirichlet series are equivalent to boundedness of a class of composition operators. Following this, we perform a careful study of such composition operators generated by polynomial symbols $\varphi$ on a scale of Bergman--type Hilbert spaces $\mathcal{D}_\alpha$. We investigate the optimal $\beta$ such that the composition operator $\mathcal{C}_\varphi$ maps $\mathcal{D}_\alpha$ boundedly into $\mathcal{D}_\beta$. We also prove a new embedding theorem for the non-Hilbertian Hardy space $\mathcal H^p$ into a Bergman space in the half-plane and use it to consider composition operators generated by polynomial symbols on $\mathcal{H}^p$, finding the first non-trivial results of this type. The embedding also yields a new result for the functional associated to the multiplicative Hilbert matrix.