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

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 $\phi$ on a scale of Bergman--type Hilbert spaces ${\mathcal{D}}_{\alpha }$. We investigate the optimal $\beta$ such that the composition operator ${\mathcal{C}}_{\phi }$ 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.

This paper has been accepted for publication in Mathematische Zeitschrift

References in corpus (4)

• The multiplicative Hilbert matrix
• An inequality of Hardy--Littlewood type for Dirichlet polynomials
• Composition Operators on Bohr-Bergman Spaces of Dirichlet Series
• Compact composition operators with non-linear symbols on the $H^2$ space of Dirichlet series

Cited by in corpus (1)

• Idempotent Fourier multipliers acting contractively on $H^p$ spaces