Volterra operators between Hardy spaces of vector-valued Dirichlet series

TL;DR

The paper investigates Volterra operators between Hardy spaces of vector-valued Dirichlet series, focusing on maps from $\mathscr{H}^{\text{weak}}_p(X)$ to $\mathscr{H}^+_p(X)$ for $2\le p<\infty$ and infinite-dimensional Banach spaces $X$. It proves that, when $X$ is $p$-uniformly PL-convex, no nontrivial bounded $T_g$ can exist unless the symbol $g$ is constant, using a vector-valued Littlewood–Paley inequality for Dirichlet series and a synthesis of harmonic-analytic tools. A key contribution is the embedding $\mathscr{H}^+_p(X)\subset \mathscr{D}^p_{p-1}(X)$ (McCarthy-Dirichlet spaces) together with a Dvoretzky-based construction and multiplier theory to force rigidity of $g$. These results highlight a sharp separation between $\mathscr{H}^+_p(X)$ and $\mathscr{H}^{\text{weak}}_p(X)$ in the infinite-dimensional setting and provide techniques for operator theory on vector-valued Dirichlet spaces.

Abstract

Let $2\leq p<\infty$ and $X$ be a complex infinite-dimensional Banach space. It is proved that if $X$ is $p$-uniformly PL-convex, then there is no nontrivial bounded Volterra operator from the weak Hardy space $\mathscr{H}^{\text{weak}}_p(X)$ to the Hardy space $\mathscr{H}^+_p(X)$ of vector-valued Dirichlet series. To obtain this, a Littlewood--Paley inequality for Dirichlet series is established.

Theorems & Definitions (17)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Lemma 2.2
• Proposition 2.3
• proof
• proof : Proof of Theorem \ref{['L-P']}
• Theorem 2.4
• Lemma 2.5
• Lemma 2.6