Linear space properties of $H^p$ spaces of Dirichlet series

TL;DR

This paper investigates the structure and properties of $ ext{H}^p$ spaces of Dirichlet series, revealing a contractive symmetry and providing new coefficient and partial sum estimates across different p-ranges.

Contribution

It establishes the equivalence of two definitions of $ ext{H}^p$, explores linear functionals and symmetries, and extends estimates for coefficients and partial sums to the range $0<p extless 1$.

Findings

Contractive symmetry between $ ext{H}^p$ and $ ext{H}^{4/p}$ contrasts classical duality.

New coefficient estimates based on multiplicative structure and one-variable bounds.

Extended partial sum operator estimates for $0<p extless 1$, complementing classical results.

Abstract

We study $H^p$ spaces of Dirichlet series, called $\mathcal{H}^p$, for the range $0<p< \infty$. We begin by showing that two natural ways to define $\mathcal{H}^p$ coincide. We then proceed to study some linear space properties of $\mathcal{H}^p$. More specifically, we study linear functionals generated by fractional primitives of the Riemann zeta function; our estimates rely on certain Hardy--Littlewood inequalities and display an interesting phenomenon, called contractive symmetry between $\mathcal{H}^p$ and $\mathcal{H}^{4/p}$, contrasting the usual $L^p$ duality. We next deduce general coefficient estimates, based on an interplay between the multiplicative structure of $\mathcal{H}^p$ and certain new one variable bounds. Finally, we deduce general estimates for the norm of the partial sum operator $\sum_{n=1}^\infty a_n n^{-s}\mapsto \sum_{n=1}^N a_n n^{-s}$ on $\mathcal{H}^p$ with $0< p \le 1$, supplementing a classical result of Helson for the range $1<p<\infty$. The results for the coefficient estimates and for the partial sum operator exhibit the traditional schism between the ranges $1\le p \le \infty$ and $0<p<1$.