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

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 with 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 $0<p<1$.