Bounded Sequence-to-Function Hausdorff Transformations

Abstract

Let \[ \left ( {Ta} \right )\left ( y \right ) = \sum \limits _{n = 0}^\infty {{{\left ( { - y} \right )}^n}} \frac {{{g^{\left ( n \right )}}\left ( y \right )}}{{n!}}{a_{n,\quad }}\quad y \geq 0\] be the sequence-to-function Hausdorff transformation generated by the completely monotone function $g$ or, what is equivalent, the Laplace transform of a finite positive measure $\sigma$ on $[0,\infty )$. It is shown that for $1 \leq p \leq \infty$, $T$ is a bounded transformation of ${l^p}$ with weight $\Gamma \left ( {n + s + 1} \right ) / n!$ into ${L^p}[0,\infty )$ with weight ${y^s},s > - 1$, whose norm $\left \| T \right \| = \int _0^\infty {{t^{ - \left ( {1 + s} \right ) / p}}} d\sigma \left ( t \right ) = C\left ( {p,s} \right )$ if and only if $C\left ( {p,s} \right ) < \infty$, and that for $1 < p < \infty ,{\left \| {Ta} \right \|_{p,s}} < C\left ( {p,s} \right ){\left \| a \right \|_{p,s}}$ unless ${a_n}$ is a null sequence. Furthermore, if $1 < p < r < \infty , \;0 < \lambda < 1$ and $\sigma$ is absolutely continuous with derivatives $\psi$ such that the function ${\psi _r}\left ( t \right ) = {t^{ - 1 / r}}\psi \left ( t \right )$ belongs to ${L^{1 / \lambda }}[0,\infty )$, then the transformation $\left ( {{T_\lambda }a} \right )\left ( y \right ) = {y^{1 - \lambda }}\left ( {Ta} \right )\left ( y \right )$ is bounded from ${l^p}$ to ${L^r}[0,\infty )$ and has norm $\left \| {{T_\lambda }} \right \| \leq {\left \| {{\psi _r}} \right \|_{1 / \lambda }}$. The transformation $T$ includes in particular the Borel transform and that of generalized Abel means. These results constitute an improved analogue of a theorem of Hardy concerning the discrete Hausdorff transformation on ${l^p}$ which corresponds to a totally monotone sequence, and lead to improved forms of some inequalities of Hardy and Littlewood for power series and moment sequences.