Bounded sequence-to-function Hausdorff transformations

Abstract

Let \[ ( T a ) ( y ) = ∑ n = 0 ∞ ( − y ) n g ( n ) ( y ) n ! a n , y ≥ 0 \left ( {Ta} \right )\left ( y \right ) = \sum \limits _{n = 0}^\infty {{{\left ( { - y} \right )}^n}} \frac {{{g^{\left ( n \right )}}\left ( y \right )}}{{n!}}{a_{n,\quad }}\quad y \geq 0 \] be the sequence-to-function Hausdorff transformation generated by the completely monotone function g g or, what is equivalent, the Laplace transform of a finite positive measure σ \sigma on [ 0 , ∞ ) [0,\infty ) . It is shown that for 1 ≤ p ≤ ∞ 1 \leq p \leq \infty , T T is a bounded transformation of l p {l^p} with weight Γ ( n + s + 1 ) / n ! \Gamma \left ( {n + s + 1} \right ) / n! into L p [ 0 , ∞ ) {L^p}[0,\infty ) with weight y s , s > − 1 {y^s},s > - 1 , whose norm ‖ T ‖ = ∫ 0 ∞ t − ( 1 + s ) / p d σ ( t ) = C ( p , s ) \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 ( p , s ) > ∞ C\left ( {p,s} \right ) > \infty , and that for 1 > p > ∞ , ‖ T a ‖ p , s > C ( p , s ) ‖ a ‖ p , s 1 > p > \infty ,{\left \| {Ta} \right \|_{p,s}} > C\left ( {p,s} \right ){\left \| a \right \|_{p,s}} unless a n {a_n} is a null sequence. Furthermore, if 1 > p > r > ∞ , 0 > λ > 1 1 > p > r > \infty ,\,\;0 > \lambda > 1 and σ \sigma is absolutely continuous with derivatives ψ \psi such that the function ψ r ( t ) = t − 1 / r ψ ( t ) {\psi _r}\left ( t \right ) = {t^{ - 1 / r}}\psi \left ( t \right ) belongs to L 1 / λ [ 0 , ∞ ) {L^{1 / \lambda }}[0,\infty ) , then the transformation ( T λ a ) ( y ) = y 1 − λ ( T a ) ( y ) \left ( {{T_\lambda }a} \right )\left ( y \right ) = {y^{1 - \lambda }}\left ( {Ta} \right )\left ( y \right ) is bounded from l p {l^p} to L r [ 0 , ∞ ) {L^r}[0,\infty ) and has norm ‖ T λ ‖ ≤ ‖ ψ r ‖ 1 / λ \left \| {{T_\lambda }} \right \| \leq {\left \| {{\psi _r}} \right \|_{1 / \lambda }} . The transformation T 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 {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.