Bounded sequence-to-function Hausdorff transformations

Abstract

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 <r on [0, oo). It is shown that for 1 < p < oo, T is a bounded transformation of V with weight T(n + s + l)/n! into Lp[0, oo) with weight y3, s > -1, whose norm ||T|| = /o°° t-(i+«)/P do{t) = C{p,s) if and only if C(p, s) < oo, and that for 1 < p < oo, ||To||Pl4 < C(p, s)||a||p,s unless an is a null sequence.Furthermore, if 1 < p < r < oo, 0 < X < 1, 1/r = 1/p + A -1 and o is absolutely continuous with derivatives ip such that the function ipr(t) = t~l'ril>(t) belongs to L'/^JO, oo), then the transformation (T),a)(y) = y1~x(Ta)(y) is bounded from lp to Lr[0, oo) and has norm ||7\|| < ||^r IIi/a■ 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 V 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.Hardy showed that the discrete Hausdorff transformation (Ha)n = J2 UJ(An~V*K, Kp< oo,where pn is a totally monotone sequence or, equivalently, pn -fQ tn da(t), where a is a probability measure on [0,1], such that <r({0}) = 0 and Cv = /0 t~llv dcr(t) < oo, is a bounded transformation on lp and ||./ïd||p < Cp||a||p. Moreover, Cp is the best possible constant and exact equality occurs only if an is a null sequence or T is the identity transformation [2] or [3]. Our aim here is to establish the analogue of Hardy's Theorem for the Hausdorff transformation generated by a completely monotone function and to strengthen it in the case of the measure a is an absolutely continuous with derivative in Lv. Then we show how these results lead to certain Lp inequalities for the Borel transform and the transformation of generalized Abel means. These in turn give rise to improved

Bounded sequence-to-function generalized Hausdorff transformations

“…The analogues to inequalities A 0 , B 0 , C 0 , and D 0 in [2] are Setting α = 0 in the results of this paper yields the corresponding results from [2].…”

“…Part (a) is part (a) of Lemma 1 of [2]. The other parts are proved in the same way as their counterparts in [2].…”

“…Georgakis (1988) n g (n) (y) n! an, y ≥ 0, considered as an operator from the sequence space p , with weights Let {µ n } denote a real or complex sequence.…”