Bounded sequence-to-function generalized Hausdorff transformations

Abstract:

Georgakis (1988) obtained the norm of the transformation \begin{equation*}(Ta)(y) = \sum ^{\infty }_{n=0} (-y)^{n} \frac {g^{(n)}(y)}{n!} a_{n},\quad y\geq 0, \end{equation*} considered as an operator from the sequence space $\ell ^{p}$, with weights $\Gamma (n+s+1)/n!$ to $L^{p}[0{,}\infty )$, with weight $y^{s}, s>-1$. As corollaries he obtained inequality statements for Borel and generalized Abel transformations. He also obtained the best constants possible for several weighted norm inequalities of Hardy and Littlewood. In this paper Georgakis’ results are extended to the Endl generalized Hausdorff matrices.