arXiv subjects
Publications and source records attributed to Krzysztof Stempak.
Connections between Hankel transforms of different order for $L^p$-functions are examined. Well known are the results of Guy [Guy] and Schindler [Sch]. Further relations result from projection formulae for Bessel functions of different order. Consequences for Hankel multipliers are exhibited and implications for radial Fourier multipliers on Euclidean spaces of different dimensions indicated.
The aim of this note is to provide a fractional integration theorem in the framework of Laguerre expansions. The method of proof consists of establishing an asymptotic estimate for the involved kernel and then applying a method of Hedberg \cite{pro}. We combine this result with sufficient $(p,p)$ multiplier criteria of Stempak and Trebels \cite{ST}. The resulting sufficient $(p,q)$ multiplier criteria are comparable with necessary ones of Gasper and Trebels \cite{laguerre}.
Using the standard square--function method (based on the Poisson semigroup), multiplier conditions of Hörmander type are derived for Laguerre expansions in $L^p$--spaces with power weights in the $A_p$-range; this result can be interpreted as an ``upper end point'' multiplier criterion which is fairly good for $p$ near $1$ or near $\infty $. A weighted generalization of Kanjin's \cite{kan} transplantation theorem allows to obtain a ``lower end point'' multiplier criterion whence by interpolation nearly ``optimal'' multiplier criteria (in dependance of $p$, the order of the Laguerre polynomial, the weight).