$L^p$-results for fractional integration and multipliers for the Jacobi transform

Details $L^p$-results for fractional integration and multipliers for the Jacobi transform $L^p$-results for fractional integration and multipliers for the Jacobi transform

2011-08-17

arxiv.org/abs/1108.3478v1

Mathematics

Classical Analysis and ODEs

The present preprint will not be submitted for publication since the main result is a special case of results from Bloom/Xu: "

Scientific paper

We use precise asymptotic expansions for Jacobi functions $\phi^{(\alpha,\beta)}_\lambda$ parameters $\alpha$, $\beta$ satisfying $\alpha>1/2$, $\alpha>\beta>-1/2$, to generalizing classical H\"ormander-type multiplier theorem for the spherical transform on a rank one Riemannian symmetric space (by Clerc/Stein and Stanton/Tomas) to the framework of Jacobi analysis. In particular, multiplier results for the spherical transform on Damek--Ricci spaces are subsumed by this approach, and it yields multiplier results for the hypergeometric `Heckman--Opdam transform' associated with a rank one root system. We obtain near-optimal $L^p-L^q$ estimates for the integral operator associated with the convolution kernel $m_a:\lambda\mapsto(\lambda^2+\rho^2)^{-a/2}$, $a>0$.

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