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

Abstract

We use precise asymptotic expansions for Jacobi functions ${\varphi }_{\lambda }^{(\alpha ,\beta )}$ parameters $\alpha$, $\beta$ satisfying $\alpha >1/2$, $\alpha >\beta >-1/2$, to generalizing classical Hörmander-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 $m_a:\lambda \mapsto ({\lambda }^2+{\rho }^2{)}^{-a/2}$, $a>0$.

arXiv commentThe present preprint will not be submitted for publication since the main result is a special case of results from Bloom/Xu: "Fourier multipliers for L^p Chebli-Trimeche hypergroups", Proc. London Math. Soc. 80 (2000), 643-664. The techniques in the proofs are different and are used elsewhere, so we decided to upload the preprint for future reference. Read the disclaimer in Section 0

References in corpus (1)

• On a Class of Non-Integrable Multipliers for the Jacobi Transform

Cited by in corpus (1)

• Spectral multipliers for functions of fixed $K$-type on $L^p(SL(2,\mathbb{R}))$