Potential operators associated with Jacobi and Fourier-Bessel expansions

Abstract

We study potential operators (Riesz and Bessel potentials) associated with classical Jacobi and Fourier-Bessel expansions. We prove sharp estimates for the corresponding potential kernels. Then we characterize those $1\le p,q\le \infty$, for which the potential operators are of strong type $(p,q)$, of weak type $(p,q)$ and of restricted weak type $(p,q)$. These results may be thought of as analogues of the celebrated Hardy-Littlewood-Sobolev fractional integration theorem in the Jacobi and Fourier-Bessel settings. As an ingredient of our line of reasoning, we also obtain sharp estimates of the Poisson kernel related to Fourier-Bessel expansions.

28 pages, 4 figures; v2 (some comments on Bessel potentials added)

References in corpus (5)

• Negative powers of Laguerre operators
• Sharp estimates of the potential kernel for the harmonic oscillator with applications
• Analysis related to all admissible type parameters in the Jacobi setting
• Fractional integrals on compact Riemannian symmetric spaces of rank one
• Harmonic analysis operators related to symmetrized Jacobi expansions

Cited by in corpus (4)

• Vector-valued extensions for fractional integrals of Laguerre expansions
• Sharp estimates for potential operators associated with Laguerre and Dunkl-Laguerre expansions
• Potential operators associated with Hankel and Hankel-Dunkl transforms
• Hardy spaces for Fourier--Bessel expansions