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2014•arXiv (Cornell University)Open access

Sharp estimates for potential operators associated with Laguerre and\n Dunkl-Laguerre expansions

Adam Nowak, Krzysztof Stempak

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Abstract

We study potential operators associated with Laguerre function expansions of\nconvolution and Hermite types, and with Dunkl-Laguerre expansions. We prove\nqualitatively sharp estimates of the corresponding potential kernels. Then we\ncharacterize those $1 \\le p,q \\le \\infty$, for which the potential operators\nare $L^p-L^q$ bounded. These results are sharp analogues of the classical\nHardy-Littlewood-Sobolev fractional integration theorem in the Laguerre and\nDunkl-Laguerre settings.\n

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We study potential operators associated with Laguerre function expansions of\nconvolution and Hermite types, and with Dunkl-Laguerre expansions. We prove\nqualitatively sharp estimates of the corresponding potential kernels. Then we\ncharacterize those $1 \\le p,q \\le \\infty$, for which the potential operators\nare $L^p-L^q$ bounded. These results are sharp analogues of the classical\nHardy-Littlewood-Sobolev fractional integration theorem in the Laguerre and\nDunkl-Laguerre settings.\n

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Available abstract

We study potential operators associated with Laguerre function expansions of\nconvolution and Hermite types, and with Dunkl-Laguerre expansions. We prove\nqualitatively sharp estimates of the corresponding potential kernels. Then we\ncharacterize those $1 \\le p,q \\le \\infty$, for which the potential operators\nare $L^p-L^q$ bounded. These results are sharp analogues of the classical\nHardy-Littlewood-Sobolev fractional integration theorem in the Laguerre and\nDunkl-Laguerre settings.\n

Key concepts: Laguerre polynomials, Hermite polynomials, Mathematics, Bounded function, Laguerre's method, Convolution (computer science), Function (biology), Pure mathematics

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