3,242 research outputs found
Sharp estimates of the potential kernel for the harmonic oscillator with applications
We prove qualitatively sharp estimates of the potential kernel for the
harmonic oscillator. These bounds are then used to show that the
estimates of the associated potential operator obtained recently by Bongioanni
and Torrea are in fact sharp.Comment: 10 pages, 1 figure; v2 (corrections in Section 3 concerning Theorem
3.1 and its proof and Figure 1
Potential operators associated with Jacobi and Fourier-Bessel expansions
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 , for which the potential operators are of strong type , of weak
type and of restricted weak type . 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.Comment: 28 pages, 4 figures; v2 (some comments on Bessel potentials added
Sharp estimates for potential operators associated with Laguerre and Dunkl-Laguerre expansions
We study potential operators associated with Laguerre function expansions of
convolution and Hermite types, and with Dunkl-Laguerre expansions. We prove
qualitatively sharp estimates of the corresponding potential kernels. Then we
characterize those , for which the potential operators
are bounded. These results are sharp analogues of the classical
Hardy-Littlewood-Sobolev fractional integration theorem in the Laguerre and
Dunkl-Laguerre settings.Comment: 25 pages, 2 figure
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