Riesz potential for (𝑘, 1)-generalized Fourier transform
https://doi.org/10.22405/2226-8383-2021-22-4-114-135
Abstract
In spaces with weight |𝑥|−1𝑣𝑘(𝑥), where 𝑣𝑘(𝑥) is the Dunkl weight, there is the (𝑘, 1)- generalized Fourier transform. Harmonic analysis in these spaces is important, in particular, in
problems of quantum mechanics. We define the Riesz potential for the (𝑘, 1)-generalized Fourier transform and prove for it, a (𝐿𝑞,𝐿𝑝)-inequality with radial power weights, which is an analogue of the well-known Stein–Weiss inequality for the classical Riesz potential. For the Riesz potential we calculate the sharp value of the 𝐿𝑝-norm with radial power weights. The sharp value of the 𝐿𝑝-norm with radial power weights for the classical Riesz potential was obtained independently by W. Beckner and S. Samko.
Supporting agencies: The research was supported by a grant from the Russian Science Foundation number 18-11-00199, https://rscf. ru/project/18-11-00199/.
About the Author
Valerii Ivanovich Ivanov
Tula State University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences, professor
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Review
For citations:
Ivanov V.I. Riesz potential for (𝑘, 1)-generalized Fourier transform. Chebyshevskii Sbornik. 2021;22(4):114-135. (In Russ.) https://doi.org/10.22405/2226-8383-2021-22-4-114-135
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