Lebesgue boundedness of Riesz potential for $(k,1)$-generalized Fourier transform with radial piecewise power weights

V. I. Ivanov

Čebyševskij sbornik, Tome 23 (2022) no. 4, pp. 92-104.

Résumé

In spaces with weight $|x|^{-1}v_k(x)$, where $v_k(x)$ 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. Recently, for the $(k,1)$-generalized Fourier transform, the Riesz potential was defined and the $(L^p,L^q)$-inequality with radial power weights was proved for it, which is an analogue of the well-known Stein–Weiss inequality for the classical Riesz potential and the Dunkl–Riesz potential. In the paper, this result is generalized to the case of radial piecewise power weights. Previously, a similar inequality was proved for the Dunkl–Riesz potential.

Détail

Keywords: $(k,1)$-generalized Fourier transform, Riesz potential.

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