Riesz potential for $(k,1)$-generalized Fourier transform

V. I. Ivanov

Čebyševskij sbornik, Tome 22 (2021) no. 4, pp. 114-135.

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. We define the Riesz potential for the $(k,1)$-generalized Fourier transform and prove for it, a $(L^q,L^p)$-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 $L^p$-norm with radial power weights. The sharp value of the $L^p$-norm with radial power weights for the classical Riesz potential was obtained independently by W. Beckner and S. Samko.

Détail

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

Bibliographie

[1] Hardy G. H., Littelwood J. E., “Some properties of fractional integrals, I”, Math. Zeit., 27 (1928), 565–606 | DOI | MR

[2] Soboleff S., “On a theorem in functional analysis”, Rec. Math. [Mat. Sbornik] N.S., 4(46):3 (1938), 471–497 | Zbl

[3] Stein E. M., Weiss G., “Fractional integrals on n-dimensional Euclidean space”, J. Math. Mech., 7:4 (1958), 503–514 | MR | Zbl

[4] Gorbachev D.V., Ivanov V.I., “Vesovye neravenstva dlya potentsiala Danklya – Rissa”, Chebyshevckii sbornik, 20:1 (2019), 131–147 | DOI | Zbl

[5] Herbst I. W., “Spectral theory of the operator $(p^2+m^2)^{1/2}-Ze^2/r$”, Comm. Math. Phys., 53 (1977), 285–294 | DOI | MR | Zbl

[6] Beckner W., “Pitt's inequality with sharp convolution estimates”, Proc. Amer. Math. Soc., 136:5 (2008), 1871–1885 | DOI | MR | Zbl

[7] Samko S., “Best constant in the weighted Hardy inequality: the spatial and spherical version”, Fract. Calc. Anal. Appl., 8 (2005), 39–52 | MR | Zbl

[8] Dunkl C. F., “Hankel transforms associated to finite reflections groups”, Contemp. Math., 138, 1992, 123–138 | DOI | MR | Zbl

[9] Rösler M., “Dunkl operators. Theory and applications”, Orthogonal Polynomials and Special Functions, Lecture Notes in Math., 1817, Springer-Verlag, 2003, 93–135 | DOI | MR | Zbl

[10] Thangavelu S., Xu Y., “Riesz transform and Riesz potentials for Dunkl transform”, J. Comput. Appl. Math., 199 (2007), 181–195 | DOI | MR | Zbl

[11] Gorbachev D. V., Ivanov V. I., Tikhonov S. Yu., “Riesz potential and maximal function for Dunkl transform”, Potential Analysis, 55:4 (2021), 513–538 | DOI | MR | Zbl

[12] Salem Ben Saïd, Kobayashi T., Ørsted B., “Laguerre semigroup and Dunkl operators”, Compos. Math., 148:4 (2012), 1265–1336 | DOI | MR | Zbl

[13] Salem Ben Sa\"{i}d, Deleaval L., “Translation operator and maximal function for the $(k,1)$-generalized Fourier transform”, Journal of Functional Analysis, 279:8 (2020), 108706 | DOI | MR | Zbl

[14] Gorbachev D. V., Ivanov V. I., Tikhonov S. Yu., “Pitt's Inequalities and Uncertainty Principle for Generalized Fourier Transform”, International Mathematics Research Notices, 2016:23 (2016), 7179–7200 | DOI | MR | Zbl

[15] Ivanov V. I., “Ogranichennyi operator sdviga dlya (k, 1)-obobschennogo preobrazovaniya Fure”, Chebyshevskii sbornik, 21:4 (2020), 85–96 | DOI | MR | Zbl

[16] Gorbachev D. V., Ivanov V. I., Tikhonov S. Yu., “Positive $L^p$-bounded Dunkl-type generalized translation operator and its applications”, Constr. Approx., 49:3 (2019), 555–605 | DOI | MR | Zbl

[17] Ivanov V. I., “Svoistva i primenenie polozhitelnogo operatora sdviga dlya (k, 1)-obobschennogo preobrazovaniya Fure”, Chebyshevskii sbornik, 22:4 (2021)

[18] Beitmen G., Erdeii A., Tablitsy integralnykh preobrazovanii, v. 2, Preobrazovaniya Besselya. Integraly, Nauka, M., 1970, 328 pp.

[19] Vatson G.N., Teoriya besselevykh funktsii, IL, M., 1949, 799 pp.

[20] Sinnamon G, Stepanov V. D., “The weighted Hardy inequality: new proofs and the case $p=1$”, J. London Math. Soc., 54:2 (1996), 89–101 | DOI | MR | Zbl

[21] Kufner A., Opic B., Xardy-type inequalities, Pitman Research Notes in Mathematics Series, Longman Scientific and Technical, Harlow, 1990, 333 pp. | MR

[22] Kufner A., Persson L. E., Weighted inequalities of Xardy type, World Scientific hrblishing Co. Pte. Ltd, Singopure-London, 2003, 358 pp. | MR

[23] Beitmen G., Erdeii A., Vysshie transtsendentnye funktsii. Gipergemetricheskaya funktsiya. funktsiya Lezhandra, Nauka, M., 1965, 296 pp.