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Hardy-Type Inequalities for Fractional Powers of the Dunkl–Hermite Operator

Published online by Cambridge University Press:  02 April 2018

Óscar Ciaurri
Affiliation:
Departamento de Matemáticas y Computación, Universidad de La Rioja, 26006 Logroño, Spain (oscar.ciaurri@unirioja.es)
Luz Roncal*
Affiliation:
Departamento de Matemáticas y Computación, Universidad de La Rioja, 26006 Logroño, Spain and BCAM – Basque Center for Applied Mathematics, 48009 Bilbao, Spain (lroncal@bcamath.org)
Sundaram Thangavelu
Affiliation:
Department of Mathematics, Indian Institute of Science, 560 012 Bangalore, India (veluma@math.iisc.ernet.in)
*
*Corresponding author.

Abstract

We prove Hardy-type inequalities for a fractional Dunkl–Hermite operator, which incidentally gives Hardy inequalities for the fractional harmonic oscillator as well. The idea is to use h-harmonic expansions to reduce the problem in the Dunkl–Hermite context to the Laguerre setting. Then, we push forward a technique based on a non-local ground representation, initially developed by Frank et al. [‘Hardy–Lieb–Thirring inequalities for fractional Schrödinger operators, J. Amer. Math. Soc.21 (2008), 925–950’] in the Euclidean setting, to obtain a Hardy inequality for the fractional-type Laguerre operator. The above-mentioned method is shown to be adaptable to an abstract setting, whenever there is a ‘good’ spectral theorem and an integral representation for the fractional operators involved.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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References

1.Amri, B., Riesz transforms for Dunkl Hermite expansions, J. Math. Anal. Appl. 423 (2015), 646659.CrossRefGoogle Scholar
2.Beckner, W., Pitt's inequality and the uncertainty principle, Proc. Amer. Math. Soc. 123 (1995), 18971905.Google Scholar
3.Boggarapu, P., Roncal, L. and Thangavelu, S., Mixed norm estimates for the Cesàro means associated with Dunkl–Hermite expansions, to appear in Trans. Amer. Math. Soc. 369(1) (2017), 70217047.CrossRefGoogle Scholar
4.Branson, T. P., Fontana, L. and Morpurgo, C., Moser–Trudinger and Beckner–Onofri's inequalities on the CR sphere, Ann. Math. 177(2) (2013), 152.CrossRefGoogle Scholar
5.Chang, S-Y. A. and González, M. d. M., Fractional Laplacian in conformal geometry, Adv. Math. 226 (2011), 14101432.CrossRefGoogle Scholar
6Ciaurri, Ó and Roncal, L., Vector-valued extensions for fractional integrals of Laguerre expansions, Studia Math. 240(1) (2018), 6999.CrossRefGoogle Scholar
7.Cowling, M. and Haagerup, U., Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one, Invent. Math. 96 (1989), 507549.Google Scholar
8.Dai, F. and Xu, Y., Approximation theory and harmonic analysis on spheres and balls, Springer Monographs in Mathematics (Springer, 2013).Google Scholar
9.Dunkl, C. F., Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311(1) (1989), 167183.Google Scholar
10.Dunkl, C. F. and Xu, Y., Orthogonal polynomials of several variables, Encyclopedia of Mathematics and its Applications, Volume 81 (Cambridge University Press, Cambridge, 2001).CrossRefGoogle Scholar
11.Frank, R. L., Lieb, E. H. and Seiringer, R., Hardy–Lieb–Thirring inequalities for fractional Schrödinger operators, J. Amer. Math. Soc. 21 (2008), 925950.Google Scholar
12.Frank, R. L. and Seiringer, R., Non-linear ground state representations and sharp Hardy inequalities, J. Funct. Anal. 255 (2008), 34073430.Google Scholar
13.Gorbachev, D. V., Ivanov, V. I. and Tikhonov, S. Yu., Sharp Pitt inequality and logarithmic uncertainty principle for Dunkl transform in L 2, J. Approx. Theory 202 (2016), 109118.Google Scholar
14.Gradshteyn, I. S. and Ryzhik, I. M., Table of integrals, series and products, 7th edition (Elsevier/Academic Press, Amsterdam, 2007).Google Scholar
15.Graham, C. R. and Zworski, M., Scattering matrix in conformal geometry, Invent. Math. 152(1) (2003), 89118.Google Scholar
16.Helgason, S., Differential geometry, Lie groups and symmetric spaces (Academic Press, New York, 1978).Google Scholar
17.Herbst, I. W., Spectral theory of the operator (p 2 + m 2)1/2Ze 2/r, Commun. Math. Phys. 53 (1977), 285294.Google Scholar
18.Knapp, A. W. and Stein, E. M., Intertwining operators for semisimple groups, Ann. Math. 93 (1971), 489578.Google Scholar
19.Lebedev, N. N., Special functions and its applications (Dover, New York, 1972).Google Scholar
20.Nikiforov, A. F. and Uvarov, V. B., Special functions of mathematical physics. A unified introduction with applications. Translated from the Russian and with a preface by Boas, Ralph P.. With a foreword by Samarskiĭ, A. A. (Birkhäuser Verlag, Basel, 1988).Google Scholar
21.Olde, A. B., Confluent hypergeometric functions, in NIST handbook of mathematical functions (ed. Olver, F. W. J., Lozier, D. W., Boisvert, R. F. and Clark, C. W.), Chapter 13, (National Institute of Standards and Technology, Washington, DC, and Cambridge University Press, Cambridge, 2010). Available online at http://dlmf.nist.gov/13.Google Scholar
22.Prudnikov, A. P., Brychkov, A. Y. and Marichev, O. I., Integrals and series, Elementary Functions, Volume 1 (Gordon and Breach Science Publishers, New York, 1986).Google Scholar
23.Prudnikov, A. P., Brychkov, A. Y. and Marichev, O. I., Integrals and series, Elementary Functions, Volume 2 (Gordon and Breach Science Publishers, New York, 1986).Google Scholar
24.Roncal, L. and Thangavelu, S., Hardy's inequality for fractional powers of the sublaplacian on the Heisenberg group, Adv. Math. 302 (2016), 106158.Google Scholar
25.Rösler, M., Generalized Hermite polynomials and the heat equation for Dunkl operators, Comm. Math. Phys. 192 (1998), 519542.Google Scholar
26.Rösler, M., One-parameter semigroups related to abstract quantum models of Calogero types, in Infinite dimensional harmonic analysis (Kioto, 1999), Gräbner, Altendorf, 2000, 290305.Google Scholar
27.Rösler, M., Dunkl operators: theory and applications, Orthogonal Polynomials and Special Functions (Leuven, 2002), Lecture Notes in Mathematics, Volume 1817, pp. 93135 (Springer, Berlin, 2003).Google Scholar
28.Stinga, P. R. and Torrea, J. L., Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations 35 (2010), 20922122.Google Scholar
29.Thangavelu, S., Lectures on Hermite and Laguerre expansions, Mathematical Notes, Volume 42 (Princeton University Press, Princeton, NJ, 1993).Google Scholar
30.Thangavelu, S., Harmonic analysis on the Heisenberg group, Progress in Mathematics, Volume 159 (Birkhäuser, Boston, MA, 1998).CrossRefGoogle Scholar
31.Yafaev, D., Sharp constants in the Hardy–Rellich inequalities, J. Funct. Anal. 168 (1999), 121144.CrossRefGoogle Scholar