Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-10T09:27:45.804Z Has data issue: false hasContentIssue false
  • English
  • Français

Sharp estimates of the potential kernel for the harmonic oscillator with applications

Published online by Cambridge University Press:  11 January 2016

Adam Nowak  and
Krzysztof Stempak
Adam Nowak
Affiliation:
Instytut Matematyczny Polska Akademia Nauk, Sniadeckich 8, 00-956 Warszawa, Poland, Adam.Nowak@impan.pl
Krzysztof Stempak
Affiliation:
Instytut Matematyki i Informatyki, Politechnika Wrocławska, Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland, and Katedra Matematyki i Zastosowań Informatyki, Politechnika Opolska, Mikołajczyka 5, 45-271 Opole, Poland, Krzysztof.Stempak@pwr.wroc.pl
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove qualitatively sharp estimates of the potential kernel for the harmonic oscillator. These bounds are then used to show that the Lp–Lq estimates of the associated potential operator obtained recently by Bongioanni and Torrea are in fact sharp.

Keywords

47G4031C15
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 212 , December 2013 , pp. 1 - 17
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2013

References

[1] Bennett, y and Sharpley, R., Interpolation of Operators, Pure Appl. Math. 129, Academic Press, Boston, 1988. MR 0928802.Google Scholar
[2] Bongioanni, B. and Torrea, J. L., Sobolev spaces associated to the harmonic oscillator, Proc. Indian Acad. Sci. Math. Sci. 116 (2006), 337–360. MR 2256010. DOI 10.1007/ BF02829750.Google Scholar
[3] Nowak, A. and Stempak, K., Riesz transforms and conjugacy for Laguerre function expansions of Hermite type, J. Funct. Anal. 244 (2007), 399–443. MR 2297030. DOI 10.1016/j.jfa.2006.12.010.Google Scholar
[4] Nowak, A. and Stempak, K., Negative powers of Laguerre operators, Canad. J. Math. 64 (2012), 183–216. MR 2932174. DOI 10.4153/CJM-2011-040-7.CrossRefGoogle Scholar
[5] Stempak, K. and Torrea, J. L., Poisson integrals and Riesz transforms for Hermite function expansions with weights, J. Funct. Anal. 202 (2003), 443–472. MR 1990533. DOI 10.1016/S0022-1236(03)00083-1.Google Scholar
[6] Stempak, K. and Torrea, J. L., BMO results for operators associated to Hermite expansions, Illinois J. Math. 49 (2005), 1111–1131. MR 2210354.CrossRefGoogle Scholar