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On the images of Dunkl–Sobolev spaces under the Schrödinger semigroup associated to Dunkl operators

  • C. Sivaramakrishnan
  • D. Sukumar
  • D. Venku NaiduEmail author
Article
  • 87 Downloads

Abstract

In this article, we consider the Schrödinger semigroup related to the Dunkl–Laplacian \(\Delta _{\mu }\) (associated to finite reflection group G) on \(\mathbb {R}^n\). We characterize the image of \(L^2(\mathbb {R}^n, e^{u^2} h_{\mu }(u) du)\) under the Schrödinger semigroup as a reproducing kernel Hilbert space. We define Dunkl–Sobolev space in \(L^2(\mathbb {R}^n, e^{u^2} h_{\mu }(u) du)\) and characterize it’s image under the Schrödinger semigroup associated to \(G=\mathbb {Z}_2^n\) as a reproducing kernel Hilbert space up to equivalence of norms. Also we provide similar results for Schrödinger semigroup associated to Dunkl–Hermite operator.

Keywords

Segal–Bargmann transform Schrödinger semigroup Weighted Bergman space Dunkl–Sobolev space 

Mathematics Subject Classification

Primary 46E35 Secondary 46F12 47D06 35B65 35J10 

Notes

Acknowledgements

The authors wish to thank G.B. Folland for giving clarification to their questions related to weighted Sobolev spaces. The first author thanks University Grant Commission, India for the financial support. We thank anonymous referee for thorough and careful review of our manuscript and helping us to improve the manuscript and to bring it to the present form.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology HyderabadHyderabadIndia

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