Lp-L2 estimates for solutions of the wave equation associated to the Grushin operator

Sundaram Thangavelu 1  and Venku Naidu Dogga 2
  • 1 Department of Mathematics, Indian Institute of Science, 560012, Bangalore, India
  • 2 Department of Mathematics, Indian Institute of Technology Hyderabad, 502285, Kandi, India
Sundaram Thangavelu and Venku Naidu Dogga
  • Corresponding author
  • Department of Mathematics, Indian Institute of Technology Hyderabad, Kandi, 502285, Sangareddy, Telangana, India
  • Email
  • Search for other articles:
  • degruyter.comGoogle Scholar

Abstract

This article deals with Lp-L2 estimates for solutions of the wave equation associated to the Grushin operator G:=-Δ-|x|22t2 on n+1. We make use of the Lp boundedness of Riesz transforms for the Grushin operator in studying the wave equation.

  • [1]

    K. Jotsaroop, P. K. Sanjay and S. Thangavelu, Riesz transforms and multipliers for the Grushin operator, J. Anal. Math. 119 (2013), 255–273.

    • Crossref
    • Export Citation
  • [2]

    K. Jotsaroop and S. Thangavelu, L p L^{p} estimates for the wave equation associated to the Grushin operator, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 13 (2014), no. 3, 775–794.

  • [3]

    D. Müller and E. M. Stein, L p L^{p}-estimates for the wave equation on the Heisenberg group, Rev. Mat. Iberoam. 15 (1999), no. 2, 297–334.

  • [4]

    R. Strichartz, Convolutions with kernels having singularities on a sphere, Trans. Amer. Math. Soc. 148 (1970), 461–471.

    • Crossref
    • Export Citation
  • [5]

    S. Thangavelu, Riesz transforms and the wave equation for the Hermite operator, Comm. Partial Differential Equations 15 (1990), no. 8, 1199–1215.

    • Crossref
    • Export Citation
  • [6]

    S. Thangavelu, Lectures on Hermite and Laguerre Expansions, Math. Notes 42, Princeton University Press, Princeton, 1993.

  • [7]

    G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 1944.

  • [8]

    J. Zhong, The L p {L^{p}}-L q {L^{q}} estimates for the wave equation with a nonnegative potential, Comm. Partial Differential Equations 20 (1995), no. 1–2, 315–334.

    • Crossref
    • Export Citation
Purchase article
Get instant unlimited access to the article.
Log in
Already have access? Please log in.


or
Log in with your institution

Journal + Issues

Search