A.e. convergence of anisotropic partial Fourier integrals on Euclidean spaces and Heisenberg groups

Volume 118 / 2010

D. Müller, E. Prestini Colloquium Mathematicum 118 (2010), 333-347 MSC: 22E30, 43A50. DOI: 10.4064/cm118-1-18

Abstract

We define partial spectral integrals $S_R$ on the Heisenberg group by means of localizations to isotropic or anisotropic dilates of suitable star-shaped subsets $V$ containing the joint spectrum of the partial sub-Laplacians and the central derivative. Under the assumption that an $L^2$-function $f$ lies in the logarithmic Sobolev space given by $\log(2+L_\alpha)f\in L^2,$ where $L_\alpha$ is a suitable “generalized” sub-Laplacian associated to the dilation structure, we show that $S_Rf(x)$ converges a.e. to $f(x)$ as $R\to\infty.$

Authors

  • D. MüllerMathematisches Seminar
    C.A.-Universität Kiel
    Ludewig-Meyn-Str. 4
    D-24098 Kiel, Germany
    e-mail
  • E. PrestiniDipartimento di Matematica
    Università di Roma “Tor Vergata”
    Via della Ricerca Scientifica
    00133 Roma, Italy
    e-mail

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