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

D. Müller ¹   ; E. Prestini ²

¹ Mathematisches Seminar C.A.-Universität Kiel Ludewig-Meyn-Str. 4 D-24098 Kiel, Germany
² Dipartimento di Matematica Università di Roma “Tor Vergata” Via della Ricerca Scientifica 00133 Roma, Italy

Colloquium Mathematicum, Tome 118 (2010) no. 1, pp. 333-347

Résumé

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 $R\to\infty.$

Détail

DOI : 10.4064/cm118-1-18

Keywords: define partial spectral integrals heisenberg group means localizations isotropic anisotropic dilates suitable star shaped subsets containing joint spectrum partial sub laplacians central derivative under assumption function lies logarithmic sobolev space given log alpha where alpha suitable generalized sub laplacian associated dilation structure converges infty