A.e. convergence of anisotropic partial Fourier integrals on Euclidean spaces and Heisenberg groups
Volume 118 / 2010
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 $R\to\infty.$
