AE CONVERGENCE OF ANISOTROPIC PARTIAL FOURIER INTEGRALS ON EUCLIDEAN SPACES AND HEISENBERG GROUPS

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 is an element of L(2), where L(alpha) is a suitable "generalized" sub-Laplacian associated to the dilation structure, we show that S(R)f(x) converges a.e. to f(x) as R -> infinity.