Lp-Lq ESTIMATES FOR SOME CONVOLUTION OPERATORS WITH SINGULAR MEASURES ON THE HEISENBERG GROUP

We consider the Heisenberg group H-n = C-n x R. Let v be the Borel measure on H-n defined by v(E) = integral C-n chi(E)(omega, phi(omega))eta(omega) = d omega, where phi(omega) = (Sigma(n)(j=1) a(j) vertical bar omega(j)vertical bar(2),omega = (omega(1), ..., omega(n)) is an element of C-n, a(j) is an element of R, and eta(omega) = eta(0) (vertical bar omega vertical bar(2)) eta(0) is an element of C-c(infinity)(R). We characterize the set of pairs (p, q) such that the convolution operator with v is L-p(H-n)-L-q(H-n) bounded. We also obtain L-p-improving properties of measures supported on the graph of the function phi(omega) = vertical bar omega vertical bar(2m).