An estimate for bounded solutions of the Hermite heat equation

Abstract

An estimate result on the partial derivatives of the Mehler kernel E(x; ;t ) for t > 0 is rst established. Particularly for 0 < t < 1, it extends the estimate result given by S. Thangavelu in his monograph A lecture notes on Hermite and Laguerre expansions on the order of the partial derivative of the Mehler kernel with respect to the space variable. Furthermore, for each m 2 N0, a growth estimate on the partial derivative @ m U(x;t) @xm of all bounded solutions U(x;t) of the Cauchy Dirichlet problem for the Hermite heat equation is established.