Pointwise convergence to initial data of heat and Laplace equations

Abstract

Let L be either the Hermite or the Ornstein-Uhlenbeck operator on Rd. We find optimal integrability conditions on a function f for the existence of its heat and Poisson integrals, e−tLf(x) and e−t √Lf(x), solutions respectively of Ut = −LU and Utt = LU on Rd+1 + with initial datum f. As a consequence we identify the most general class of weights v(x) for which such solutions converge a.e. to f for all f ∈ Lp(v), and each p ∈ [1,∞). Moreover, if 1 <p< ∞ we additionally show that for such weights the associated local maximal operators are strongly bounded from Lp(v) → Lp(u) for some other weight u(x).