Pointwise convergence to initial data of heat and Laplace equations

Abstract

Let $L$ be either the Hermite or the Ornstein-Uhlenbeck operator on $\mathbb {R}^d$. We find optimal integrability conditions on a function $f$ for the existence of its heat and Poisson integrals, $e^{-tL}f(x)$ and $e^{-t\sqrt L}f(x)$, solutions respectively of $U_t = -LU$ and $U_{tt} = LU$ on $\mathbb {R}^{d+1}_+$ 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 $f\in L^p(v)$, and each $p\in [1,\infty )$. Moreover, if $1\!<\!p\!<\!\infty$ we additionally show that for such weights the associated local maximal operators are strongly bounded from $L^p(v)\to L^p(u)$ for some other weight $u(x)$.