Pointwise Convergence to Initial Data of Heat and Poisson Equations

TERESA, SIGNES; HATZSTEIN, SILVIA; BEATRIZ, VIVIANI; GARRIGOS, GUSTAVO; JOSE LUIS, TORREA

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY

AMER MATHEMATICAL SOC

Lugar: Providence; Año: 2016 vol. 368 p. 6575 - 6600

0002-9947

Let $L$ be either the Hermite or the Ornstein-Uhlenbeck operator on $SRd$. We find optimal integrability conditions on the initial data $f$ for the existence of the solutions $e^{-tL}f(x)$ and $e^{-tsqrt L}f(x)$, of the heat and Laplace equations $U_t = -LU$ and $U_{tt} = LU$ on $SR^{d+1}_+$. As a consequence we identify the most general class of weights $v(x)$ for which such solutions converge a.e. to $f$ for all $fin L^p(v)$, and each $pin[1,infty)$. Moreover, if $1!