Let L be either the Hermite or the Ornstein-Uhlenbeck operator on 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 √ L f (x), solutions respectively of Ut = −LU and Utt = LU on 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 ∈ L p (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 L p (v) → L p (u) for some other weight u(x).

Let 0 < γ < 1, b be a BMO function and I m γ,b the commutator of order m for the fractional integral. We prove two type of weighted L p inequalities for I m γ,b in the context of the spaces of homogeneous type. The first one establishes that, for A ∞ weights, the operator I m γ,b is bounded in the weighted L p norm by the maximal operator M γ (M m ), where M γ is the fractional maximal operator and M m is the HardyLittlewood maximal operator iterated m times. The second inequality is a consequence of the first one and shows that the operator, where [(m + 1)p] is the integer part of (m + 1)p and no condition on the weight w is required. From the first inequality we also obtain weighted L p -L q estimates for I m γ,b generalizing the classical results of Muckenhoupt and Wheeden for the fractional integral operator.

We characterize the weighted Lebesgue spaces, L p (R n , v(x)dx), for which the solutions of the Heat and Poisson problems have limits a.e. when the time t tends to zero.

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