No abstract

We consider real interpolation methods defined by means of slowly varying functions and rearrangement invariant spaces, for which we present a collection of reiteration theorems for interpolation and extrapolation spaces. As an application we obtain interpolation formulas for Lorentz-Karamata type spaces, for Zygmund spaces LlogL, L exp and for the grand and small Lebesgue spaces.

We find the sharp range of boundedness for transplantation operators associated with Laguerre function expansions in L p spaces with power weights. Namely, the operators interchangingThis improves a previous partial result by Stempak and Trebels, which was only sharp for ρ 0. Our approach is based on new multiplier estimates for Hermite expansions, weighted inequalities for local singular integrals and a careful analysis of Kanjin's original proof of the unweighted case. As a consequence we obtain new results on multipliers, Riesz transforms and g-functions for Laguerre expansions in L p (y δp ).

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).