From restricted type to strong type estimates on quasi-Banach rearrangement invariant spaces, preprint

Abstract. Let X be a quasi-Banach rearrangement invariant space and let T be a (ε, δ)-atomic operator for which a restricted type estimate of the form ‖TχE‖X ≤ D(|E|) for some positive function D and every measurable set E is known. Then, we show that this estimate can be extended to the set of all positive functions f ∈ L1 such that ‖f‖ ∞ ≤ 1, in the sense that ‖Tf‖X ≤ D(‖f‖1). This inequality allows us to obtain strong type estimates for T on several classes of spaces as soon as some information about the Galb of the space X is known. This paper will consider the case of weighted Lorentz spaces X = Λq(w) and their weak version. 1