Some New Light on a Few Classical Results

The purpose of this paper is to describe a unified approach to proving vector-valued inequalities without relying on the full strength of weighted theory. Our applications include the Fefferman-Stein and Cordoba-Fefferman inequalities, as well as the vector-valued Carleson operator. Using this approach we also produce a proof of the boundedness of the classical bi-parameter multiplier operators, that does not rely on product theory. Our arguments are inspired by the vector valued restricted type interpolation used in [1]. 1. The general principle In this paper we describe an alternative approach to a few well known vector-valued inequalities. One of them leads to an alternative way to estimate bi-parameter linear operators. This approach has already played a crucial role in recent work in the linear setting [1] but also in the context of bilinear operators [13], where weighted estimates were not available. At its core lies restricted type vector valued interpolation as encoded by the following principle: Theorem 1.1 (The general principle [1]). Let p0, p1 ∈ (1,∞) be such that p0 < p1 and let {Tj}j be a (possibly finite) sequence of sublinear operators on R n which are uniformly bounded on L. Assume that for p ∈ {p0, p1} there is Cp > 0 with the following property: (P ) for each finite nonzero measure sets H,G ⊂ R there exist subsets H ′ ⊂ H and G ⊂ G with |H | ≥ 1 2 |H|, |G| ≥ 1 2 |G|, (1) such that ∫ |Tj(f1H′)| 1G′ ≤ Cp ( |G| |H| )1− 2 p ∫ |f | (2) for each j and each f ∈ L(R). Then ‖( ∑ j |Tjfj |) ‖q .q ‖( ∑