Elementary reverse Hölder type inequalities with application to operator interpolation theory

Abstract

Abstract. We give a very elementary proof of the reverse Hölder type inequality for the classes of weights which characterize the boundedness on L p of the Hardy operator for nonincreasing functions. The same technique is applied to Calderón operator involved in the theory of interpolation for general Lorentz spaces. This allows us to obtain further consequences for intermediate interpolation spaces. IntroductionAriño and Muckenhoupt characterized the class of weights, ω, such that the Hardy operator is bounded on L p (ω) for nonnegative and nonincreasing functions (see [AM]). This class, say (AM ) p , is composed of those weights for which there is a constant C > 0 such that for every t > 0(see also [L] for an earlier version of this formula). A crucial step in their proof is the following reverse inequality: if ω ∈ (AM ) p , then ω ∈ (AM ) p− for some > 0.In this note we will see that this fact, the reverse type inequality, has a very elementary proof by means of some reiteration procedure. Furthermore, we will show how similar ideas can be applied in the context of operator interpolation theory for Lorentz spaces, in order to prove that weak type interpolation spaces are the same as restricted weak type interpolation ones. These concepts require some notations which will be introduced below but, briefly, in a very particular case, we say that a rearrangement invariant Banach space, X, is a weak