On weak-type $(1,\,1)$ for averaging type operators

Abstract

It is known that, due to the fact that L 1,∞ is not a Banach space, if (Tj)j is a sequence of bounded operators so thatwith norm less than or equal to ||Tj || and j ||Tj || < ∞, nothing can be said about the operator T = j Tj . This is the origin of many difficult and open problems. However, if we assume thatwith norm less than or equal to ϕ(||u||A 1 )||Tj ||, where ϕ is a nondecreasing function and A1 the Muckenhoupt class of weights, then we prove that, essentially,, ∀u ∈ A1. We shall see that this is the case of many interesting problems in Harmonic Analysis.