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

Abstract

It is known that, due to the fact that L1,∞ is not a Banach space, if (Tj)j is a sequence of bounded operators so thatTj:L1⟶L1,∞, with norm less than or equal to ||Tj|| and ∑j||Tj||<∞, nothing can be said about the operator T=∑jTj. This is the origin of many difficult and open problems. However, if we assume thatTj:L1(u)⟶L1,∞(u),∀u∈A1, with norm less than or equal to φ(||u||A1)||Tj||, where φ is a nondecreasing function and A1 the Muckenhoupt class of weights, then we prove that, essentially,T:L1(u)⟶L1,∞(u),∀u∈A1. We shall see that this is the case of many interesting problems in Harmonic Analysis.