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

It is known that, due to the fact that L1,infinity is not a Banach space, if (Tj)j is a sequence of bounded operators so that Tj : L1--> L1,infinity, with norm less than or equal to ||Tj|| and E nothing can be said about the operator T = j ||Tj|| < infinity, j Tj. This is the origin of many difficult and open problems. However, if we assume that Tj : L1(u) --> L1,infinity(u), for all u is an element of A1, with norm less than or equal to phi(||u||A1)||Tj||, where phi is a nondecreasing function and A1 the Muckenhoupt class of weights, then we prove that, essentially, T : L1(u) --> L1,infinity(u), for all u is an element of A1. We shall see that this is the case of many interesting problems in Harmonic Analysis. (c) 2023 Elsevier Inc. All rights reserved.