Weighted Weak Type (1,1) Bounds for Rough Operators

In this work, we study three, by now classical, operators from the point of view of their weak type (1,1) behaviour with respect to weights. The first one is the Bochner‐Riesz multiplier operator of index 1 2 ( n − 1). We show that it is of weak type (1,1) with respect to any weight in the A 1 ‐Muckenhoupt class. The other two are the maximal operator and the homogeneous singular integral (the former in any dimension and the latter in dimension 2) with rough kernel. We prove that they are of weak‐type (1,1) with respect to weights( ω ,[ ℳ ~ ( ω β ) ]1 / β)for β > 1, whereℳ ~is a composition of the Hardy‐Littlewood maximal operator and the maximal operator with rough kernel.