Weak and Strong Type A 1 –A ∞ Estimates for Sparsely Dominated Operators

We consider operators T satisfying a sparse domination property (Formula presented.)with averaging exponents (Formula presented.). We prove weighted strong type boundedness for (Formula presented.) and use new techniques to prove weighted weak type (Formula presented.) boundedness with quantitative mixed (Formula presented.)–(Formula presented.) estimates, generalizing results of Lerner, Ombrosi, and Pérez and Hytönen and Pérez. Even in the case (Formula presented.) we improve upon their results as we do not make use of a Hörmander condition of the operator T. Moreover, we also establish a dual weak type (Formula presented.) estimate. In a last part, we give a result on the optimality of the weighted strong type bounds including those previously obtained by Bernicot, Frey, and Petermichl.