Weak and Strong Type [Formula: see text] –[Formula: see text] Estimates for Sparsely Dominated Operators

We consider operators T satisfying a sparse domination property [Formula: see text] with averaging exponents [Formula: see text] . We prove weighted strong type boundedness for [Formula: see text] and use new techniques to prove weighted weak type [Formula: see text] boundedness with quantitative mixed [Formula: see text] –[Formula: see text] estimates, generalizing results of Lerner, Ombrosi, and Pérez and Hytönen and Pérez. Even in the case [Formula: see text] 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: see text] 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.