Weak and Strong Type - Estimates for Sparsely Dominated Operators

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