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

We consider operators T satisfying a sparse domination property | ⟨ T f , g ⟩ | ≤ c ∑ Q ∈ S ⟨ f ⟩ p 0 , Q ⟨ g ⟩ q 0 ' , Q | Q | with averaging exponents 1 ≤ p 0 < q 0 ≤ ∞ . We prove weighted strong type boundedness for p 0 < p < q 0 and use new techniques to prove weighted weak type ( p 0 , p 0 ) boundedness with quantitative mixed A 1 - A ∞ estimates, generalizing results of Lerner, Ombrosi, and Pérez and Hytönen and Pérez. Even in the case p 0 = 1 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 ( q 0 ' , q 0 ' ) 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.