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

We consider operators T satisfying a sparse domination property $\begin{array}{c}\left|⟨Tf,g⟩\right|\le c\sum _{Q\in S}{⟨f⟩}_{p_0,Q}{⟨g⟩}_{q_0^{\text{'}},Q}\left|Q\right|\end{array}$ with averaging exponents $1\le p_0<q_0\le \infty$ . 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 $A_1$ - $A_{\infty }$ 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 $(q_0^{\text{'}},q_0^{\text{'}})$ 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.