  We consider operators $T$ satisfying a sparse domination property \[ |\langle
Tf,g\rangle|\leq c\sum_{Q\in\mathscr{S}}\langle f\rangle_{p_0,Q}\langle
g\rangle_{q_0',Q}|Q| \] with averaging exponents $1\leq p_0<q_0\leq\infty$.
  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 P\'erez and Hyt\"onen and P\'erez. Even in the case $p_0=1$ we
improve upon their results as we do not make use of a H\"ormander 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.
