Weak and Strong Type $$A_1$$–$$A_\infty $$ Estimates for Sparsely Dominated Operators

Abstract

AbstractWe consider operators T satisfying a sparse domination property $$\begin{aligned} |\langle Tf,g\rangle |\le c\sum _{Q\in \mathscr {S}}\langle f\rangle _{p_0,Q}\langle g\rangle _{q_0',Q}|Q| \end{aligned}$$ | ⟨ T f , g ⟩ | ≤ c ∑ Q ∈ S ⟨ f ⟩ p 0 , Q ⟨ g ⟩ q 0 ′ , Q | Q | with averaging exponents $$1\le p_0<q_0\le \infty $$ 1 ≤ p 0 < q 0 ≤ ∞ . We prove weighted strong type boundedness for $$p_0<p<q_0$$ p 0 < p < q 0 and use new techniques to prove weighted weak type $$(p_0,p_0)$$ ( p 0 , p 0 ) boundedness with quantitative mixed $$A_1$$ A 1 –$$A_\infty $$ A ∞ estimates, generalizing results of Lerner, Ombrosi, and Pérez and Hytönen and Pérez. Even in the case $$p_0=1$$ 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 $$(q_0',q_0')$$ ( 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.