On pointwise $\ell^r$-sparse domination in a space of homogeneous type

Emiel Lorist

Published 2019-07-01Version 1

We prove a general sparse domination theorem in spaces of homogeneous type, in which we control a vector-valued operator pointwise by a positive, local expression called a sparse operator. We use the structure of the operator to get sparse domination in which the $\ell^1$-sum in the sparse operator is replaced by an $\ell^r$-sum. This sparse domination theorem is applicable to many operators from both harmonic analysis and PDE and yields simple and unified proofs for the (sharp) weighted $L^p$-boundedness of these operators. As an illustration of the versatility of our theorem we prove the $A_2$-theorem for vector-valued Calder\'on--Zygmund operators in a space of homogeneous type and sharp weighted norm inequalities for Littlewood--Paley operators and both the lattice Hardy--Littlewood and the Rademacher maximal operator.