On Pointwise $$\ell ^r$$-Sparse Domination in a Space of Homogeneous Type

We prove a general sparse domination theorem in a space of homogeneous type, in which a vector-valued operator is controlled pointwise by a positive, local expression called a sparse operator. We use the structure of the operator to get sparse domination in which the usual $$\ell ^1$$ ℓ 1 -sum in the sparse operator is replaced by an $$\ell ^r$$ ℓ r -sum. This sparse domination theorem is applicable to various operators from both harmonic analysis and (S)PDE. Using our main theorem, we prove the $$A_2$$ A 2 -theorem for vector-valued Calderón–Zygmund operators in a space of homogeneous type, from which we deduce an anisotropic, mixed-norm Mihlin multiplier theorem. Furthermore, we show quantitative weighted norm inequalities for the Rademacher maximal operator, for which Banach space geometry plays a major role.