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

Abstract

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.

Sparse domination implies vector-valued sparse domination

“…In this theorem a localized r -estimate is imposed on T to deduce (6.3) with q 0 = r . The localized r -estimate for T becomes weaker for smaller r , so [35,Theorem 3.5] also yields the result of Conjecture 6.2. • One of the main results in [10] is (6.3) with q 0 = 1 for rough homogeneous singular operators T , see also [27] for an alternative proof.…”

Operator-free sparse domination

“…Our key novel point is the language in which our main results are written. This language unifies (1.1) with all of the results contained in [37, 39]. More important, it allows us to deal with many non-operator objects, which have not yet been investigated using sparse domination techniques.…”

“…Primarily motivated by sharp quantitative weighted norm inequalities, sparse domination has quickly transformed into a very active area, dealing with various operators within and beyond the Calderón–Zygmund theory. During the last five years, a number of sparse domination principles (that is, general results establishing sparse domination for a given class of operators) have appeared, for example, in the works [2, 5, 6, 8, 13, 14, 34, 35, 37, 39].…”

“…The proof of the pointwise sparse domination result for T was subsequently simplified by the first author in [34] and the first and third authors in [37], in which a general sparse domination principle was established, allowing one to deal with a vast number of ‘smooth’ operators. The main result of [37] was then extended by the second author [39] into several directions, including the setting of vector-valued functions on spaces of homogeneous type, along with the concept of -sparse domination.…”

“…In the present article, we return sparse domination to its roots, using functions rather than operators. We will essentially use the techniques developed in [37, 39]. Our key novel point is the language in which our main results are written.…”