Some Remarks on the Pointwise Sparse Domination

Abstract

We obtain an improved version of the pointwise sparse domination principle established by the first author in [19]. This allows us to determine nearly minimal assumptions on a singular integral operator T for which it admits a sparse domination.where f p p,Q = 1 |Q| Q |f | p , and S is a sparse family of cubes of R n . Recall that the family of cubes S is called η-sparse, 0 < η ≤ 1, if for every cube Q ∈ S, there exists a measurable set E Q ⊂ Q such that |E Q | ≥ η|Q|, and the sets {E Q } Q∈S are pairwise disjoint.Localization and sparseness are two main ingredients which make sparse bounds especially effective in quantitative weighted norm inequalities.The literature about sparse bounds is too extensive to be given here in more or less adequate form. We mention only that sparse bounds for Calderón-Zygmund operators can be found in [7,15,16,18,19,20]. Also, there are several general sparse domination principles [5,6,8,19].In [19], a sparse domination principle was obtained in terms of the grand maximal truncated operatordefined for a given operator T .2010 Mathematics Subject Classification. 42B20, 42B25.

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

“…This sparse domination principle was further generalized in the recent paper [58] by Lerner and Ombrosi, in which the authors showed that the weak L p 2 -boundedness of the more flexible operator…”

“…The main motivation to generalize the results in [58] comes from the application in the recent work [64] by Veraar and the author, in which Calderón-Zygmund theory is developed for stochastic singular integral operators. In particular, in [64,Theorem 6.4] Theorem 1.1 is applied with p 1 = p 2 = r = 2 to prove a stochastic version of the vector-valued A 2 -theorem for Calderón-Zygmund operators, which yields new results in the theory of maximal regularity for stochastic partial differential equations.…”

Iterated commutators under a joint condition on the tuple of multiplying functions

“…This leads us to consider joint conditions involving Young functions that can be made strictly weaker than S 2+ε + T 2+ε for all ε > 0. Hence, we prove the commutator upper bound with these updated conditions with a version of the sparse domination principle introduced in Lerner and Ombrosi [10].…”

“…Next, our focus will be on Calderón-Zygmund operators satisfying the Dini condition. We begin with partially recalling, with only minor modifications, a sparse domination from Lerner [9] (also see Lerner, Ombrosi [10]). See also Ibánez-Firnkorn -Rivera-Ríos [7].…”

“…refined argument for the sparse domination in Theorem 4.5 without Lemma 4.4, see the latest version of the sparse domination principle in Lerner, Ombrosi[10].…”