Sparse domination and the strong maximal function

Abstract

We study the problem of dominating the dyadic strong maximal function by (1, 1)-type sparse forms based on rectangles with sides parallel to the axes, and show that such domination is impossible. Our proof relies on an explicit construction of a pair of maximally separated point sets with respect to an appropriately defined notion of distance.

Singular Brascamp–Lieb inequalities with cubical structure

“…While the classical strong 2 m−1 -maximal operator, which is the case α = 0 of M α , does not satisfy a weak type 2 m−1 bound, the operator M α with α > 0 does, as we will show in Lemma 3.1. Related to this we also recall that the classical strong maximal operator cannot be sparsely dominated [1], while the result of Theorem 1.3 with α > 0 allows us to obtain the sparse bounds of Theorem 1.2.…”

Two-weight commutator estimates: general multi-parameter framework

“…For example, let m = 3, w = 1, I 1 = {1, 2}, and I 2 = {3}. Then b ∈ bmo {{1,2},{3}} if b ∈ BMO (1,3) and b ∈ BMO (2,3) . Remark 2.7.…”

“…Here the recent progress is most often based on the so-called representation theorems as sparse domination methods essentially currently work in oneparameter only (although see ). In fact, Barron-Conde-Alonso-Ou-Rey ( [1]) show that one of the simplest bi-parameter model operators -the dyadic bi-parameter maximal function -cannot satisfy the most natural or useful candidate for bi-parameter sparse domination. A representation theorem represents SIOs by some dyadic model operators (DMOs).…”

Product Space Singular Integrals with Mild Kernel Regularity

“…the one-parameter Beurling kernel 1/(x − y) 2 with the bi-parameter kernel 1/[(x 1 − y 1 )(x 2 − y 2 )]the product of Hilbert kernels in both coordinate directions. In general, the product space analysis is quite different from one-parameter analysis and seems to resist many techniques-in part due to the failure of bi-parameter sparse domination methods, see [3] (see also [4] however), representation theorems are even more important in bi-parameter than in one-parameter. For example, the dyadic representation methods have proved very fruitful in connection with bi-parameter commutators and weighted analysis, see Holmes-Petermichl-Wick [19], Ou-Petermichl-Strouse [36] and [28].…”

“…Then we have[T 1 , [T 2 , b]] L p (μ)→L p (λ) b bmo I (ν) .Here bmo I (ν) is the following weighted little product BMO space:b bmo I (ν) = sup ū b BMO ū prod (ν) ,where ū = (u i ) 2 i=1 is such that u i ∈ I i and BMO ū prod (ν) is the natural weighted bi-parameter product BMO space on the parameters ū. For example, b BMO(1,3) prod (ν) := supx 2 ∈R d 2 ,x 4 ∈R d 4 b(•, x 2 , •, x 4 ) BMO prod (ν(•,x 2 ,•,x 4 )) ,…”