Weighted weak-type (1, 1) estimates for radial Fourier multipliers via extrapolation theory

A sparse domination principle for rough singular integrals

“…The separation condition (2.3) tells us that if the 7-fold dilates of two cubes L, L ′ belonging to the same stopping collection intersect nontrivially, then L, L ′ must be neighbors. We also recall the notation L for the 2 5 -fold dilate of L. I : let f j ∈ L p j (R d ), j = 1, 2, with compact support be xed. By suitably choosing the dyadic lattice D, we may nd Q 0 ∈ D such that supp f 1 ⊂ Q 0 , supp f 2 ⊂ 3Q 0 and s Q 0 is larger than the largest nonzero scale occurring in the kernel.…”

“…For the singular integrals (1.2) no sparse domination results were known prior to this article, although some quantitative weighted estimates were established in the recent works [17,28]; see below for details. For the Bochner-Riesz means (1.3), the recent results of [1] and [5] are far from being optimal at the critical exponent.…”

On weak-type $(1,\,1)$ for averaging type operators

“…Thereby, in virtue of Theorem 1.6, we completely answer the open question formulated in [7] about the restricted weak-type boundedness of B n−1 2 . Corollary 4.2.…”

“…Same estimates can be obtained for a large list of operators such as those appearing in [2,7,9]: rough operators, Hörmander multipliers, radial Fourier multipliers, square functions, etc.…”

“…, introduced by S. Bochner in [4] and defined as follows (see [7] for some partial results in this context): let a + = max{a, 0} denote the positive part of a ∈ R and given λ > 0, the Bochner-Riesz operator B λ on R n is defined by 28,35]). For every n > 1,…”