A two weight theorem for $\alpha$-fractional singular integrals with an energy side condition

Abstract

Let σ and ω be locally finite positive Borel measures on R n (possibly having common point masses), and let T α be a standard α-fractional Calderón-Zygmund operator on R n with 0 ≤ α < n. Suppose that Ω : R n → R n is a globally biLipschitz map, and refer to the images ΩQ of cubes Q as quasicubes. Furthermore, assume as side conditions the A α 2 conditions, punctured A α 2 conditions, and certain α-energy conditions taken over quasicubes. Then we show that T α is bounded from L 2 (σ) to L 2 (ω) if the quasicube testing conditions hold for T α and its dual, and if the quasiweak boundedness property holds for T α .Conversely, if T α is bounded from L 2 (σ) to L 2 (ω), then the quasitesting conditions hold, and the quasiweak boundedness condition holds. If the vector of α-fractional Riesz transforms R α σ (or more generally a strongly elliptic vector of transforms) is bounded from L 2 (σ) to L 2 (ω), then both the A α 2 conditions and the punctured A α 2 conditions hold. We do not know if our quasienergy conditions are necessary when n ≥ 2, except for certain situations in which one of the measures is one-dimensional, or both measures are sufficiently dispersed.

A Good-λ Lemma, Two Weight T1 Theorems Without Weak Boundedness, and a Two Weight Accretive Global Tb Theorem

“…Note that these definitions are the analogues of the corresponding conditions with 'holes' introduced by Hytönen [Hyt2] in dimension n = 1 -the supports of the measures 1 Q c σ and 1 Q ω in the definition of A α 2 are disjoint, and so the common point masses of σ and ω do not appear simultaneously in each factor. Note also that, unlike in [SaShUr5], where common point masses were not permitted, we can no longer assert the equivalence of A α 2 with holes taken over quasicubes with A α 2 with holes taken over cubes.…”

“…Good/bad technology. First we recall the good/bad cube technology of Nazarov, Treil and Volberg [Vol] as in [SaShUr7], but with a small simplification introduced in the real line by Hytönen in [Hyt2]. This simplification does not impact the validity of the arguments in [SaShUr6], but will facilitate the use of NTV surgery in later subsections.…”

“…The extension to higher dimensions we use here is that of 'strong quasienergy condition' defined in [SaShUr9] and recalled below. We define a quasicube K (not necessarily in ΩD) to be an alternate ΩD-quasicube if it is a union of 2 n ΩD-quasicubes K ′ with side length ℓ (K ′ ) = 1 2 ℓ (K) (such quasicubes were called shifted in [SaShUr5], but that terminology conflicts with the more familiar notion of shifted quasigrid). Thus for any ΩD-quasicube L there are exactly 2 n alternate ΩD-quasicubes of twice the side length that contain L, and one of them is of course the ΩD-parent of L. We denote the collection of alternate ΩD-quasicubes by AΩD.…”

The boundedness of multilinear Calderón-Zygmund operators on weighted and variable Hardy spaces

“…In the linear case, fractional Calderón-Zygmund operators have been studied by a number of authors. See, for instance, the recent papers [17,18].…”

The two-weight inequality for the Hilbert transform with general measures

“…Incidentally, an condition of a similar nature has appeared in the work of Belov, Mengestie and Seip [, equation (1.5)] on discrete Hilbert transforms on sparse sequences in the plane, and we strongly feel that working with this type of weight conditions, rather than or its stronger versions, will be instrumental in the further development of the two‐weight theory. Indeed, following the circulation of the present paper, this framework and some related techniques have already been adopted to study a number of higher dimensional extensions of the two‐weight inequalities in .…”