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

Abstract

The two‐weight inequality for the Hilbert transform is characterized for an arbitrary pair of positive Radon measures σ and w on double-struckR. In particular, the possibility of common point masses is allowed, lifting a restriction from the recent solution of the two‐weight problem by Lacey, Sawyer, Shen, and Uriarte‐Tuero. Our characterization is in terms of Sawyer‐type testing conditions and a variant of the two‐weight A2 condition, where σ and w are integrated over complementary intervals only. A key novelty of the proof is a two‐weight inequality for the Poisson integral with ‘holes’.

Bilinear Calderón–Zygmund theory on product spaces

“…It is possible to first prove a representation theorem in a certain finite set up, where no a priori boundedness is needed. Reductions of this type in the linear one-parameter situation appear in [28] and [35]. We omit the technical details in our setting as they are similar.…”

The $\theta $-bump theorem for product fractional integrals

“…We also define D bad ≡ D \ D good . Parameterizations of dyadic grids: Here we recall a construction from [SaShUr10] that was in turn based on that of Hytönen in [Hyt2]. Momentarily fix a large positive integer M ∈ N, and consider the tiling of R by the family of intervals D M ≡ I M α α∈Z having side length 2 −M and given by…”

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

“…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.…”

“…For example, Cotlar and Sadosky gave a beautiful function theoretic characterization of the weight pairs (σ, ω) for which H is bounded from L 2 (R; σ) to L 2 (R; ω), namely a two-weight extension of the Helson-Szegö theorem, which illuminated a deep connection between two quite different function theoretic conditions, but failed to shed much light on when either of them held 1 . On the other hand, the two weight inequality for positive fractional integrals, Poisson integrals and maximal functions were characterized using testing conditions by one of us in [Saw] (see also [Hyt2] for the Poisson inequality with 'holes') and [Saw1], but relying in a very strong way on the positivity of the kernel, something the Hilbert kernel lacks. In a groundbreaking series of papers including [NTV1], [NTV2] and [NTV4], Nazarov, Treil and Volberg used weighted Haar decompositions with random grids, introduced their 'pivotal' condition, and proved that the Hilbert transform is bounded from L 2 (R; σ) to L 2 (R; ω) if and only if a variant of the A 2 condition 'on steroids' held, and the norm inequality and its dual held when tested locally over indicators of cubes -but only under the side assumption that their pivotal conditions held.…”

“…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.…”