Related papers: A Two Weight Inequality for the Hilbert transform …

We consider the two weight problem for the Hilbert transform, namely the question of finding real-variable characterization of those pair of weights for which the Hilbert transform acts boundedly on $ L ^2 $ of the weights. Such a…

A conjecture of Nazarov--Treil--Volberg on the two weight inequality for the Hilbert transform is verified. Given two non-negative Borel measures u and w on the real line, the Hilbert transform $H_u$ maps $L^2(u)$ to $L^2(w)$ if and only if…

Recent work of Lacey-Sawyer-Shen-Uriarte-Tuero and Lacey have established a conjecture of Nazarov-Treil-Volberg, giving a real-variable characterization of the two weight inequality for the Hilbert transform, provided the pair of weights do…

The two weight inequality for the Hilbert transform arises in the settings of analytic function spaces, operator theory, and spectral theory, and what would be most useful is a characterization in the simplest real-variable terms. We show…

In the case (4/3)<p<4, and assuming a pair of locally finite positive Borel measures on the real line have no common point masses, we prove variants of two conjectures of T. Hyt\"onen and E. Vuorinen from 2018 on two weight testing theorems…

As a corollary to our main theorem we give a new proof of the result that the norm of the Hilbert transform on L^2(w) has norm bounded by a the A_2 characteristic of a weight to the first power, a theorem of one of us. This new proof begins…

The two-weight inequality for the Hilbert transform is characterized for an arbitrary pair of positive Radon measures $\sigma$ and $w$ on $\mathbb R$. In particular, the possibility of common point masses is allowed, lifting a restriction…

This article was written in 2005 and subsequently lost (at least by the third author). Recently it resurfaced due to one of the colleagues to whom a hard copy has been sent in 2005. We consider here a problem of finding necessary and…

We characterize two-weight inequalities for certain maximal truncations of the Hilbert transform in terms of testing conditions on simpler functions. For 1<p<2 and two positive Borel measures u, v on R, we assume that u is doubling, and we…

We provide an essentially complete dictionary of all implications among the basic and fundamental conditions in weighted theory such as the doubling, one weight A_p(w), A_\infty and C_p conditions as well as the two weight A_p and the…

In proving the local $T_b$ Theorem for two weights in one dimension [SaShUT] Sawyer, Shen and Uriarte-Tuero used a basic theorem of Hyt\"{o}nen [Hy] to deal with estimates for measures living in adjacent intervals. Hyt\"{o}nen's theorem…

This paper is a sequel to our paper Rev. Mat. Iberoam. 32 (2016), no. 1, 79-174. Let T be a standard fractional Calderon Zygmund operator. Assume appropriate Muckenhoupt and quasienergy side conditions. Then we show that T is bounded from…

We prove a two weight theorem for fractional singular integrals in higher dimensions assuming energy side conditions on the weights. The testing conditions are taken over quasicubes, namely globally biLipschitz images of cubes.

Let $(X,d,\mu )$ be a space of homogeneous type in the sense of Coifman and Weiss, i.e. $d$ is a quasi metric on $X$ and $\mu $ is a positive measure satisfying the doubling condition. Suppose that $u$ and $v$ are two locally finite…

We prove that the energy conditions in arXiv:1302.5093v7 are implied by the fractional A2 conditions and testing conditions for the vector of fractional Riesz transforms when one measure is supported on a line. Then we apply the main…

Multiple methods for deriving the energy-momentum tensor for a physical theory exist in the literature. The most common methods are to use Noether's first theorem with the 4-parameter Poincar\'{e} translation, or to write the action in a…

We extend our previous work in arXiv:1302.5093v10 to obtain a T1 theorem with an energy side condition that allows for common point masses.

We prove a T1 theorem for fractional vector Riesz transforms mapping one weighted Sobolev space to another, where the weights are doubling measures on Euclidean space. Boundedness is characterized by the classical A_2 condition and two dual…

We give a necessary and sufficient condition for the two weight $L^p$-estimates for paraproducts in non-homogeneous settings, $1<p<\infty$. We are mainly interested in the case $p\ne 2$, since the case $p=2$ is a well-known and easy…

For statistical inference on an infinite-dimensional Hilbert space $\H $ with no moment conditions we introduce a new class of energy distances on the space of probability measures on $\H$. The proposed distances consist of the integrated…