Related papers: Local Tb theorem with L2 testing conditions and ge…

We announce a local $T(b)$ theorem, an inductive scheme, and $L^p$ extrapolation results for $L^2$ square function estimates related to the analysis of integral operators that act on Ahlfors-David regular sets of arbitrary codimension in…

Let $X$ be a space of homogeneous type and let $L$ be a sectorial operator with bounded holomorphic functional calculus on $L^2(X)$. We assume that the semigroup $\{e^{-tL}\}_{t>0}$ satisfies Davies-Gaffney estimates. Associated to $L$ are…

In the present work we extend a local Tb theorem for square functions of Christ and Hofmann to the multilinear setting. We also present new BMO type interpolation result for square functions associated to multilinear operators. These square…

Let $\mu$ be a Borel measure on $R^d$ which may be non doubling. The only condition that $\mu$ must satisfy is $\mu(B(x,r))\leq C r^n$ for all $x\in R^d$, $r>0$, and for some fixed $0<n\leq d$. In this paper, we develop Littlewood-Paley…

We show that any Littlewood--Paley square function $S$ satisfying a minimal local testing condition is dominated by a sparse form, \begin{equation*} \langle (Sf)^2,g \rangle\le C \sum_{I \in \mathscr{S}} \langle \lvert f\rvert\rangle_I^2…

We prove a Tb theorem on quasimetric spaces equipped with what we call an upper doubling measure. This is a property that encompasses both the doubling measures and those satisfying the upper power bound \mu(B(x,r)) \le Cr^d. Our spaces are…

We establish square function estimates for integral operators on uniformly rectifiable sets by proving a local $T(b)$ theorem and applying it to show that such estimates are stable under the so-called big pieces functor. More generally, we…

We prove a Leibniz rule for BV functions in a complete metric space that is equipped with a doubling measure and supports a Poincar\'e inequality. Unlike in previous versions of the rule, we do not assume the functions to be locally…

We prove weak type inequalities for a large class of noncommutative square functions. In conjunction with BMO type estimates, interpolation and duality, we will obtain the corresponding equivalences in the whole Lp scale. The main novelty…

In this paper, we propose a general method for testing inequality restrictions on nonparametric functions. Our framework includes many nonparametric testing problems in a unified framework, with a number of possible applications in auction…

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…

We obtain a local two weight $Tb$ theorem with an energy side condition for higher dimensional fractional Calder\'{o}n-Zygmund operators. The proof follows the general outline of the proof for the corresponding one-dimensional $Tb$ theorem…

We investigate the unconditional basis property of martingale differences in weighted $L^2$ spaces in the non-homogeneous situation (i.e. when the reference measure is not doubling). Specifically, we prove that finiteness of the quantity…

We give a proof of a so-called "local $Tb$" Theorem for singular integrals whose kernels satisfy the standard Calder\'on-Zygmund conditions. The present theorem, which extends an earlier result of M. Christ \cite{Ch}, was proved in…

We prove a Tb Theorem that characterizes all Calderon-Zygmund operators that extend compactly on L^p(R^n), 1<p<\infty . The result, whose proof does not require the property of accretivity, can be used to prove compactness of the Double…

We demonstrate and develop dyadic-probabilistic methods in connection with non-homogeneous bilinear operators, namely singular integrals and square functions. We develop the full non-homogeneous theory of bilinear singular integrals using a…

This paper develops tests for inequality constraints of nonparametric regression functions. The test statistics involve a one-sided version of $L_p$-type functionals of kernel estimators $(1 \leq p < \infty)$. Drawing on the approach of…

We obtain a two weight local Tb theorem for any elliptic and gradient elliptic fractional singular integral operator T on the real line, and any pair of locally finite positive Borel measures on the line. This includes the Hilbert transform…

This paper establishes a necessary and sufficient condition for $L^p$-boundedness of a class of multilinear functionals which includes both the Brascamp-Lieb inequalities and generalized Radon transforms associated to algebraic incidence…

We consider a non-negative self-adjoint operator L satisfying generalized Gaussian estimates on a doubling metric measure space, and show that if L has a spectral gap then the local and global Hardy spaces defined by means of appropriate…