Related papers: An elementary proof of the $A_2$ Bound

We consider the decreasing and the increasing $r$-excessive functions $\varphi_r$ and $\psi_r$ that are associated with a one-dimensional conservative regular continuous strong Markov process $X$ with values in an interval with endpoints…

For a matrix A_2 weight W on R^p, we introduce a new notion of W-Calder\'on-Zygmund matrix kernels, following earlier work in by Isralowitz. We state and prove a T1 theorem for such operators and give a representation theorem in terms of…

This paper is concerned with the hard thresholding operator which sets all but the $k$ largest absolute elements of a vector to zero. We establish a {\em tight} bound to quantitatively characterize the deviation of the thresholded solution…

We prove weighted weak-type $(r,r)$ estimates for operators satisfying $(r,s)$ limited-range sparse domination of $\ell^q$-type. Our results contain improvements for operators satisfying limited-range and square function sparse domination.…

We prove the sharp weighted-$L^2$ bounds for the strong-sparse operators introduced in \cite{KaragulyanM}. The main contribution of the paper is the construction of a weight that is a lacunary mixture of dual power weights. This weights…

We prove strong-type $A_p$-$A_\infty$ estimate for square functions, improving on the $ A_p$ bound due to Lerner. Entropy bounds, in the recent innovation of Treil-Volberg, are then proved. The techniques of proof include parallel stopping…

For any operator $T$ whose bilinear form can be dominated by a sparse bilinear form, we prove that $T$ is bounded as a map from $L^1(\widetilde{M}w)$ into weak--$L^1(w)$. Our main innovation is that $\widetilde{M}$ is a maximal function…

We prove a general sparse domination theorem in a space of homogeneous type, in which a vector-valued operator is controlled pointwise by a positive, local expression called a sparse operator. We use the structure of the operator to get…

Let $\mathcal{H}^{\mathbb{T}}$ denote the Hilbert transform on the circle. The paper contains the proofs of the sharp estimates \begin{equation*} \frac{1}{2\pi}|\{ \xi\in\mathbb{T} : \mathcal{H}^{\mathbb{T}}f(\xi) \geq 1 \}| \leq…

It is well-known that dyadic martingale transforms are a good model for Calder\'on-Zygmund singular integral operators. In this paper we extend some results on weighted norm inequalities to vector-valued functions. We prove that, if $W$ is…

We investigate the almost sure asymptotic properties of vector martingale transforms. Assuming some appropriate regularity conditions both on the increasing process and on the moments of the martingale, we prove that normalized moments of…

We obtain mixed $A_p$--$A_\infty$ estimates for a large family of multilinear maximal and sparse operators. Operators from this family are known to control for instance multilinear Calder\'on--Zygmund operators, square functions, fractional…

We use a method of rotations to study the $L^p$ boundedness, $1<p<\infty$, of Fourier multipliers which arise as the projection of martingale transforms with respect to symmetric $\alpha$-stable processes, $0<\alpha<2$. Our proof does not…

For an L ^2-bounded Calderon-Zygmund Operator T, and a weight w \in A_2, the norm of T on L ^2 (w) is dominated by A_2 characteristic of the weight. The recent theorem completes a line of investigation initiated by Hunt-Muckenhoupt-Wheeden…

We show some simple sufficient conditions for which the multilinear embedding theorem holds for fractional sparse operators. By verifying these conditions, we establish the theorem for power weights. We also provide Morrey-type sufficient…

We give a self-contained proof of the $A_2$ conjecture, which claims that the norm of any Calderon-Zygmund operator is bounded by the first degree of the $A_2$ norm of the weight. The original proof of this result by the first author relied…

This paper gives the pointwise sparse dominations for variation operators of singular integrals and commutators with kernels satisfying the $L^r$-H\"{o}rmander conditions. As applications, we obtain the strong type quantitative weighted…

We use the Bellman function method to give an elementary proof of a sharp weighted estimate for the Haar shifts, which is linear in the $A_2$ norm of the weight and in the complexity of the shift. Together with the representation of a…

Following the ideas of A. Lerner, F. Nazarov, S. Ombrosi from [12] we prove that there is a sequence of weights $w\in A^d_1$ such that $[w]^d_{A_1}\to \infty$, and martingale transforms $T$ such that with an absolute positive $c$ $\|T:…

Consider averages along the prime integers $ \mathbb P $ given by \begin{equation*} \mathcal{A}_N f (x) = N ^{-1} \sum_{ p \in \mathbb P \;:\; p\leq N} (\log p) f (x-p). \end{equation*} These averages satisfy a uniform scale-free $ \ell…