Related papers: Martingale transform and Square function: some wea…

Consider a root system of type $BC_1$ on the real line $\mathbb R$ with general positive multiplicities. The Cherednik-Opdam transform defines a unitary operator from an $L^2$-space on $\mathbb R$ to a $L^2$-space of $\mathbb C^2$-valued…

Quantitative weighted estimates are obtained for the Littlewood-Paley square function $S$ associated with a lacunary decomposition of ${\mathbb R}$ and for the Marcinkiewicz multiplier operator. In particular, we find the sharp dependence…

This paper extends the results from arXiv:1702.04569 about sharp $A_2$-$A_\infty$ estimates with matrix weights to the non-homogeneous situation.

In this article, with introducing concepts of variable scalar $\mathcal{A}_{p(\cdot),\infty}$ weights and variable matrix $\mathscr{A}_{p(\cdot),\infty}$ weights, we seek a comprehensive theory of $A_\infty$ weights within the framework of…

Working in combinatorial model $\mathrm{W_{co}}(d)$, $d=1,2,\dots$, of P\'olya's random walker in $\mathbb{Z}^d$, we prove two theorems on recurrence to a vertex. We obtain an effective version of the first theorem if $d=2$. Using a…

We prove an estimate for weighted $p$-th moments of the pathwise $r$-variation of a martingale in terms of the $A_{p}$ characteristic of the weight. The novelty of the proof is that we avoid real interpolation techniques.

We show that for every nondecreasing concave function w:R+ --> R+ with w(0)=0, either every finite metric space embeds with distortion arbitrarily close to 1 into a metric space of the form (X,w o d) for some metric d on X, or there exists…

We consider weighted Radon transforms $R_W$ along hyperplanes in $R^3$ with strictly positive weights $W$. We construct an example of such a transform with non-trivial kernel $\mathrm{Ker}R_W$ in the space of infinitely smooth compactly…

We consider two weight $L^{p}\to L^{q}$-inequalities for dyadic shifts and the dyadic square function with general exponents $1<p,q<\infty$. It is shown that if a so-called quadratic $\mathscr{A}_{p,q}$-condition related to the measures…

Let $w$ denote a weight in $\mathbb{R}^n$ which belongs to the Muckenhoupt class $A_\infty$ and let $\mathsf{M}_w$ denote the uncentered Hardy-Littlewood maximal operator defined with respect to the measure $w(x)dx$. The \emph{sharp…

In this paper we consider a weighted version of one dimensional discrete Hardy's Inequality on half-line with power weights of the form $n^\alpha$. Namely we consider: \begin{equation} \sum_{n=1}^\infty |u(n)-u(n-1)|^2 n^\alpha \geq…

We provide a quantitative two weight estimate for the dyadic paraproduct $\pi_b$ under certain conditions on a pair of weights $(u;v)$ and $b$ in $Carl_{u,v}$, a new class of functions that we show coincides with BMO when $u = v \in A^d_2$.…

In this paper we present a theorem that generalizes Sawyer's classic result about mixed weighted inequalities to the multilinear context. Let $\vec{w}=(w_1,...,w_m)$ and $\nu = w_1^\frac{1}{m}...w_m^\frac{1}{m}$, the main result of the…

This paper studies plane domains satisfying a quadrature identity with respect to the singular weight $\rho_0(w)=|w|^{-2}$. These are referred to as log-weighted quadrature domains (LQDs). The logarithmic singularity at $w=0$ leads to…

It is known that the improved Cotlar's inequality $B^{*}f(z) \le C M(Bf)(z)$, $z\in\mathbb C$, holds for the Beurling transform $B$, the maximal Beurling transform $B^{*}f(z)=$ $\displaystyle\sup_{\varepsilon…

We establish some sharp weighted trace inequalities $W^{1,2}(\rho^{1-2\sigma}, M)\hookrightarrow L^{\frac{2n}{n-2\sigma}}(\pa M)$ on $n+1$ dimensional compact smooth manifolds with smooth boundaries, where $\rho$ is a defining function of…

We consider function spaces of Besov, Triebel-Lizorkin, Bessel-potential and Sobolev type on $\R^d$, equipped with power weights $w(x) = |x|^\gamma$, $\gamma>-d$. We prove two-weight Sobolev embeddings for these spaces. Moreover, we…

We obtain Marcinkiewicz-Zygmund strong laws of large numbers for weighted sums of pairwise positively quadrant dependent random variables stochastically dominated by a random variable $X \in \mathscr{L}_{p}$, $1 \leqslant p < 2$. We use our…

A quantitative two weight theorem for the Hardy-Littlewood maximal operator is proved improving the known ones. As a consequence a new proof of the main results in [HP] and in [HPR12] is obtained which avoids the use of the sharp…

In this paper, we will obtain the weak type estimates of intrinsic square functions including the Lusin area integral, Littlewood-Paley $g$-function and $g^*_\lambda$-function on the weighted Morrey spaces $L^{1,\kappa}(w)$ for $0<\kappa<1$…