Related papers: Riesz potential and maximal function for Dunkl tra…

We study a.e. convergence on $L^p$, and Lorentz spaces $L^{p,q}$, $p>\tfrac{2d}{d-1}$, for variants of Riesz means at the critical index $d(\tfrac 12-\tfrac 1p)-\tfrac12$. We derive more general results for (quasi-)radial Fourier…

In this paper, we obtained the Dunkl analogy of classical Lp Hardy inequality for $p > N + 2\gamma$ with sharp constant $\left(\frac{p-N-2\gamma}{p}\right)^{p}$, where $2\gamma$ is the degree of weight function associated with Dunkl…

We consider a class of non-doubling manifolds $\mathcal{M}$ defined by taking connected sum of finite Riemannian manifolds with dimension N which has the form $\mathbb{R}^{n_i}\times \mathcal{M}_i$ and the Euclidean dimension $n_i$ are not…

We study the boundedness of certain fractional integral operators from Hp(.) into Lq(.). We also obtain the Hp(.)- Hq(.) boundedness of the Riesz potential.

We establish a new global endpoint Sobolev inequality for measures that extends the classical theorem of Meyers-Ziemer by placing a maximal function on the right-hand side. This result has several significant consequences. It extends…

Let L=-\Delta+V be a Schr\"odinger operator on R^d, d\geq 3. We assume that V is a nonnegative, compactly supported potential that belongs to L^p(R^d), for some p>d/2. Let K_t be the semigroup generated by -L. We say that an…

We consider vector-valued magnetic Schr\"odinger operators $-\bm \Delta_{\bm a}+V$ with magnetic potential $\bm a \in L^2_{\mathrm{loc}}(\mathbb{R}^d;\mathbb{R}^d)$ and electric potential $V$ given by a matrix-valued function whose entries…

In this note, we study both the Riesz and reverse Riesz transforms on broken line. This model can be described by $(-\infty, -1] \cup [1,\infty)$ equipped with the measure $d\mu = |r|^{d_{1}-1}dr$ for $r \le -1$ and $d\mu = r^{d_{2}-1}dr$…

Let $\sigma$, $\omega$ be measures on $\mathbb{R}^d$, and let $\{\lambda_Q\}_{Q\in\mathcal{D}}$ be a family of non-negative reals indexed by the collection $\mathcal{D}$ of dyadic cubes in $\mathbb{R}^d$. We characterize the two-weight norm…

Fix an integer $ n$ and number $d$, $ 0< d\neq n-1 \leq n$, and two weights $ w$ and $ \sigma $ on $ \mathbb R ^{n}$. We two extra conditions (1) no common point masses and (2) the two weights separately are not concentrated on a set of…

We estimate in Lp the maximal Riesz transform in terms of the Riesz transform itself for p greater than 1. In the limiting case p=1 the weak L1 inequality is shown to fail. Surprisingly, the weak L1 inequality for the maximal Beurling…

We study the boundedness on $L^p$ of the Riesz transform $\nabla L^{-1/2}$, where $L$ is one of several operators defined on $\R$ or $\R_+$, endowed with the measure $r^{d-1} dr$, $d > 1$, where $dr$ is Lebesgue measure. For integer $d$,…

By using an $H^{\infty}$ joint functional calculus for strongly commuting operators, we derive a scheme to deduce the $L^p$ boundedness of certain $d$-dimensional Riesz transforms from the $L^p$ boundedness of appropriate one-dimensional…

In this paper we study the $s$-dimensional Riesz transform of a finite measure $\mu$ in $\mathbf{R}^d$, with $s\in (d-1,d)$. We show that the boundedness of the Riesz transform of $\mu$ implies that a nonlinear potential of exponential type…

We prove a weighted version of the Hardy-Littlewood-Sobolev inequality for radially symmetric functions, and show that the range of admissible power weights appearing in the classical inequality due to Stein and Weiss can be improved in…

Let $M=(0,\infty)_r\times Y$ be a $d$-dimensional ($d\ge 3$) metric cone with metric<br/>$g=dr^2+r^2h$, where $(Y,h)$ is a closed Riemannian manifold. Let<br/>$H=\Delta+V_0/r^2$ be the associated Schrodinger operator, with<br/>$V_0\in…

We introduce first weighted function spaces on Rd using the Dunkl convolution that we call Besov-Dunkl spaces. We provide characterizations of these spaces by decomposition of functions. Next we obtain in the real line and in radial case on…

We consider the Hardy-Littlewood-Sobolev inequality on mixed-norm Lebesgue spaces. We give a complete characterization of indices $\vec p$ and $\vec q$ such that the Riesz potential is bounded from $L^{\vec p}$ to $L^{\vec q}$, including…

In this paper, we establish a class of Stein-Weiss type inequality with partial variable weight functions on the upper half space using a weighted Hardy type inequality. Overcoming the impact of weighted functions, the existence of extremal…

In this paper we introduce capacitary analogues of the Hardy-Littlewood maximal function, \begin{align*} \mathcal{M}_C(f)(x):= \sup_{r>0} \frac{1}{C(B(x,r))} \int_{B(x,r)} |f|\;dC, \end{align*} for $C=$ the Hausdorff content or a Riesz…