Related papers: Riesz transforms through reverse H\"older and Poin…

If T is a fractional vector Riesz transform, 1<p<infinity, and sigma and omega are doubling measures, then the two weight L^{p} norm inequality holds if and only if the quadratic triple testing conditions of Hyt\"onen and Vuorinen hold. We…

We prove an almost everywhere convergence result for Bochner-Riesz means of $L^p$ functions on Heisenberg-type groups, yielding the existence of a $p>2$ for which convergence holds for means of arbitrarily small order. The proof hinges on a…

We study weighted $(L^p, L^q)$-boundedness properties of Riesz potentials and fractional maximal functions for the Dunkl transform. In particular, we obtain the weighted Hardy-Littlewood-Sobolev type inequality and weighted week $(L^1,…

We consider several harmonic analysis operators in the multi-dimensional context of the Dunkl Laplacian with the underlying group of reflections isomorphic to $\mathbb{Z}_2^n$ (also negative values of the multiplicity function are…

Discrete analogues of classical operators in harmonic analysis have been widely studied, revealing deep connections with areas such as ergodic theory and analytic number theory. This line of research is commonly known as \emph{Discrete…

We study the boundedness of the Hilbert transform $H$ and the Hilbert maximal operator $H^*$ on weighted Lorentz spaces $\Lambda^p_u(w)$. We start by giving several necessary conditions that, in particular, lead us to the complete…

This paper is devoted to the study of $L^{p_1} \times L^{p_2}$ to $L^{p}$ boundedness of the bilinear Bochner-Riesz mean $\mathcal{B}^{\alpha}$ associated with the Grushin operator $\mathcal{L} = -\Delta_{x'} - |x'|^2 \Delta_{x''}$ on…

We give necessary and sufficient conditions for inhomogeneous Calder\'on-Zgymund operators to be bounded on the local hardy spaces $h^p(\mathbb{R}^n)$. We then give applications to local and truncated Riesz transforms, as well as…

We consider a complete non-compact Riemannian manifold satisfying the volume doubling property and a Gaussian upper bound for its heat kernel (on functions). Let -- $\rightarrow$ $\Delta$ k be the Hodge-de Rham Laplacian on differential…

Let $M$ be a complete Riemannian manifold satisfying the doubling volume condition for geodesic balls and $L^q$ scaled Poincar\'e inequalities on suitable remote balls for some $q<2$. We prove the inequality $\left\Vert…

We establish necessary and sufficient conditions on a weight pair $(v,w)$ governing the boundedness of the Riesz potential operator $I_{\alpha}$ defined on a homogeneous group $G$ from $L^p_{dec,r}(w, G)$ to $L^q(v, G)$, where…

As a corollary to our main result we deduce sharp A_p$ inequalities for T being either the Hilbert transform in dimension d=1, the Beurling transform in dimension d=2, or a Riesz transform in any dimension d\ge 2. For T_{\ast} the maximal…

We consider boundedness of a certain positive dyadic operator $$ T^\sigma \colon L^p(\sigma; \ \! \ell^2) \to L^p(\omega), $$ that arose during our attempts to develop a two-weight theory for the Hilbert transform in $L^p$. Boundedness of…

This paper present an overview of some of the applications of the martingale inequalities of D.L. Burkholder to $L^p$-bounds for singular integral operators, concentrating on the Hilbert transform, first and second order Riesz transforms,…

Let $M$ be a complete connected Riemannian manifold. Assuming that the Riemannian measure is doubling, we define Hardy spaces $H^p$ of differential forms on $M$ and give various characterizations of them, including an atomic decomposition.…

Let $(X, d, \mu)$ be a space of homogeneous type, i.e. the measure $\mu$ satisfies doubling (volume) property with respect to the balls defined by the metric $d$. Let $L$ be a non-negative self-adjoint operator on $L^2(X)$. Assume that the…

We prove a dimension-free $L^p(\mathbb{R}^d)$, $1<p<\infty$, estimate for the vector of maximal Riesz transforms of odd order in terms of the corresponding Riesz transforms. This implies a dimension-free $L^p(\mathbb{R}^d)$ estimate for the…

We extend Stein's maximal theorem to the bilinear setting. Let $M$ be a homogeneous space with a transitive action of a compact abelian group, and let $1 \le p,q \le 2$ and $1/2 \le r \le 1$ satisfy $1/p + 1/q = 1/r$. For a family of…

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…

Let $G = - \Delta_{\xi} - |\xi|^2 \frac{\partial^2}{\partial \eta^2}$ be the Grushin operator on $\R^n \times \R.$ We prove that the Riesz transforms associated to this operator are bounded on $L^p (\R^{n+1}), 1 < p < \infty$ and their…