Related papers: Noncommutative Riesz transforms -- a probabilistic…

Building on the author's recent work with Jan Maas and Jan van Neerven, this paper establishes the equivalence of two norms (one using a maximal function, the other a square function) used to define a Hardy space on $\R^{n}$ with the…

By defining tracial states on a non-commutative analogue of a path space, we construct Markov dilations for a class of conservative completely Markov semigroups on finite von Neumann algebras. This class includes all symmetric semigroups.…

We prove that, for totally irregular measures $\mu$ on $\mathbb{R}^{d}$ with $d\geq3$, the $(d-1)$-dimensional Riesz transform $$ T_{A,\mu}^{V}f(x) = \int_{\mathbb{R}^d} \nabla_{1}\mathcal{E}_{A}^{V}(x,y) f(y) \, d \mu(y) $$ adapted to the…

Motivated by Cuntz-Krieger-Toeplitz systems associated to undirected graphs and representations of groupoids, we obtain a generalisation of the Sz-Nagy's Dilation Theorem for operator valued partially positive semidefinite maps on…

We study the $L^p$ boundedness of Riesz transform as well as the reverse inequality on Riemannian manifolds and graphs under the volume doubling property and a sub-Gaussian heat kernel upper bound. We prove that the Riesz transform is then…

We show that the sequential closure of a family of probability measures on the canonical space of c{\`a}dl{\`a}g paths satisfying Stricker's uniform tightness condition is a weak${}^*$ compact set of semimartingale measures in the pairing…

We study the $L^{2}$-boundedness of the $3$-dimensional (Heisenberg) Riesz transform on intrinsic Lipschitz graphs in the first Heisenberg group $\mathbb{H}$. Inspired by the notion of vertical perimeter, recently defined and studied by…

A metric measure space is a complete separable metric space equipped with probability measure that has full support. Two such spaces are equivalent if they are isometric as metric spaces via an isometry that maps the probability measure on…

We prove that if $\tau$ is a large positive number, then the atomic Goldberg-type space $\mathfrak{h}^1(N)$ and the space $\mathfrak{h}_{\mathcal R_\tau}^1(N)$ of all integrable functions on $N$ whose local Riesz transform $\mathcal R_\tau$…

Let $\Gamma$ be a Zariski dense Anosov subgroup of a connected semisimple real algebraic group $G$. For a maximal horospherical subgroup $N$ of $G$, we show that the space of all non-trivial $NM$-invariant ergodic and $A$-quasi-invariant…

We propose a novel approach in noncommutative probability, which can be regarded as an analogue of good-$\lambda$ inequalities from the classical case due to Burkholder and Gundy (Acta Math {\bf124}: 249-304,1970). This resolves a…

Let $R_{1,2}$ be scalar Riesz transforms on $\mathbb{R}^2$. We prove that the $L^p$ norms of $k$-th powers of the operator $R_2+iR_1$ behave exactly as $|k|^{1-2/p}p$, uniformly in $k\in\mathbb{Z}\backslash\{0\}$, $p\geq2$. This gives a…

We consider the Grushin operator with drift which is symmetric with respect to a measure having exponential growth. For the corresponding Riesz transforms, we study strong-type $(p, p)$, $1 < p < \infty$, and weak-type $(1, 1)$ boundedness.

Let $L$ be a second order divergence form elliptic operator with complex bounded measurable coefficients. The operators arising in connection with $L$, such as the heat semigroup and Riesz transform, are not, in general, of…

Let \(\mathcal{L}_\nu\) be the Laguerre differential operator which is the self-adjoint extension of the differential operator \[ L_\nu := \sum_{i=1}^n \left[-\frac{\partial^2}{\partial x_i^2} + x_i^2 + \frac{1}{x_i^2} \left(\nu_i^2 -…

We present a nonparametric prior over reversible Markov chains. We use completely random measures, specifically gamma processes, to construct a countably infinite graph with weighted edges. By enforcing symmetry to make the edges undirected…

We derive sub-Riemannian Ricci curvature tensor for sub-Riemannian manifolds. We provide examples including the Heisenberg group, displacement group, and Martinet sub-Riemannian structure with arbitrary weighted volumes, in which we…

Let $R$ be the vector of Riesz transforms on $\mathbb{R}^n$, and let $\mu,\lambda \in A_p$ be two weights on $\mathbb{R}^n$, $1 < p < \infty$. The two-weight norm inequality for the commutator $[b, R] : L^p(\mathbb{R}^n;\mu) \to…

We introduce the manifold of {\it restricted} $n\times n$ positive semidefinite matrices of fixed rank $p$, denoted $S(n,p)^{*}$. The manifold itself is an open and dense submanifold of $S(n,p)$, the manifold of $n\times n$ positive…

In this paper we consider a complete connected noncompact Riemannian manifold M with bounded geometry and spectral gap. We prove that the Hardy type spaces X^k(M), introduced in a previous paper of the authors, have an atomic…