arXiv research

In this paper we prove mixed norm estimates for Riesz transforms related to Laplace--Beltrami operators on compact Riemannian symmetric spaces of rank one. These operators are closely related to the Riesz transforms for Jacobi polynomials expansions. The key point is to obtain sharp estimates for the kernel of the Jaco…

An locally conformally Kahler (LCK) manifold with potential is a complex manifold with a cover which admits an automorphic Kahler potential. An LCK manifold with potential can be embedded to a Hopf manifold, if its dimension is at least 3. We give a functional-analytic proof of this result based on Riesz-Schauder theor…

Gradient boosting estimates Riesz representer for causal inference.

Python package automates causal parameter estimation using Riesz regression.

The paper generalizes spectral section concepts to non-compact spaces.

We show that there exists an integrable function on the $n$-sphere $(n\ge 2)$, whose Cesàro (C,$\frac{n-1}{2}$) means with respect to the spherical harmonic expansion diverge unboundedly almost everywhere. By studying equivalence theorems, we also obtain the corresponding results for Riesz and Bochner-Riesz means. This…

The paper explores properties of functions on Teichmüller space, proving theorems about limits and non-ergodicity.

We prove a version of Smirnov type theorem and Charatheodory type theorem for a harmonic homeomorphism of the unit disk onto a Jordan surface with rectifiable boundary. Further we establish the classical isoperimetric inequality and Riesz--Zygmund inequality for Jordan harmonic surfaces without any smoothness assumptio…

Unified framework for linear attribution methods in deep learning.

Riesz regression connects to density ratio estimation for causal inference.

We investigate the boundness of the Riesz transform on $L^p$ for connected sum of manifolds where the Riesz transform is bounded on $L^p$.

Unified framework for debiased machine learning using Riesz representer and Bregman divergence.

There is an interesting potential theory associated to each degenerate elliptic, fully nonlinear equation $f(D^2u) = 0$. These include all the potential theories attached to calibrated geometries. This paper begins the study of tangents to the subsolutions in these theories, a topic inspired by the results of Kiselman …

Study proves NN matching is equivalent to Riesz regression for debiased machine learning.

Given a sequence of complete Riemannian manifolds $(M_n)$ of the same dimension, we construct a complete Riemannian manifold $M$ such that for all $p \in (1,\infty)$ the $L^p$-norm of the Riesz transform on $M$ dominates the $L^p$-norm of the Riesz transform on $M_n$ for all $n$. Thus we establish the following dichoto…

Develops a direct debiased machine learning framework using Bregman divergence.

Unified theory for causal inference using various methods.

We show a perturbation result for the boundedness of the Riesz transform : if $M$ and $M_0$ are complete Riemannian manifolds satisfying a Sobolev inequality of dimension $n$, which are isometric outside a compact set, and if the Riesz transform on $M_0$ is bounded on $L^q$, then for all $\frac{n}{n-2}, the Riesz trans…

We investigate the $L^p$-boundness of the Riesz transform on Riemannian manifolds whose Ricci curvature has quadratic decay. Two criteria for the $L^p$-unboundness of the Riesz transform are given. We recover known results about manifolds that are Euclidean or conical at infinity.

Study $L^p$ boundedness of Riesz transform on differential forms for certain manifolds.

Let $M$ be a complete non-compact Riemannian manifold. In this paper, we derive sufficient conditions on metric perturbation for stability of $L^p$-boundedness of the Riesz transform, $p\in (2,\infty)$. We also provide counter-examples regarding in-stability for $L^p$-boundedness of Riesz transform.

Two approaches to directly estimating Riesz representer are shown to be numerically equivalent under certain conditions.

The study bounds Riesz transforms on manifolds with controlled curvature.

We show that balls, circles and 2-spheres can be identified by generalized Riesz energy among compact submanifolds of the Euclidean space that are either closed or with codimension 0, where the Riesz energy is defined as the double integral of some power of the distance between pairs of points. As a consequence, we obt…

Estimates heat kernel gradients on fractal-like cable systems.

Unified framework for estimating density ratios in causal inference.

Graph continuous operators become Riesz continuous after multiplication by unitary operators.

Proposes adversarial method to estimate Riesz representer.

A reverse Riesz estimate and spectral gap imply a Poincaré inequality.

The paper establishes a Poisson integral formula for bounded pluriharmonic functions on Teichmüller space.

We study the validity of the $L^p$ inequality for the Riesz transform when $p>2$ and of its reverse inequality when $p<2$ on complete Riemannian manifolds under the doubling property and some Poincaré inequalities.

In this paper we study the Riesz transform on complete and connected Riemannian manifolds $M$ with a certain spectral gap in the $L^2$ spectrum of the Laplacian. We show that on such manifolds the Riesz transform is $L^p$ bounded for all $p \in (1,\infty)$. This generalizes a result by Mandouvalos and Marias and extend…

ScoreMatchingRiesz improves debiased machine learning and policy effects estimation.

We prove an optimal reverse Poincaré inequality for the heat semigroup generated by the sub-Laplacian on a Carnot group of any step. As an application we give new proofs of the isoperimetric inequality and of the boundedness of the Riesz transform in Carnot groups.

We construct a large class of Riemannian manifolds of arbitrary dimension with Riesz transform unbounded on $L^p(M)$ for all $p > 2$. This extends recent results for Vicsek manifolds, and in particular shows that fractal structure is not necessary for this property.

The paper proves boundedness of a Riesz transform on weighted manifolds.

By using an explicit Bellman function, we prove a bilinear embedding theorem for the Laplacian associated with a weighted Riemannian manifold $(M,μ_φ)$ having the Bakry-Emery curvature bounded from below. The embedding, acting on the cartesian product of $L^p(M,μ_φ)$ and $L^q(T^*M,μ_φ)$, $1/p+1/q=1$, involves estimates…

Improved AutoDML estimator for causal inference using outcome-adapted shared covariate representation.

This is the fourth article of our series. Here, we study weighted norm inequalities for the Riesz transform of the Laplace-Beltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Gaussian upper bounds.

Let $M$ be a smooth Riemannian manifold which is the union of a compact part and a finite number of Euclidean ends, $\RR^n \setminus B(0,R)$ for some $R > 0$, each of which carries the standard metric. Our main result is that the Riesz transform on $M$ is bounded from $L^p(M) \to L^p(M; T^*M)$ for $1 < p < n$ and unbou…

The Barankin bound is generalized to the vector case in the mean square error sense. Necessary and sufficient conditions are obtained to achieve the lower bound. To obtain the result, a simple finite dimensional real vector valued generalization of the Riesz representation theorem for Hilbert spaces is given. The bound…

DPI quantifies phase differences in 1D and multidimensional signals using Riesz transform.

Local Hardy spaces defined for Riemannian manifolds with bounded geometry.

Paper improves MMD flow efficiency with Riesz kernels for image generation.

Let $(M^\circ, g)$ be an asymptotically conic manifold, in the sense that $M^\circ$ compactifies to a manifold with boundary $M$ in such a way that $g$ becomes a scattering metric on $M$. A special case of particular interest is that of asymptotically Euclidean manifolds, where $\partial M = S^{n-1}$ and the induced me…

Prediction-powered causal inference achieves smaller asymptotic variance than traditional methods.

In this paper we study the problem of deriving further Sobolev inequalities from a given Sobolev inequality. We use several different methods, including Bessel potentials and Riesz transforms. We apply the results to the Ricci flow to extend the author's results on the $W^{1,2}$ Sobolev inequality along the Ricci flow …

One considers the class of complete non-compact Riemannian manifolds whose heat kernel satisfies Gaussian estimates from above and below. One shows that the Riesz transform is $L^p$ bounded on such a manifold, for $p$ ranging in an open interval above 2, if and only if the gradient of the heat kernel satisfies a certai…