Related papers: The measures with $L^2$-bounded Riesz transform an…

Let $\mu$ be a measure on the real line $\mathbb{R}$ such that $\int_{\mathbb{R}}\frac{d\mu(t)}{1+t^2} < \infty$ and let $a>0$. Assume that the norms $\|f\|_{L^2(\mathbb{R})}$ and $\|f\|_{L^2(\mu)}$ are comparable for functions $f$ in the…

Focusing first on the inner $\alpha$-harmonic measure $\varepsilon_y^A$ ($\varepsilon_y$ being the unit Dirac measure, and $\mu^A$ the inner $\alpha$-Riesz balayage of a Radon measure $\mu$ to $A\subset\mathbb R^n$ arbitrary), we describe…

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…

The discrete data encoded in the power moments of a positive measure, fast decaying at infinity on euclidean space, is incomplete for recovery, leading to the concept of moment indeterminateness. On the other hand, classical integral…

We identify a set of sufficient local conditions under which a significant portion of a Radon measure $\mu$ on $\mathbb{R}^{n+1}$ with compact support can be covered by an $n$-uniformly rectifiable set at the level of a ball $B\subset…

We consider an $L^2$-Wasserstein type distance $\rho$ on the configuration space $\Gamma_X$ over a Riemannian manifold $X$, and we prove that $\rho$-Lipschitz functions are contained in a Dirichlet space associated with a measure on…

In this paper we consider the symmetric Kolmogorov operator $L=\Delta +\frac{\nabla \mu}{\mu}\cdot \nabla$ on $L^2(\mathbb R^N,d\mu)$, where $\mu$ is the density of a probability measure on $\mathbb R^N$. Under general conditions on $\mu$…

A finitely-additive measure $\lambda $ on an infinite-dimensional real Hilbert space $E$ which is invariant with respect to shifts and orthogonal mappings has been defined. This measure can be considered as the analog of the Lebesgue…

Suppose $G$ is a compact semisimple Lie group, $\mu$ is the normalized Haar measure on $G$, and $A, A^2 \subseteq G$ are measurable. We show that $$\mu(A^2)\geq \min\{1, 2\mu(A)+\eta\mu(A)(1-2\mu(A))\}$$ with the absolute constant $\eta>0$…

For a map $T \colon [0,1] \to [0,1]$ with an invariant measure $\mu$, we study, for a $\mu$-typical $x$, the set of points $y$ such that the inequality $|T^n x - y| < r_n$ is satisfied for infinitely many $n$. We give a formula for the…

We prove that if $\mu$ is a self-affine measure in the plane whose defining IFS acts totally irreducibly on $\mathbb{RP}^1$ and satisfies an exponential separation condition, then its dimension is equal to its Lyapunov dimension. We also…

We study the invariant subspaces generated by inner functions for a class of $\mathcal{P}^t(\mu)$-spaces which can be identified as spaces of analytic functions in the unit disk $\mathbb{D}$, where $\mu$ is a measure supported in the closed…

We repurpose tools from the theory of quantitative rectifiability to study the qualitative rectifiability of measures in $\Bbb{R}^n$, $n\geq 2$. To each locally finite Borel measure $\mu$, we associate a function $\widetilde J_2(\mu, x)$…

A classical result of Norbert Wiener characterises doubly shift-invariant subspaces for square integrable functions on the unit circle with respect to a finite positive Borel measure $\mu$, as being the ranges of the multiplication maps…

We study regression of $1$-Lipschitz functions under a log-concave measure $\mu$ on $\mathbb{R}^d$. We focus on the high-dimensional regime where the sample size $n$ is subexponential in $d$, in which distribution-free estimators are…

Let $({\mathcal X},\,d,\,\mu)$ be a metric measure space satisfying the upper doubling condition and the geometrically doubling condition in the sense of T. Hyt\"onen. In this paper, the authors prove that the $L^p(\mu)$ boundedness with…

We study probabilistic iterated function systems (IFS), consisting of a finite or infinite number of average-contracting bi-Lipschitz maps on R^d. If our strong open set condition is also satisfied, we show that both upper and lower bounds…

Let $ T\colon[0,1]^d\to [0,1]^d $ be a piecewise expanding map with an absolutely continuous invariant measure $ \mu $. Let $ \{H_n\} $ be a sequence of hyperrectangles or hyperboloids centered at the origin. Denote by $ \mathcal R(\{H_n\})…

We study unimodular measures on the space $\mathcal M^d$ of all pointed Riemannian $d$-manifolds. Examples can be constructed from finite volume manifolds, from measured foliations with Riemannian leaves, and from invariant random subgroups…

For a regular, compact, polynomially convex circled set K in C^2, we construct a sequence of pairs {P_n,Q_n} of homogeneous polynomials in two variables with deg P_n = deg Q_n = n such that the sets K_n: = {(z,w) \in C^2 : |P_n(z,w)| \leq…