相关论文: On the absolute continuity of radial projections

In this paper we prove that for any definable subset $X\subset \mathbb{R}^{n}$ in a polynomially bounded o-minimal structure, with $dim(X)<n$, there is a finite set of regular projections (in the sense of Mostowski ). We give also a weak…

We prove existence and uniqueness of optimal maps on $RCD^*(K,N)$ spaces under the assumption that the starting measure is absolutely continuous. We also discuss how this result naturally leads to the notion of exponentiation.

We show that the spectrum of a Schr\"odinger operator on $\mathbb{R}^n$, $n\ge 3$, with a periodic smooth Riemannian metric, whose conformal multiple has a product structure with one Euclidean direction, and with a periodic electric…

In this paper, we study the smoothness of the density function of absolutely continuous measures supported on random self-similar sets on the line. We show that the natural projection of a measure with symbolic local dimension greater than…

A $*$-ring $R$ is called (strongly) $*$-clean if every element of $R$ is the sum of a projection and a unit (which commute with each other). In this note, some properties of $*$-clean rings are considered. In particular, a new class of…

Let {\mathbb{V} = V x R^l : V \in G(n-l,m-l)} be the family of m-dimensional subspaces of R^n containing {0} x R^l, and let \pi_{\mathbb{V}} : R^n --> \mathbb{V} be the orthogonal projection onto \mathbb{V}. We prove that the mapping V…

We introduce the {\em $\mu$-topological stability}. This is a type of stability depending on the measure $\mu$ different from the set-valued approach \cite{lm}. We prove that the map $f$ is $m_p$-topologically stable if and only if $p$ is a…

If $\mu$ is a finite complex measure in the complex plane $\C$ we denote by $C^\mu$ its Cauchy integral defined in the sense of principal value. The measure $\mu$ is called reflectionless if it is continuous (has no atoms) and $C^\mu=0$ at…

Suppose that $G$ is a compact Hausdorff Abelian group. We say $\mu \in M(G)$ is strongly continuous if $|\mu|(x+H)=0$ for any $x \in G$ and any $H \leq G$ that is closed and of infinite index. We prove that for any sufficiently rapidly…

We extend the celebrated rigidity of the sharp first spectral gap under $Ric\ge0$ to compact infinitesimally Hilbertian spaces with non-negative (weak, also called synthetic) Ricci curvature and bounded (synthetic) dimension i.e. to…

The lower spectral radius of a set of $d \times d$ matrices is defined to be the minimum possible exponential growth rate of long products of matrices drawn from that set. When considered as a function of a finite set of matrices of fixed…

Given positive measures $\nu,\mu$ on an arbitrary measurable space $(\Omega, \mathcal F)$, we construct a sequence of finite partitions $(\pi_n)_n$ of $(\Omega, \mathcal F)$ s.t. $$ \sum_{A\in \pi_n: \mu(A)>0} 1_{A} \frac{\nu(A)}{\mu(A)}…

In the field of radial basis functions mathematicians have been endeavouring to find infinitely differentiable and compactly supported radial functions. This kind of functions are extremely important for some reasons. First, its…

Let $f$ and $g$ be two volume preserving, Anosov diffeomorphisms on $\mathbb{T}^2$, sharing common stable and unstable cones. In this paper, we find conditions for the existence of (dissipative) neighborhoods of $f$ and $g$, $\mathcal{U}_f$…

In this paper, we consider probability measures $\mu$ and $\nu$ on a $d$--dimensional sphere in $\Rd, d \geq 1,$ and cost functions of the form $c(\x,\y)=l(\frac{|\x-\y|^2}{2})$ that generalize those arising in geometric optics where…

Let $\Lambda$ be a uniformly discrete set and $S$ be a compact set in $R$. We prove that if there exists a bounded sequence of functions in Paley--Wiener space $PW_S$, which approximates $\delta-$functions on $\Lambda$ with $l^2-$error $d$,…

We observe that, for $r>1$, $s$ in an $r$-dependent interval, $p$ a homogeneous pseudodifferential symbol of order $m$ having $C^{r}$ regularity in space, and $u\in H^{s+m-r}(\mathbb{R}^{n})$ such that $p(x,D)u\in H^{s}(\mathbb{R}^{n})$,…

We show $\ell^p\big(\mathbb Z^d\big)$ boundedness, for $p\in(1, \infty)$, of discrete singular integrals of Radon type with the aid of appropriate square function estimates, which can be thought as a discrete counterpart of the…

In this article, we give a unified proof of the end-point estimates of the totally-geodesic $k$-plane transform of radial functions on spaces of constant curvature. The problem of getting end-point estimates is not new and some results are…

We characterize Radon measures $\mu$ in $\mathbb{R}^{n}$ that are $d$-rectifiable in the sense that their supports are covered up to $\mu$-measure zero by countably many $d$-dimensional Lipschitz graphs and $\mu \ll \mathcal{H}^{d}$. The…