相关论文: Nazarov's uncertainty principles in higher dimensi…

Let $\Omega$ be an open subset of $\mathbb R^n$, and let $f: \Omega \to \mathbb R$ be differentiable $\mathcal H^k$-almost everywhere, for some nonnegative integer $k < n$, where $\mathcal H^k$ denotes the $k$=dimensional Hausdorff measure.…

A great challenge in the analysis of the discrepancy function D_N is to obtain universal lower bounds on the L-infty norm of D_N in dimensions d \geq 3. It follows from the average case bound of Klaus Roth that the L-infty norm of D_N is at…

Given an extended real-valued submeasure $\nu$ defined on a field of subsets $\Sigma$ of a given set, we provide necessary and sufficient conditions for which the pseudometric $d_\nu$ defined by $d_{\nu}(A,B):=\min\{1,\nu(A\bigtriangleup…

In the present article we provide a sufficient condition for a closed set F in R^d to have the following property which we call c-removability: Whenever a function f:R^d->R is locally convex on the complement of F, it is convex on the whole…

We show that there exist constants $\alpha,\epsilon>0$ such that for every positive integer $n$ there is a continuous odd function $f:S^m\to S^n$, with $m\geq \alpha n$, such that the $\epsilon$-expansion of the image of $f$ does not…

We prove the developability and $C^{1,1/2}$ regularity of $W^{2,2}$ isometric immersions of $n$-dimensional domains into $R^{n+1}$. As a conclusion we show that any such Sobolev isometry can be approximated by smooth isometries in the…

If $f\in L^1({\mathbb R})$ it is proved that $\lim_{S\to\infty}\lVert f-f\ast D_S\rVert=0$, where $D_S(x)=\sin(Sx)/(\pi x)$ is the Dirichlet kernel and $\lVert f\rVert = \sup_{\alpha<\beta}|\int_{\alpha}^{\beta}f(x)\,dx|$ is the Alexiewicz…

We prove that if the Hausdorff dimension of $E\subset\mathbb{R}^d$, $d\geq 2$ is greater than $\frac{d}{2}+\frac{1}{3}$, the set of gaps of $2$-chains inside $E$, $$\Delta_2(E)=\{(|x-y|, |y-z|): x, y, z\in E \}\subset\mathbb{R}^2$$ has…

Cilleruelo conjectured that if $f\in\mathbb{Z}[x]$ of degree $d\ge 2$ is irreducible over the rationals, then $\log\operatorname{lcm}(f(1),\ldots,f(N))\sim(d-1)N\log N$ as $N\to\infty$. He proved it for the case $d = 2$. Very recently,…

A graph $G$ is $m$-minor-universal if every graph with at most $m$ edges (and no isolated vertices) is a minor of $G$. We prove that the $d$-dimensional hypercube, $Q_d$, is $\Omega\left(\frac{2^d}{d}\right)$-minor-universal, and that there…

We prove three results on the existence of densities for the laws of finite dimensional functionals of the solutions of the stochastic Navier-Stokes equations in dimension 3. In particular, under very mild assumptions on the noise, we prove…

An easy consequence of Kantorovich-Rubinstein duality is the following: if $f:[0,1]^d \rightarrow \infty$ is Lipschitz and $\left\{x_1, \dots, x_N \right\} \subset [0,1]^d$, then $$ \left| \int_{[0,1]^d} f(x) dx - \frac{1}{N}…

It is shown (Theorem A and its corollary) that if g is any nonconstant nonunivalent analytic function on a half-plane H and if D is either a half-plane or a smoothly bounded Jordan domain, then there is a function f on D for which f'(D)…

We prove that if $f:\mathbb{R}^n\to\mathbb{R}$ is convex and $A\subset\mathbb{R}^n$ has finite measure, then for any $\varepsilon>0$ there is a convex function $g:\mathbb{R}^n\to\mathbb{R}$ of class $C^{1,1}$ such that $\mathcal{L}^n(\{x\in…

For sums $S_n=\sum_{k=1}^n X_k$, $n\ge 1$ of independent random variables $ X_k $ taking values in $\Z$ we prove, as a consequence of a more general result, that if (i) For some function $1\le \phi(t)\uparrow \infty $ as $t\to \infty$, and…

In [Israel J. Math, 2014], Grahl and Nevo obtained a significant improvement for the well-known normality criterion of Montel. They proved that for a family of meromorphic functions $\mathcal F$ in a domain $D\subset \mathbb C,$ and for a…

We consider densities $D_\Sigma(A)$, $\overline{D}_\Sigma(A)$ and $\underline{D}_\Sigma(A)$ for a subset $A$ of $\mathbb{N}$ with respect to a sequence $\Sigma$ of finite subsets of $\mathbb{N}$ and study Fourier coefficients of ergodic,…

This note is concerned with the Bianchi-Egnell inequality, which quantifies the stability of the Sobolev inequality, and its generalization to fractional exponents $s \in (0, \frac{d}{2})$. We prove that in dimension $d \geq 2$ the best…

Let $\Omega$ be a subanalytic bounded open subset of $\mathbb{R}^n$, with possibly singular boundary. We show that given $p\in [1,\infty)$, there is a constant $C$ such that for any $u\in W^{1,p}(\Omega)$ we have $||u-u_{\Omega}||_{L^p} \le…

We study continuity properties of Sobolev mappings $f \in W_{\mathrm{loc}}^{1,n} (\Omega, \mathbb{R}^n)$, $n \ge 2$, that satisfy the following generalized finite distortion inequality \[\lvert Df(x)\rvert^n \leq K(x) J_f(x) + \Sigma (x)\]…