Related papers: On the weak Sard property

We obtain, under an additional assumption on the subanalytic abnormal distribution constructed in [4], a proof of the minimal rank Sard conjecture in the analytic category. It establishes that from a given point the set of points accessible…

For $c\in(1,2)$ we consider the following operators \[ \mathcal{C}_{c}f(x) = \sup_{\lambda \in [-1/2,1/2)}\bigg| \sum_{n \neq 0}f(x-n) \frac{e^{2\pi i\lambda \lfloor |n|^{c} \rfloor}}{n}\bigg|\text{,}\quad \mathcal{C}^{\mathsf{sgn}}_{c}f(x)…

Let $f:\mathbb{R}^n\to\mathbb{R}$ be a function. Assume that for a measurable set $\Omega$ and almost every $x\in\Omega$ there exists a vector $\xi_x\in\mathbb{R}^n$ such that $$\liminf_{h\to 0}\frac{f(x+h)-f(x)-\langle \xi_x,…

It is shown that for every $\alpha$, where $\alpha\in [0, 1/2]$, there exists an $\alpha$-rigid transformation whose spectrum has Lebesgue component. This answers the question posed by Klemes and Reinhold in [7]. We apply a certain…

We mainly establish a monotonicity property between some special Riemann sums of a convex function $f$ on $[a,b]$, which in particular yields that $\frac{b-a}{n+1}\sum_{i=0}^n f\left(a+i\frac{b-a}{n}\right)$ is decreasing while…

For classical Bernoulli convolutions, the Rajchman property, i.e. the convergence to zero at infinity of the Fourier transform, was characterized by successive works of Erd{\"o}s [2] and Salem [12]. We prove weak forms of their results for…

Let $D$ be a domain of finite Lebesgue measure in $\bR^d$ and let $X^D_t$ be the symmetric $\alpha$-stable process killed upon exiting $D$. Each element of the set $\{\lambda_i^\alpha\}_{i=1}^\infty$ of eigenvalues associated to $X^D_t$,…

We show that the best constants appearing in the weak type (1,1) inequalities satisfied by the centered Hardy-Littlewood maximal function associated to some finite radial measures, such as the standard gaussian measure, grow exponentially…

In this short note we prove that, if (C[a,b],{A_n}) is an approximation scheme and (A_n) satisfies de La Vall\'ee-Poussin Theorem, there are instances of continuous functions on [a,b], real analytic on (a,b], which are poorly approximable…

We show that under natural technical conditions, the sum of a $C^2$ dynamically defined Cantor set with a compact set in most cases (for almost all parameters) has positive Lebesgue measure, provided that the sum of the Hausdorff dimensions…

The weak tightness $wt(X)$ of a space $X$ was introduced in [11] with the property $wt(X)\leq t(X)$. We investigate several well-known results concerning $t(X)$ and consider whether they extend to the weak tightness setting. First we give…

This note describes Fatou's lemma and Lebesgue's dominated convergence theorem for a sequence of measures converging weakly to a finite measure and for a sequence of functions whose negative parts are uniformly integrable with respect to…

The classical Morse--Sard theorem claims that for a mapping $v:\mathbb R^n\to\mathbb R^{m+1}$ of class $C^k$ the measure of critical values $v(Z_{v,m})$ is zero under condition $k\ge n-m$. Here the critical set, or $m$-critical set is…

We investigate characterizations of uniformly rectifiable (UR) metric spaces by so-called weak Carleson conditions for flatness coefficients which measure the extent to which Hausdorff measure on the metric space differs from Hausdorff…

We introduce an interesting method of proving separable reduction theorems - the method of elementary submodels. We are studying whether it is true that a set (function) has given property if and only if it has this property with respect to…

For a simple graph $\Gamma$ and for unital $C^*$-algebras with GNS-faithful states $(\mathbf{A}_v,\varphi_v)$ for $v\in V\Gamma$, we consider the reduced graph product $(\mathcal{A},\varphi)=*_{v,\Gamma}(\mathbf{A}_{v},\varphi_v)$ , and…

We prove a contraction property of Fock type spaces $\mathcal{L}_{\alpha}^p$ of log-subharmonic functions in $\mathbb{R}^n$. To prove the result, we demonstrate a certain monotonic property of measures of the superlevel set of the function…

We prove that a closed convex subset of a Banach space is (super-)weakly compact if and only if it is (super)-ergodic. As a consequence we deduce that super weakly compact sets are characterized by the fixed point property for continuous…

We prove that Allard's regularity theorem holds for rectifiable $n$-dimensional varifolds $V$ assuming a weaker condition on the first variation. This, in the special case when $V$ is a smooth manifold translates to the following: If…

We analyze the properties of weakly compact sets in Lipschitz free spaces. Prior research has established that, for a complete metric space $M$, weakly precompact sets in the Lipschitz free space $\mathcal F(M)$ are tight. In this paper, we…