In this paper we study some potential theoretical properties of solutions and super-solutions of some PDE systems (S) of type $L_{1}u=-{\mu }_{1}v$, $L_{2}v=-{\mu }_{2}u$, on a domain $D$ of ${ℝ}^d$, where ${\mu }_1$ and ${\mu }_2$ are suitable measures on $D$, and $L_1$, $L_2$ are two second order linear differential elliptic operators on $D$ with coefficients of class ${𝒞}^{\infty }$. We also obtain the integral representation of the nonnegative solutions and supersolutions of the system (S) by means of the Green kernels and Martin boundaries associated with $L_1$ and $L_2$, and...

Suppose that $\mu$ is an absolutely continuous probability measure on ${}^R$n, for large $n$. Then $\mu$ has low-dimensional marginals that are approximately spherically-symmetric. More precisely, if $n\ge {(C/\epsilon )}^{Cd}$, then there exist $d$-dimensional marginals of $\mu$ that are $\epsilon$-far from being sphericallysymmetric, in an appropriate sense. Here $C>0$ is a universal constant.

Let ${\alpha }_i,{\beta }_i>0$, 1 ≤ i ≤ n, and for t > 0 and x = (x₁,...,xₙ) ∈ ℝⁿ, let $t•x=(t^{\alpha ₁}x₁,...,t^{\alpha ₙ}xₙ)$, $t\circ x=(t^{\beta ₁}x₁,...,t^{\beta ₙ}xₙ)$ and $\left|\right|x\left|\right|={\sum }_{i=1}^n{\left|x_i\right|}^{1/{\alpha }_i}$. Let φ₁,...,φₙ be real functions in $C^{\infty }(ℝⁿ-0)$ such that φ = (φ₁,..., φₙ) satisfies φ(t • x) = t ∘ φ(x). Let γ > 0 and let μ be the Borel measure on ${ℝ}^{2n}$ given by $\mu \left(E\right)={\int }_{ℝⁿ}{\chi }_E{(x,\phi \left(x\right))\left|\right|x\left|\right|}^{\gamma -\alpha }dx$, where $\alpha ={\sum }_{i=1}^n{\alpha }_i$ and dx denotes the Lebesgue measure on ℝⁿ. Let $T_{\mu }f=\mu \ast f$ and let $\left|\right|T_{\mu }{\left|\right|}_{p,q}$ be the operator norm of $T_{\mu }$ from $L^p\left({ℝ}^{2n}\right)$ into $L^q\left({ℝ}^{2n}\right)$, where the $L^p$ spaces are taken with respect to the Lebesgue measure. The type set $E_{\mu }$ is defined by $E_{\mu }=(1/p,1/q):\left|\right|T_{\mu }{\left|\right|}_{p,q}<\infty ,1\le p,q\le \infty$. In the case ${\alpha }_i\ne {\beta }_k$ for 1 ≤ i,k ≤ n we characterize the...

Let ${\mu }_A$ be the singular measure on the Heisenberg group ${ℍ}^n$ supported on the graph of the quadratic function $\varphi \left(y\right)=y^{t}Ay$, where $A$ is a $2n\times 2n$ real symmetric matrix. If $det(2A\pm J)\ne 0$, we prove that the operator of convolution by ${\mu }_A$ on the right is bounded from $L^{\frac{(2n+2)}{(2n+1)}}\left({ℍ}^n\right)$ to $L^{2n+2}\left({ℍ}^n\right)$. We also study the type set of the measures $\mathrm{d}{\nu }_{\gamma }(y,s)=\eta \left(y\right){\left|y\right|}^{-\gamma }\mathrm{d}{\mu }_A(y,s)$, for $0\le \gamma <2n$, where $\eta$ is a cut-off function around the origin on ${ℝ}^{2n}$. Moreover, for $\gamma =0$ we characterize the type set of ${\nu }_0$.

Motivated by the fundamental theorem of calculus, and based on the works of W. Feller as well as M. Kac and M. G. Kreĭn, given an atomless Borel probability measure $\eta$ supported on a compact subset of $ℝ$ U. Freiberg and M. Zähle introduced a measure-geometric approach to define a first order differential operator ${\nabla }_{\eta }$ and a second order differential operator ${\Delta }_{\eta }$, with respect to $\eta$. We generalize this approach to measures of the form $\eta :=\nu +\delta$, where $\nu$ is non-atomic and $\delta$ is finitely supported. We determine...

We prove boundedness and continuity for solutions to the Dirichlet problem for the equation $$-\mathrm{div}\left(a(x,\nabla u)\right)=h(x,u)+\mu ,\text{in}\Omega \subset {ℝ}^N,$$ where the left-hand side is a Leray-Lions operator from $W_0^{1,p}\left(\Omega \right)$ into $W^{-1,p^{\text{'}}}\left(\Omega \right)$ with $1<p<N$, $h(x,s)$ is a Carathéodory function which grows like ${\left|s\right|}^{p-1}$ and $\mu$ is a finite Radon measure. We prove that renormalized solutions, though not globally bounded, are Hölder-continuous far from the support of $\mu$.

The author proved in 2018 that if $G$ is an open subset of a Hilbert space, $f_1,f_2:G\to ℝ$ continuous functions and $\omega$ a nontrivial modulus such that $f_1\le f_2$, $f_1$ is locally semiconvex with modulus $\omega$ and $f_2$ is locally semiconcave with modulus $\omega$, then there exists $f\in C_{\text{loc}}^{1,\omega }\left(G\right)$ such that $f_1\le f\le f_2$. This is a generalization of Ilmanen’s lemma (which deals with linear modulus and functions on an open subset of ${ℝ}^n$). Here we extend the mentioned result from Hilbert spaces to some superreflexive spaces, in particular to $L^p$ spaces, $p\in [2,\infty )$. We...

We consider $C^2$ families $t\mapsto f_t$ of $C^4$ unimodal maps $f_t$ whose critical point is slowly recurrent, and we show that the unique absolutely continuous invariant measure ${\mu }_t$ of $f_t$ depends differentiably on $t$, as a distribution of order $1$. The proof uses transfer operators on towers whose level boundaries are mollified via smooth cutoff functions, in order to avoid artificial discontinuities. We give a new representation of ${\mu }_t$ for a Benedicks-Carleson map $f_t$, in terms of a single smooth function and the...

The class of $H^{\left(N\right)}$-sets forms an important subclass of the class of sets of uniqueness for trigonometric series. We investigate the size of this class which is reflected by the family of measures (called polar) annihilating all sets from the class. The main aim of this paper is to answer in the negative a question stated by Lyons, whether the polars of the classes of $H^{\left(N\right)}$-sets are the same for all N ∈ ℕ. To prove our result we also present a new description of $H^{\left(N\right)}$-sets.

For a Banach space $E$ and a probability space $(X,𝒜,\lambda )$, a new proof is given that a measure $\mu :𝒜\to E$, with $\mu \ll \lambda$, has RN derivative with respect to $\lambda$ iff there is a compact or a weakly compact $C\subset E$ such that ${\left|\mu \right|}_C:𝒜\to [0,\infty ]$ is a finite valued countably additive measure. Here we define ${\left|\mu \right|}_C\left(A\right)=sup\{{\sum }_k|\langle \mu \left(A_k\right),f_k\rangle \left|\right\}$ where $\left\{A_k\right\}$ is a finite disjoint collection of elements from $𝒜$, each contained in $A$, and $\left\{f_k\right\}\subset E^{\text{'}}$ satisfies ${sup}_k\left|f_k\left(C\right)\right|\le 1$. Then the result is extended to the case when $E$ is a Frechet space.

Let ${\Phi }_1,...,{\Phi }_{k+1}$ (with $k\ge 1$) be vector fields of class $C^k$ in an open set $U{\subset }^{N+m}$, let $𝕄$ be a $N$-dimensional $C^k$ submanifold of $U$ and define $$𝕋:=\{z\in 𝕄:{\Phi }_1\left(z\right),...,{\Phi }_{k+1}\left(z\right)\in T_z𝕄\}$$ where $T_z𝕄$ is the tangent space to $𝕄$ at $z$. Then we expect the following property, which is obvious in the special case when $z_0$ is an interior point (relative to $𝕄$) of $𝕋$: If $z_0\in 𝕄$ is a $(N+k)$-density point (relative to $𝕄$) of $𝕋$ then all the iterated Lie brackets of order less or equal to $k$ $${\Phi }_{i_1}\left(z_0\right),[{\Phi }_{i_1},{\Phi }_{i_2}]\left(z_0\right),[[{\Phi }_{i_1},{\Phi }_{i_2}],{\Phi }_{i_3}]\left(z_0\right),...(h,i_h\le k+1)$$ belong to $T_{z_0}𝕄$. Such a property has been proved in [9] for $k=1$ and its proof in the...

It is well-known that a probability measure $\mu$ on the circle $𝕋$ satisfies $\parallel {\mu }^n{*f-\int f\mathrm{d}m\parallel }_p\to 0$ for every $f\in L_p$, every (some) $p\in [1,\infty )$, if and only if $|\widehat{\mu }\left(n\right)|lt;1$ for every non-zero $n\in \mathbb{Z}$ ($\mu$ is strictly aperiodic). In this paper we study the a.e. convergence of ${\mu }^n*f$ for every $f\in L_p$ whenever $pgt;1$. We prove a necessary and sufficient condition, in terms of the Fourier–Stieltjes coefficients of $\mu$, for the strong sweeping out property (existence of a Borel set $B$ with $lim\; sup{\mu }^n*1_B=1$ a.e. and $lim\; inf{\mu }^n*1_B=0$ a.e.). The results are extended to general compact Abelian groups...

We prove a density version of the Carlson–Simpson Theorem. Specifically we show the following. For every integer $k\ge 2$ and every set $A$ of words over $k$ satisfying $\lim {\sup }_{n\to \infty }|A\cap {\left[k\right]}^n|/k^n>0$ there exist a word $c$ over $k$ and a sequence $\left(w_n\right)$ of left variable words over $k$ such that the set $c\cup \{c^{}{w}_0{\left(a_0\right)}^{}..{.}^{}{w}_n\left(a_n\right):n\in \mathbb{N}\text{and}{a}_0,...,a_n\in \left[k\right]\}$ is contained in $A$. While the result is infinite-dimensional its proof is based on an appropriate finite and quantitative version, also obtained in the paper.

We consider the equation $$-y^{\text{'}}\left(x\right)+q\left(x\right)y(x-\varphi \left(x\right))=f\left(x\right),x\in ℝ,$$ where $\varphi$ and $q$ ($q\ge 1$) are positive continuous functions for all $x\in ℝ$ and $f\in C\left(ℝ\right)$. By a solution of the equation we mean any function $y$, continuously differentiable everywhere in $ℝ$, which satisfies the equation for all $x\in ℝ$. We show that under certain additional conditions on the functions $\varphi$ and $q$, the above equation has a unique solution $y$, satisfying the inequality $$\parallel y^{\text{'}}{\parallel }_{C\left(ℝ\right)}+{\parallel qy\parallel }_{C\left(ℝ\right)}\le c{\parallel f\parallel }_{C\left(ℝ\right)},$$ where the constant $c\in (0,\infty )$ does not depend on the choice of $f$.