An example of a non-zero non-atomic translation-invariant Borel measure ${\nu }_p$ on the Banach space ${\ell }_p\left(1\le p\le \infty \right)$ is constructed in Solovay’s model. It is established that, for 1 ≤ p < ∞, the condition "${\nu }_p$-almost every element of ${\ell }_p$ has a property P" implies that “almost every” element of ${\ell }_p$ (in the sense of [4]) has the property P. It is also shown that the converse is not valid.

Let $S_i:{ℝ}^d\to {ℝ}^d$ for i = 1,..., N be contracting similarities, let $(p₁,...,p_N,p)$ be a probability vector and let ν be a probability measure on ${ℝ}^d$ with compact support. It is well known that there exists a unique inhomogeneous self-similar probability measure μ on ${ℝ}^d$ such that $\mu ={\sum }_{i=1}^N{p}_i\mu \circ S_i^{-1}+p\nu$. We give satisfactory estimates for the lower and upper bounds of the $L^q$ spectra of inhomogeneous self-similar measures. The case in which there are a countable number of contracting similarities and probabilities is considered. In particular,...

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 establish an explicit connection between the perimeter measure of an open set $E$ with $C^1$ boundary and the spherical Hausdorff measure $S^{Q-1}$ restricted to $\partial E$, when the ambient space is a stratified group endowed with a left invariant sub-Riemannian metric and $Q$ denotes the Hausdorff dimension of the group. Our formula implies that the perimeter measure of $E$ is less than or equal to $S^{Q-1}\left(\partial E\right)$ up to a dimensional factor. The validity of this estimate positively answers a conjecture raised by Danielli,...

Let $\beta >1$ be a non-integer. We consider expansions of the form ${\sum }_{i=1}^{\infty }{d}_i/{\beta }^i$, where the digits ${\left(d_i\right)}_{i\ge 1}$ are generated by means of a Borel map $K_{\beta }$ defined on ${\{0,1\}}^{\mathbb{N}}\times [0,\lfloor \beta \rfloor \left(\beta -1\right)]$. We show existence and uniqueness of a $K_{\beta }$-invariant probability measure, absolutely continuous with respect to $m_p\otimes \lambda$, where $m_p$ is the Bernoulli measure on ${\{0,1\}}^{\mathbb{N}}$ with parameter $p$ ($0<p<1$) and $\lambda$ is the normalized Lebesgue measure on $[0,\lfloor \beta \rfloor (\beta -1\left)\right]$. Furthermore, this measure is of the form $m_p\otimes {\mu }_{\beta ,p}$, where ${\mu }_{\beta ,p}$ is equivalent to $\lambda$. We prove that the measure of maximal entropy and $m_p\otimes \lambda$ are mutually...

We prove that for normal operators $N_1,N_2\in ℒ\left(\mathscr{H}\right),$ the generalized commutator $[N_1,N_2;X]$ approaches zero when $[N_1,N_2;[N_1,N_2;X]]$ tends to zero in the norm of the Schatten-von Neumann class ${𝒞}_p$ with $p>1$ and $X$ varies in a bounded set of such a class.

Let $\mu$ be a probability measure on $[0,1]$ which is invariant and ergodic for $T_a\left(x\right)=ax\mathrm{𝚖𝚘𝚍}1$, and $0<\mathrm{𝚍𝚒𝚖}\mu <1$. Let $f$ be a local diffeomorphism on some open set. We show that if $E\subseteq ℝ$ and $\left(f\mu \right){\left|{}_E\sim \mu \right|}_E$, then $f^{\text{'}}\left(x\right)\in \left\{\pm a^r:r\in \mathbb{Q}\right\}$ at $\mu$-a.e. point $x\in f^{-1}E$. In particular, if $g$ is a piecewise-analytic map preserving $\mu$ then there is an open $g$-invariant set $U$ containing supp $\mu$ such that $g\left|{}_U\right.$ is piecewise-linear with slopes which are rational powers of $a$. In a similar vein, for $\mu$ as above, if $b$ is another integer and $a,b$ are not powers of a common integer, and if $\nu$ is...

Abstract. We study a Neumann problem for the heat equation in a cylindrical domain with $C^1$-base and data in $h_c^1$, a subspace of $L$1. We derive our results, considering the action of an adjoint operator on $B_{T}MOC$, a predual of $h_c^1$, and using known properties of this last space.

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...

In many recent articles, medians have been used as a replacement of integral averages when the function fails to be locally integrable. A point $x$ in a metric measure space $(X,d,\mu )$ is called a generalized Lebesgue point of a measurable function $f$ if the medians of $f$ over the balls $B(x,r)$ converge to $f\left(x\right)$ when $r$ converges to $0$. We know that almost every point of a measurable, almost everywhere finite function is a generalized Lebesgue point and the same is true for every point of a continuous function....

Let β ∈ (1,2) and x ∈ [0,1/(β-1)]. We call a sequence ${\left({ϵ}_i\right)}_{i=1}^{\infty }\in {0,1}^{\mathbb{N}}$ a β-expansion for x if $x={\sum }_{i=1}^{\infty }{ϵ}_i{\beta }^{-i}$. We call a finite sequence ${\left({ϵ}_i\right)}_{i=1}^n\in {0,1}^n$ an n-prefix for x if it can be extended to form a β-expansion of x. In this paper we study how good an approximation is provided by the set of n-prefixes. Given $\Psi :\mathbb{N}\to {ℝ}_{\ge 0}$, we introduce the following subset of ℝ: $W_{\beta }\left(\Psi \right):={\bigcap }_{m=1}^{\infty }{\bigcup }_{n=m}^{\infty }{\bigcup }_{{\left({ϵ}_i\right)}_{i=1}^n\in {0,1}^n}[{\sum }_{i=1}^n\left({ϵ}_i\right)/\left({\beta }^i\right),{\sum }_{i=1}^n\left({ϵ}_i\right)/\left({\beta }^i\right)+\Psi \left(n\right)]$In other words, $W_{\beta }\left(\Psi \right)$ is the set of x ∈ ℝ for which there exist infinitely many solutions to the inequalities $0\le x-{\sum }_{i=1}^n\left({ϵ}_i\right)/\left({\beta }^i\right)\le \Psi \left(n\right)$. When ${\sum }_{n=1}^{\infty }{2}^n\Psi \left(n\right)<\infty$, the Borel-Cantelli lemma tells us that the Lebesgue measure...

We consider the perturbed Neumann problem ⎧ -Δu + α(x)u = α(x)f(u) + λg(x,u) a.e. in Ω, ⎨ ⎩ ∂u/∂ν = 0 on ∂Ω, where Ω is an open bounded set in ${ℝ}^N$ with boundary of class C², $\alpha \in L^{\infty }\left(\Omega \right)$ with $essin{f}_{\Omega }\alpha >0$, f: ℝ → ℝ is a continuous function and g: Ω × ℝ → ℝ, besides being a Carathéodory function, is such that, for some p > N, $su{p}_{\left|s\right|\le t}\left|g(\cdot ,s)\right|\in L^p\left(\Omega \right)$ and $g(\cdot ,t)\in L^{\infty }\left(\Omega \right)$ for all t ∈ ℝ. In this setting, supposing only that the set of global minima of the function $1/2\xi ²-{\int }_0^{\xi }f\left(t\right)dt$ has M ≥ 2 bounded connected components, we prove that, for all λ ∈ ℝ small enough,...

We consider the exclusion process in the one-dimensional discrete torus with $N$ points, where all the bonds have conductance one, except a finite number of slow bonds, with conductance $N^{-\beta }$, with $\beta \in [0,\infty )$. We prove that the time evolution of the empirical density of particles, in the diffusive scaling, has a distinct behavior according to the range of the parameter $\beta$. If $\beta \in [0,1)$, the hydrodynamic limit is given by the usual heat equation. If $\beta =1$, it is given by a parabolic equation involving an operator...

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 $(X,d,\mu )$ be a metric measure space satisfying the doubling condition and an $L^2$-Poincaré inequality. Consider the nonnegative operator $ℒ$ generalized by a Dirichlet form on $X$. We will show that a solution $u$ to $(-{\partial }_t^2+ℒ)u=0$ on $X\times {ℝ}_{+}$ satisfies an $\alpha$-Carleson condition if and only if $u$ can be represented as the Poisson integral of the operator $ℒ$ with the trace in the generalized Morrey space $L^{2,\alpha }\left(X\right)$, where $\alpha$ is a nonnegative function defined on a class of balls in $X$. This result extends the analogous characterization...

We present two results on existence of infinitely many positive solutions to the Neumann problem ⎧ $-{\Delta }_{p}u+\lambda \left(x\right){\left|u\right|}^{p-2}u=\mu f(x,u)$ in Ω, ⎨ ⎩ ∂u/∂ν = 0 on ∂Ω, where $\Omega \subset {ℝ}^N$ is a bounded open set with sufficiently smooth boundary ∂Ω, ν is the outer unit normal vector to ∂Ω, p > 1, μ > 0, $\lambda \in L^{\infty }\left(\Omega \right)$ with $essin{f}_{x\in \Omega }\lambda \left(x\right)>0$ and f: Ω × ℝ → ℝ is a Carathéodory function. Our results ensure the existence of a sequence of nonzero and nonnegative weak solutions to the above problem.

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 ${\nu }_0$.

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...