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 ${\alpha }_i,{\beta }_i>0$, 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 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...

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...

Let $H\subseteq Z\subseteq 2^{\omega }$. For n ≥ 2, we prove that if Selivanovski measurable functions from $2^{\omega }$ to Z give as preimages of H all Σₙ¹ subsets of $2^{\omega }$, then so do continuous injections.

We show that any ${\Sigma }_s$-product of at most $𝔠$-many $L\Sigma (\le \omega )$-spaces has the $L\Sigma (\le \omega )$-property. This result generalizes some known results about $L\Sigma (\le \omega )$-spaces. On the other hand, we prove that every ${\Sigma }_s$-product of monotonically monolithic spaces is monotonically monolithic, and in a similar form, we show that every ${\Sigma }_s$-product of Collins-Roscoe spaces has the Collins-Roscoe property. These results generalize some known results about the Collins-Roscoe spaces and answer some questions due to Tkachuk [Lifting the Collins-Roscoe...

For each ordinal 1 ≤ α < ω₁ we present separable metrizable spaces $X_{\alpha }$, $Y_{\alpha }$ and $Z_{\alpha }$ such that (i) $f{X}_{\alpha },f{Y}_{\alpha },f{Z}_{\alpha }=\omega ₀$, where f is either trdef or ₀-trsur, (ii) $A\left(\alpha \right)-trind{X}_{\alpha }=\infty$ and $M\left(\alpha \right)-trind{X}_{\alpha }=-1$, (iii) $A\left(\alpha \right)-trind{Y}_{\alpha }=-1$ and $M\left(\alpha \right)-trind{Y}_{\alpha }=\infty$, and (iv) $A\left(\alpha \right)-trind{Z}_{\alpha }=M\left(\alpha \right)-trind{Z}_{\alpha }=\infty$ and $A(\alpha +1)\cap M(\alpha +1)-trind{Z}_{\alpha }=-1$. We also show that there exists no separable metrizable space $W_{\alpha }$ with $A\left(\alpha \right)-trind{W}_{\alpha }\ne \infty$, $M\left(\alpha \right)-trind{W}_{\alpha }\ne \infty$ and $A\left(\alpha \right)\cap M\left(\alpha \right)-trind{W}_{\alpha }=\infty$, where A(α) (resp. M(α)) is the absolutely additive (resp. multiplicative) Borel class.

We show that every operator from ${\ell }_s$ to ${\ell }_p\otimes ̂{\ell }_q$ is compact when 1 ≤ p,q < s and that every operator from ${\ell }_s$ to ${\ell }_p\widehat{\otimes ̂}{\ell }_q$ is compact when 1/p + 1/q > 1 + 1/s.

For Banach spaces $X$ and $Y$, let $K_{w^{*}}(X^{*},Y)$ denote the space of all $w^{*}-w$ continuous compact operators from $X^{*}$ to $Y$ endowed with the operator norm. A Banach space $X$ has the $wGP$ property if every Grothendieck subset of $X$ is relatively weakly compact. In this paper we study Banach spaces with property $wGP$. We investigate whether the spaces $K_{w^{*}}(X^{*},Y)$ and $X{\otimes }_{ϵ}Y$ have the $wGP$ property, when $X$ and $Y$ have the $wGP$ property.

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 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...

A space $X$ is said to have the Rothberger property (or simply $X$ is Rothberger) if for every sequence $\langle {𝒰}_n:n\in \omega \rangle$ of open covers of $X$, there exists $U_n\in {𝒰}_n$ for each $n\in \omega$ such that $X={\bigcup }_{n\in \omega }{U}_n$. For any $n\in \omega$, necessary and sufficient conditions are obtained for $C_p{(\Psi \left(𝒜\right),2)}^n$ to have the Rothberger property when $𝒜$ is a Mrówka mad family and, assuming CH (the Continuum Hypothesis), we prove the existence of a maximal almost disjoint family $𝒜$ for which the space $C_p{(\Psi \left(𝒜\right),2)}^n$ is Rothberger for all $n\in \omega$.

Let ${}_s$ be the family of open rectangles in the plane ℝ² with a side of angle s to the x-axis. We say that a set S of directions is an R-set if there exists a function f ∈ L¹(ℝ²) such that the basis ${}_s$ differentiates the integral of f if s ∉ S, and $D{̅}_{s}f\left(x\right)=limsu{p}_{diam\left(R\right)\to 0,x\in R{\in }_s}{\left|R\right|}^{-1}{\int }_{R}f=\infty$ almost everywhere if s ∈ S. If the condition $D{̅}_{s}f\left(x\right)=\infty$ holds on a set of positive measure (instead of a.e.) we say that S is a WR-set. It is proved that S is an R-set (resp. a WR-set) if and only if it is a $G_{\delta }$ (resp. a $G_{\delta \sigma }$).

We prove that if $X$ is a first countable space with property $\left(DC\left({\omega }_1\right)\right)$ and with a $G_{\delta }$-diagonal then the cardinality of $X$ is at most $𝔠$. We also show that if $X$ is a first countable, DCCC, normal space then the extent of $X$ is at most $𝔠$.

Let $D$ be a ${𝒞}^d$ $q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, ${𝒞}^k$-estimates, $k=2,3,\cdots ,\infty$, for solutions to the $\overline{\partial }$-equation with small loss of smoothness are obtained for $E$-valued $(0,s)$-forms on $D$ when $n-q\le s\le n$. In addition, we solve the $\overline{\partial }$-equation with a support condition in ${𝒞}^k$-spaces. More precisely, we prove that for a $\overline{\partial }$-closed form $f$ in ${𝒞}_{0,q}^k(X\setminus D,E)$, $1\le q\le n-2$, $n\ge 3$, with compact support and for $\epsilon$ with $0<\epsilon <1$ there...

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...

Let X be a Polish space, and let C₀ and C₁ be disjoint coanalytic subsets of X. The pair (C₀,C₁) is said to be complete if for every pair (D₀,D₁) of disjoint coanalytic subsets of ${\omega }^{\omega }$ there exists a continuous function $f:{\omega }^{\omega }\to X$ such that $f^{-1}\left(C₀\right)=D₀$ and $f^{-1}\left(C₁\right)=D₁$. We give several explicit examples of complete pairs of coanalytic sets.

Let $f^j={\sum }_k{a}_{k}f(2^{j+1}x-2k)$, where the sum is taken over the lattice of all points k in ${ℝ}^n$ having integer-valued components, j∈ℕ and $a_k\in ℂ$. Let $A_{pq}^s$ be either $B_{pq}^s$ or $F_{pq}^s$ (s ∈ ℝ, 0 < p < ∞, 0 < q ≤ ∞) on ${ℝ}^n.$ The aim of the paper is to clarify under what conditions $\parallel f^j|A_{pq}^s\parallel$ is equivalent to $2^{j(s-n/p)}({\sum }_k|a_k{|}^p{)}^{1/p}\parallel f|A_{pq}^s\parallel$.