We consider the Heisenberg group ℍⁿ = ℂⁿ × ℝ. Let ν be the Borel measure on ℍⁿ defined by $\nu \left(E\right)={\int }_{ℂⁿ}{\chi }_E(w,\phi \left(w\right))\eta \left(w\right)dw$, where $\phi \left(w\right)={\sum }_{j=1}^n{a}_j\left|w_j\right|²$, w = (w₁,...,wₙ) ∈ ℂⁿ, $a_j\in ℝ$, and η(w) = η₀(|w|²) with $\eta ₀\in C_c^{\infty }\left(ℝ\right)$. We characterize the set of pairs (p,q) such that the convolution operator with ν is $L^p\left(ℍⁿ\right)-L^q\left(ℍⁿ\right)$ bounded. We also obtain $L^p$-improving properties of measures supported on the graph of the function $\phi \left(w\right)={\left|w\right|}^{2m}$.

For the convolution operators $A_a^{\alpha }$ with symbols ${a\left(\right|\xi \left|\right)\left|\xi \right|}^{-\alpha }expi\left|\xi \right|$, 0 ≤ Re α < n, $a\left(\right|\xi \left|\right)\in L_{\infty }$, we construct integral representations and give the exact description of the set of pairs (1/p,1/q) for which the operators are bounded from $L_p$ to $L_q$.

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 ${\nu }_0$.

A convolution operator, bounded on $L^q({ℝ}^n)$, is bounded on $L^p({ℝ}^n)$, with the same operator norm, if $p$ and $q$ are conjugate exponents. It is well known that this fact is false if we replace ${ℝ}^n$ with a general non-commutative locally compact group $G$. In this paper we give a simple construction of a convolution operator on a suitable compact group $G$, wich is bounded on $L^q(G)$ for every $q\in [2,\mathrm{\infty })$ and is unbounded on $L^p(G)$ if $p\in [1,2)$.

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 φ:ℝ ² → ℝ be a homogeneous polynomial function of degree m ≥ 2, let μ be the Borel measure on ℝ ³ defined by $\mu \left(E\right)={\int }_D{\chi }_E(x,\phi \left(x\right))dx$ with D = x ∈ ℝ ²:|x| ≤ 1 and let $T_{\mu }$ be the convolution operator with the measure μ. Let $\phi =\phi {₁}^{e₁}\cdots \phi {ₙ}^{eₙ}$ be the decomposition of φ into irreducible factors. We show that if $e_i\ne m/2$ for each ${\phi }_i$ of degree 1, then the type set $E_{\mu }:=(1/p,1/q)\in [0,1]\times [0,1]:\left|\right|T_{\mu }{\left|\right|}_{p,q}<\infty$ can be explicitly described as a closed polygonal region.

Let $\mu \in {\left({ℝ}^d\right)}^{\text{'}}$ be an analytic functional and let $T_{\mu }$ be the corresponding convolution operator on Sato’s space $\left({ℝ}^d\right)$ of hyperfunctions. We show that $T_{\mu }$ is surjective iff $T_{\mu }$ admits an elementary solution in $\left({ℝ}^d\right)$ iff the Fourier transform μ̂ satisfies Kawai’s slowly decreasing condition (S). We also show that there are $0\ne \mu \in {\left({ℝ}^d\right)}^{\text{'}}$ such that $T_{\mu }$ is not surjective on $\left({ℝ}^d\right)$.

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 $L_G\left(s\right)$ to be the $L$-function attached to a particular automorphic form $G$ on $GL\left(6\right)$. We establish an upper bound for the mean square estimate on the critical line of Rankin-Selberg $L$-function $L_{G\times G}\left(s\right)$. As an application of this result, we give an asymptotic formula for the discrete sum of coefficients of $L_{G\times G}\left(s\right)$.

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...

A classification of weakly compact multiplication operators on $L\left(L_p\right),$1<p<$,isgiven.ThisanswersaquestionraisedbySaksmanandTylliin1992.Theclassificationinvolvestheconceptof$p$-strictlysingularoperators,andwealsoinvestigatethestructureofgeneral$p$-strictlysingularoperatorson$Lp$.Themainresultisthatifanoperator$T$on$Lp$,$1<p<2$,is$p$-strictlysingularand$T|X$isanisomorphismforsomesubspace$X$of$Lp$,then$X$embedsinto$Lr$forall$r<2$,but$X$neednotbeisomorphictoaHilbertspace.$It is also shown that if $T$ is convolution by a biased coin on $L_p$ of the Cantor group, $1\le p<2$, and $T_{|X}$ is an isomorphism for some reflexive subspace $X$ of $L_p$, then $X$ is isomorphic to a Hilbert space. The case $p=1$ answers a question asked by Rosenthal in 1976.

We study the chemotaxis system with singular sensitivity and logistic-type source: $u_t=\Delta u-\chi \nabla ·(u\nabla v/v)+ru-\mu u^k$, $0=\Delta v-v+u$ under the non-flux boundary conditions in a smooth bounded domain $\Omega \subset {ℝ}^n$, $\chi ,r,\mu >0$, $k>1$ and $n\ge 1$. It is shown with $k\in (1,2)$ that the system possesses a global generalized solution for $n\ge 2$ which is bounded when $\chi >0$ is suitably small related to $r>0$ and the initial datum is properly small, and a global bounded classical solution for $n=1$.

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.

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...

A convolution operator, bounded on $L^q({ℝ}^n)$, is bounded on $L^p({ℝ}^n)$, with the same operator norm, if $p$ and $q$ are conjugate exponents. It is well known that this fact is false if we replace ${ℝ}^n$ with a general non-commutative locally compact group $G$. In this paper we give a simple construction of a convolution operator on a suitable compact group $G$, wich is bounded on $L^q(G)$ for every $q\in [2,\mathrm{\infty })$ and is unbounded on $L^p(G)$ if $p\in [1,2)$.

Given d ≥ 2 consider the family of polynomials $P_c\left(z\right)=z^d+c$ for c ∈ ℂ. Denote by $J_c$ the Julia set of $P_c$ and let ${ℳ}_d=c|J_{c}isconnected$ be the connectedness locus; for d = 2 it is called the Mandelbrot set. We study semihyperbolic parameters $c₀\in \partial {ℳ}_d$: those for which the critical point 0 is not recurrent by $P_{c₀}$ and without parabolic cycles. The Hausdorff dimension of $J_c$, denoted by $HD\left(J_c\right)$, does not depend continuously on c at such $c₀\in \partial {ℳ}_d$; on the other hand the function $c\mapsto HD\left(J_c\right)$ is analytic in $ℂ-{ℳ}_d$. Our first result asserts that there is still some...

In this note we shall prove that for a continuous function $\varphi :\Delta \to {ℝ}^n$, where $\Delta \subset ℝ$,  the paratingent of $\varphi$ at $a\in \Delta$ is a non-empty and compact set in ${ℝ}^n$ if and only if $\varphi$ satisfies Lipschitz condition in a neighbourhood of $a$. Moreover, in this case the paratingent is a connected set.

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 $\mu$ 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 WF⁎ be the wave front set with respect to $C^{\infty }$, quasi analyticity or analyticity, and let K be the kernel of a positive operator from $C{₀}^{\infty }$ to ’. We prove that if ξ ≠ 0 and (x,x,ξ,-ξ) ∉ WF⁎(K), then (x,y,ξ,-η) ∉ WF⁎(K) and (y,x,η,-ξ) ∉ WF⁎(K) for any y,η. We apply this property to positive elements with respect to the weighted convolution $u{\ast }_B\phi \left(x\right)=\int u(x-y)\phi \left(y\right)B(x,y)dy$, where $B\in C^{\infty }$ is appropriate, and prove that if $(u{\ast }_B\phi ,\phi )\ge 0$ for every $\phi \in C{₀}^{\infty }$ and (0,ξ) ∉ WF⁎(u), then (x,ξ) ∉ WF⁎(u) for any x.

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...

Let $A$ be the class of analytic functions in the unit disc $U$ of the complex plane $ℂ$ with the normalization $f\left(0\right)=f^{{}^{\text{'}}}\left(0\right)-1=0$. We introduce a subclass $S_M^{*}(\alpha ,b)$ of $A$, which unifies the classes of bounded starlike and convex functions of complex order. Making use of Salagean operator, a more general class $S_M^{*}(n,\alpha ,b)$ ($n\ge 0$) related to $S_M^{*}(\alpha ,b)$ is also considered under the same conditions. Among other things, we find convolution conditions for a function $f\in A$ to belong to the class $S_M^{*}(\alpha ,b)$. Several properties of the class $S_M^{*}(n,\alpha ,b)$ are investigated. ...