For a real number $q>1$ and a positive integer $m$, let $Y_m\left(q\right):=\left\{{\sum }_{i=0}^n{ϵ}_i{q}^i:{ϵ}_i\in \left\{0,\pm 1,...,\pm m\right\},n=0,1,...\right\}$. In this paper, we show that $Y_m\left(q\right)$ is dense in $ℝ$ if and only if $q<m+1$ and $q$ is not a Pisot number. This completes several previous results and answers an open question raised by Erdös, Joó and Komornik [8].

We give new results on square functions $${\parallel x\parallel }_F=\parallel {\left({\int }_0^{\infty }{\left|F\left(tA\right)x\right|}^2\frac{\mathrm{d}t}{t}\right)}^{1/2}{\parallel }_p$$ associated to a sectorial operator $A$ on $L^p$ for $1\&lt;p\&lt;\infty$. Under the assumption that $A$ is actually $R$-sectorial, we prove equivalences of the form $K^{-1}{\parallel x\parallel }_G\le {\parallel x\parallel }_F\le K{\parallel x\parallel }_G$ for suitable functions $F,G$. We also show that $A$ has a bounded $H^{\infty }$ functional calculus with respect to ${\parallel .\parallel }_F$. Then we apply our results to the study of conditions under which we have an estimate $\parallel {({\int }_0^{\infty }|C{\mathrm{e}}^{-tA}{\left(x\right)|}^2\mathrm{d}t{)}^{1/2}\parallel }_q\le M{\parallel x\parallel }_p$, when $-A$ generates a bounded semigroup ${\mathrm{e}}^{-tA}$ on $L^p$ and $C:D\left(A\right)\to L^q$ is a linear mapping.

Let $P\left(z\right)={\sum }_{\nu =0}^n{a}_{\nu }{z}^{\nu }$ be a polynomial of degree at most $n$ which does not vanish in the disk $\left|z\right|<1$, then for $1\le p<\infty$ and $R>1$, Boas and Rahman proved $${∥P\left(Rz\right)∥}_p\le ({∥R^n+z∥}_p/{∥1+z∥}_p){∥P∥}_p.$$ In this paper, we improve the above inequality for $0\le p<\infty$ by involving some of the coefficients of the polynomial $P\left(z\right)$. Analogous result for the class of polynomials $P\left(z\right)$ having no zero in $\left|z\right|>1$ is also given.

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...

We consider functions $u\in W_0^{m,1}\left(\Omega \right)$, where $\Omega \subset {ℝ}^N$ is a smooth bounded domain, and $m\ge 2$ is an integer. For all $j\ge 0,1\le k\le m-1$, such that $1\le j+k\le m$, we prove that $\frac{{\partial }^{i}u\left(x\right)}{d{\left(x\right)}^{m-j-k}}\in W_0^{k,1}\left(\Omega \right)$ with $∥{\partial }^k{\left(\frac{{\partial }^{i}u\left(x\right)}{d{\left(x\right)}^{m-j-k}}\right)}_{L^1\left(\Omega \right)}\le C∥u∥{}_{W^{m,1}\left(\Omega \right)}$, where $d$ is a smooth positive function which coincides with dist$(x,\partial \Omega )$ near $\partial \Omega$, and ${\partial }^l$ denotes any partial differential operator of order $l$.

We consider the integral functional $J\left(u\right)={\int }_{\Omega }\left[f\right(\left|Du\right|\left)-u\right]dx$, $u\in W_0^{1,1}\left(\Omega \right)$, where $\Omega \subset {ℝ}^n$, $n\ge 2$, is a nonempty bounded connected open subset of ${ℝ}^n$ with smooth boundary, and $ℝ\ni s\mapsto f\left(\right|s\left|\right)$ is a convex, differentiable function. We prove that if $J$ admits a minimizer in $W_0^{1,1}\left(\Omega \right)$ depending only on the distance from the boundary of $\Omega$, then $\Omega$ must be a ball.

Let $\Omega \in L^s\left({\mathrm{S}}^{n-1}\right)$ for $s\ge 1$ be a homogeneous function of degree zero and $b$ a BMO function. The commutator generated by the Marcinkiewicz integral ${\mu }_{\Omega }$ and $b$ is defined by $${\displaystyle [b,{\mu }_{\Omega }]\left(f\right)\left(x\right)=({\int }_0^{\infty }{\left|{\int }_{|x-y|\le t}\frac{\Omega (x-y)}{{|x-y|}^{n-1}}[b\left(x\right)-b\left(y\right)]f\left(y\right)\mathrm{d}y{|}^2\frac{\mathrm{d}t}{t^3}\right)}^{1/2}.}$$ In this paper, the author proves the $(L^{p(·)}\left({ℝ}^n\right),L^{p(·)}\left({ℝ}^n\right))$-boundedness of the Marcinkiewicz integral operator ${\mu }_{\Omega }$ and its commutator $[b,{\mu }_{\Omega }]$ when $p(·)$ satisfies some conditions. Moreover, the author obtains the corresponding result about ${\mu }_{\Omega }$ and $[b,{\mu }_{\Omega }]$ on Herz spaces with variable exponent.

Let $𝒜$ and $ℬ$ be abelian categories with enough projective and injective objects, and $T:𝒜\to ℬ$ a left exact additive functor. Then one has a comma category $\left(ℬ\downarrow T\right)$. It is shown that if $T:𝒜\to ℬ$ is $𝒳$-exact, then ${(}^{\perp }𝒳,𝒳)$ is a (hereditary) cotorsion pair in $𝒜$ and ${(}^{\perp }𝒴,𝒴)$) is a (hereditary) cotorsion pair in $ℬ$ if and only if $\left(\left(\genfrac{}{}{0pt}{}{{}^{\perp }𝒳}{{}^{\perp }𝒴}\right)\right),\langle 𝐡(𝒳,𝒴)\rangle )$ is a (hereditary) cotorsion pair in $\left(ℬ\downarrow T\right)$ and $𝒳$ and $𝒴$ are closed under extensions. Furthermore, we characterize when special preenveloping classes in abelian categories $𝒜$ and $ℬ$ can induce special preenveloping...

The aim of this paper is to establish an existence and uniqueness result for a class of the set functional differential equations of neutral type$$\left\{\begin{array}{c}{D}_{H}X\left(t\right)=F(t,X_t,D_H{X}_t),\hfill \\ {X|}_{[-r,0]}=\Psi ,\hfill \end{array}\right.$$ where $F:[0,b]\times {𝒞}_0\times {𝔏}_0^1\to K_c\left(E\right)$ is a given function, $K_c\left(E\right)$ is the family of all nonempty compact and convex subsets of a separable Banach space $E$, ${𝒞}_0$ denotes the space of all continuous set-valued functions $X$ from $[-r,0]$ into $K_c\left(E\right)$, ${𝔏}_0^1$ is the space of all integrally bounded set-valued functions $X:[-r,0]\to K_c\left(E\right)$, $\Psi \in {𝒞}_0$ and $D_H$ is the Hukuhara derivative. The continuous dependence of solutions on initial...

We use the genus theory to prove the existence and multiplicity of solutions for the fractional $p$-Kirchhoff problem $$\left\{\begin{array}{c}{\displaystyle -{\left[M\left({\int }_Q{\displaystyle \frac{{|u\left(x\right)-u\left(y\right)|}^p}{{|x-y|}^{N+ps}}}\mathrm{d}x\mathrm{d}y\right)\right]}^{p-1}{(-\Delta )}_p^{s}u=\lambda h(x,u)\text{in}\Omega ,}\hfill \\ u=0\text{on}{ℝ}^N\setminus \Omega ,\hfill \end{array}\right.$$ where $\Omega$ is an open bounded smooth domain of ${ℝ}^N$, $p>1$, $N>ps$ with $s\in (0,1)$ fixed, $Q={ℝ}^{2N}\setminus (C\Omega \times C\Omega )$, $\lambda >0$ is a numerical parameter, $M$ and $h$ are continuous functions.

CONTENTSIntroduction.......................................................................................61. Definition of certain special forms...........................................62. Statement of results...................................................................83. Proof of Theorem 2.....................................................................94. Preliminaries for Theorem 1.....................................................105. Further preliminaries for Theorem...

In the present paper the author discusses certain multiple integrals $I(u)$ of the calculus of variations satisfying convexity conditions, and no growth property, and the corresponding Serrin integrals $\Im (u)$, to which the existence theorems in [3,4,5] do not apply. However, in the present paper, the integrals $I(u)$ and $\Im (u)$ are reduced to simpler form $H(v)$ and $\mathscr{H}(v)$ to which the existence theorems above apply. Thus, we derive that $I(u)\le \Im (u)$, $H(v)\le \mathscr{H}(v)$, we obtain the existence of the absolute minimum for the Serrin forms $\Im (u)$...

We characterise the boundedness of the $H^{\infty }$ calculus of a sectorial operator in terms of dilation theorems. We show e. g. that if $-A$ generates a bounded analytic $C_0$ semigroup $\left(T_t\right)$ on a UMD space, then the $H^{\infty }$ calculus of $A$ is bounded if and only if $\left(T_t\right)$ has a dilation to a bounded group on $L^2([0,1],X)$. This generalises a Hilbert space result of C.LeMerdy. If $X$ is an $L^p$ space we can choose another $L^p$ space in place of $L^2([0,1],X)$.

Let $A$ be a closed set of $M\simeq {\mathbb{R}}^n$, whose conormai cones $x+y_x^{*}\left(A\right)$, $x\in A$, have locally empty intersection. We first show in §1 that $\text{dist}\left(x,A\right)$, $x\in M\setminus A$ is a $C^1$ function. We then represent the n microfunctions of ${\mathcal{C}}_{A|X}$, $X\simeq {\mathbb{C}}^n$, using cohomology groups of ${\mathcal{O}}_X$ of degree 1. By the results of § 1-3, we are able to prove in §4 that the sections of ${\left.{\mathcal{C}}_{A|X}\right|}_{{\dot{\pi }}^{-1}\left(x_0\right)}$, $x_0\in \partial A$, satisfy the principle of the analytic continuation in the complex integral manifolds of ${\left\{H\left({\varphi }_i^C\right)\right\}}_{i=1,\mathrm{\dots },m}$, $\left\{{\varphi }_i\right\}$ being a base for the linear hull of ${\gamma }_{x_0}^{*}\left(A\right)$ in $T_{x_0}^{*}M$; in particular we get ${\left.{\mathrm{\Gamma }}_{A\times {}_{M}T{}^{*}{}_{M}X}\left({\mathcal{C}}_{A|X}\right)\right|}_{\partial A\times {}_M\dot{T}{}^{*}{}_{M}X}=0$. When $A$is a half space with...