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].

Let $u$ be a weight on $(0,\infty )$. Assume that $u$ is continuous on $(0,\infty )$. Let the operator $S_u$ be given at measurable non-negative function $\varphi$ on $(0,\infty )$ by $$S_u\varphi \left(t\right)=\underset{0<\tau \le t}{sup}u\left(\tau \right)\varphi \left(\tau \right).$$ We characterize weights $v,w$ on $(0,\infty )$ for which there exists a positive constant $C$ such that the inequality $${\left({\int }_0^{\infty }{\left[S_u\varphi \left(t\right)\right]}^{q}w\left(t\right)dt\right)}^{\frac{1}{q}}\lesssim {\left({\int }_0^{\infty }{\left[\varphi \left(t\right)\right]}^{p}v\left(t\right)dt\right)}^{\frac{1}{p}}$$ holds for every $0<p,q<\infty$. Such inequalities have been used in the study of optimal Sobolev embeddings and boundedness of certain operators on classical Lorenz spaces.

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.

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 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.

This paper discusses the existence and multiplicity of solutions for a class of $p\left(x\right)$-Kirchhoff type problems with Dirichlet boundary data of the following form $$\left\}\begin{array}{cc}-\left(a+b{\int }_{\Omega }\frac{1}{p\left(x\right)}{\left|\nabla u\right|}^{p\left(x\right)}dx\right){\mathrm{div}\left(\right|\nabla u|}^{p\left(x\right)-2}\nabla u)=f(x,u),\hfill & in\Omega \hfill \\ u=0\hfill & on\partial \Omega ,\hfill \end{array}\right.$$ where $\Omega$ is a smooth open subset of ${ℝ}^N$ and $p\in C\left(\overline{\Omega }\right)$ with $N<p^{-}={inf}_{x\in \Omega }p\left(x\right)\le p^{+}={sup}_{x\in \Omega }p\left(x\right)<+\infty$, $a$, $b$ are positive constants and $f:\overline{\Omega }\times ℝ\to ℝ$ is a continuous function. The proof is based on critical point theory and variable exponent Sobolev space theory.

In this paper, characterizations of the embeddings between weighted Copson function spaces ${\mathrm{Cop}}_{p_1,q_1}(u_1,v_1)$ and weighted Cesàro function spaces ${\mathrm{Ces}}_{p_2,q_2}(u_2,v_2)$ are given. In particular, two-sided estimates of the optimal constant $c$ in the inequality $$\begin{array}{cc}\hfill d({\int }_0^{\infty }& {\left({\int }_0^{t}f{\left(\tau \right)}^{p_2}{v}_2\left(\tau \right)\mathrm{d}\tau \right)}^{q_2/p_2}{u}_2\left(t\right)\mathrm{d}t{)}^{1/q_2}\hfill \\ & \le c{\left({\int }_0^{\infty }{\left({\int }_t^{\infty }f{\left(\tau \right)}^{p_1}{v}_1\left(\tau \right)\mathrm{d}\tau \right)}^{q_1/p_1}{u}_1\left(t\right)\mathrm{d}t\right)}^{1/q_1},\hfill \end{array}d$$ where $p_1,p_2,q_1,q_2\in (0,\infty )$, $p_2\le q_2$ and $u_1$, $u_2$, $v_1$, $v_2$ are weights on $(0,\infty )$, are obtained. The most innovative part consists of the fact that possibly different parameters $p_1$ and $p_2$ and possibly different inner weights $v_1$ and $v_2$ are allowed. The proof is based on the combination of duality techniques...

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 give a quantitative characterization of the pairs of weights $(w,v)$ for which the dyadic version of the one-sided Hardy-Littlewood maximal operator satisfies a restricted weak $(p,p)$ type inequality for $1\le p<\infty$. More precisely, given any measurable set $E_0$, the estimate $$w\left(\{x\in {ℝ}^n:M^{+,d}\left({𝒳}_{E_0}\right)\left(x\right)>t\}\right)\le \frac{C{\left[(w,v)\right]}_{A_p^{+,d}\left(ℛ\right)}^p}{t^p}v\left(E_0\right)$$ holds if and only if the pair $(w,v)$ belongs to $A_p^{+,d}\left(ℛ\right)$, that is, $$\frac{\left|E\right|}{\left|Q\right|}\le {\left[(w,v)\right]}_{A_p^{+,d}\left(ℛ\right)}{\left(\frac{v\left(E\right)}{w\left(Q\right)}\right)}^{1/p}$$ for every dyadic cube $Q$ and every measurable set $E\subset Q^{+}$. The proof follows some ideas appearing in S. Ombrosi (2005). We also obtain a similar quantitative characterization for the...

We prove that almost all positive even integers $n$ can be represented as $p_2^2+p_3^3+p_4^4+p_5^5$ with $|p_k^k-\frac{1}{4}N|\le N^{1-1/54+\epsilon }$ for $2\le k\le 5$. As a consequence, we show that each sufficiently large odd integer $N$ can be written as $p_1+p_2^2+p_3^3+p_4^4+p_5^5$ with $|p_k^k-\frac{1}{5}N|\le N^{1-1/54+\epsilon }$ for $1\le k\le 5$.

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.

The paper considers a scalar differential equation of an advance-delay type $$\dot{y}\left(t\right)=-\left(a_0+\frac{a_1}{t}\right)y(t-\tau )+\left(b_0+\frac{b_1}{t}\right)y(t+\sigma ),$$ where constants $a_0$, $b_0$, $\tau$ and $\sigma$ are positive, and $a_1$ and $b_1$ are arbitrary. The behavior of its solutions for $t\to \infty$ is analyzed provided that the transcendental equation $$\lambda =-a_0{\mathrm{e}}^{-\lambda \tau }+b_0{\mathrm{e}}^{\lambda \sigma }$$ has a positive real root. An exponential-type function approximating the solution is searched for to be used in proving the existence of a semi-global solution. Moreover, the lower and upper estimates are given for such a solution.