ContentsIntroduction.............................................................................................................................31. Definitions, notations and some auxiliary lemmas...................................................42. The definition of the spaces $H_{p,q;Y}(\Omega ,ℬ)$..........................................................73. Some properties of the spaces $H_{p,q;Y}(\Omega ,ℬ)$...................................................104. Some examples of the spaces $H_{p,q;Y}(\Omega ,ℬ)$....................................................155....

We prove the $L^p$ boundedness of the Marcinkiewicz integral operators ${\mu }_{\Omega }$ on ${ℝ}^{n₁}\times \cdots \times {ℝ}^{n_k}$ under the condition that $\Omega \in L{\left(logL\right)}^{k/2}{(}^{n₁-1}\times \cdots {\times }^{n_k-1})$. The exponent k/2 is the best possible. This answers an open question posed by Y. Ding.

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...

We consider a random, uniformly elliptic coefficient field $a$ on the lattice ${\mathbb{Z}}^d$. The distribution $\langle ·\rangle$ of the coefficient field is assumed to be stationary. Delmotte and Deuschel showed that the gradient and second mixed derivative of the parabolic Green’s function $G(t,x,y)$ satisfy optimal annealed estimates which are $L^2$ and $L^1$, respectively, in probability, i.e., they obtained bounds on $\langle |{\nabla }_{x}G(t,x,y){{|}^2\rangle }^{1/2}$ and $\langle \left|{\nabla }_x{\nabla }_{y}G(t,x,y)\right|\rangle$. In particular, the elliptic Green’s function $G(x,y)$ satisfies optimal annealed bounds. In their recent work,...

Let $P\left(z\right)$ be a polynomial of degree $n$ having no zeros in $\left|z\right|<k$, $k\le 1$, and let $Q\left(z\right):=z^n\overline{P(1/\overline{z})}$. It was shown by Govil that if ${max}_{\left|z\right|=1}\left|P^{\text{'}}\left(z\right)\right|$ and ${max}_{\left|z\right|=1}\left|Q^{\text{'}}\left(z\right)\right|$ are attained at the same point of the unit circle $\left|z\right|=1$, then $$\underset{\left|z\right|=1}{max}|P^{\text{'}}\left(z\right)|\le \frac{n}{1+k^n}\underset{\left|z\right|=1}{max}\left|P\left(z\right)\right|.$$ The main result of the present article is a generalization of Govil’s polynomial inequality to a class of entire functions of exponential type.

Let p be a prime, ℤₚ be the finite field in p elements, k be a positive integer, and A be the multiplicative subgroup of nonzero kth powers in ℤₚ. The goal of this paper is to determine, for a given positive integer s, a value tₛ such that if |A| ≫ tₛ then every element of ℤₚ is a sum of s kth powers. We obtain $t₄=p^{22/39+ϵ}$, $t₅=p^{15/29+ϵ}$ and for s ≥ 6, $tₛ=p^{(9s+45)/(29s+33)+ϵ}$. For s ≥ 24 further improvements are made, such as $t_{32}=p^{5/16+ϵ}$ and $t_{128}=p^{1/4}$.

We consider $N$, the number of solutions $(x,y,u,v)$ to the equation ${(-1)}^{u}r{a}^x+{(-1)}^{v}s{b}^y=c$ in nonnegative integers $x,y$ and integers $u,v\in \{0,1\}$, for given integers $a\&gt;1$, $b\&gt;1$, $c\&gt;0$, $r\&gt;0$ and $s\&gt;0$. When $gcd(ra,sb)=1$, we show that $N\le 3$ except for a finite number of cases all of which satisfy $max(a,b,r,s,x,y)\&lt;2·{10}^{15}$ for each solution; when $gcd(a,b)\&gt;1$, we show that $N\le 3$ except for three infinite families of exceptional cases. We find several different ways to generate an infinite number of cases giving $N=3$ solutions.

In this work, we study the existence of nonnegative and nontrivial solutions for the quasilinear Schrödinger equation $$-{\Delta }_{N}u+b{\left|u\right|}^{N-2}u-{\Delta }_N\left(u^2\right)u=h\left(u\right),x\in {ℝ}^N,$$ where ${\Delta }_N$ is the $N$-Laplacian operator, $h\left(u\right)$ is continuous and behaves as $exp\left(\alpha \right|u{|}^{N/(N-1)})$ when $\left|u\right|\to \infty$. Using the Nehari manifold method and the Schwarz symmetrization with some special techniques, the existence of a nonnegative and nontrivial solution $u\left(x\right)\in W^{1,N}\left({ℝ}^N\right)$ with $u\left(x\right)\to 0$ as $\left|x\right|\to \infty$ is established.

Let ℳ be a type II₁ von Neumann algebra, τ a trace in ℳ, and L²(ℳ,τ) the GNS Hilbert space of τ. If L²(ℳ,τ)₊ is the completion of the set ${ℳ}_{sa}$ of selfadjoint elements, then each element ξ ∈ L²(ℳ,τ)₊ gives rise to a selfadjoint unbounded operator $L_{\xi }$ on L²(ℳ,τ). In this note we show that the exponential exp: L²(ℳ,τ)₊ → L²(ℳ,τ), $exp\left(\xi \right)=e^{i{L}_{\xi }}$, is continuous but not differentiable. The same holds for the Cayley transform $C\left(\xi \right)=(L_{\xi }-i){(L_{\xi }+i)}^{-1}$. We also show that the unitary group $U_{ℳ}\subset L²(ℳ,\tau )$ with the strong operator topology is not an...

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 $G_{\delta \sigma }$).

By a sub-super solution argument, we study the existence of positive solutions for the system ⎧$-{\Delta }_{p}u=a₁\left(x\right)F₁(x,u,v)$ in Ω, ⎪$-{\Delta }_{q}v=a₂\left(x\right)F₂(x,u,v)$ in Ω, ⎨u,v > 0 in Ω, ⎩u = v = 0 on ∂Ω, where Ω is a bounded domain in ${ℝ}^N$ with smooth boundary or $\Omega ={ℝ}^N$. A nonexistence result is obtained for radially symmetric solutions.

Let $f\left(x\right)$ be a cubic, monic and separable polynomial over a field of characteristic $p\ge 3$ and let $E$ be the elliptic curve given by $y^2=f\left(x\right)$. In this paper we prove that the coefficient at $x^{\frac{1}{2}p(p-1)}$ in the $p$–th division polynomial of $E$ equals the coefficient at $x^{p-1}$ in $f{\left(x\right)}^{\frac{1}{2}(p-1)}$. For elliptic curves over a finite field of characteristic $p$, the first coefficient is zero if and only if $E$ is supersingular, which by a classical criterion of Deuring (1941) is also equivalent to the vanishing of the second coefficient. So the...

Let $X$ be a Banach space, $ℬ\left(X\right)$ the algebra of bounded linear operators on $X$ and $(J,\parallel ·{\parallel }_J)$ an admissible Banach ideal of $ℬ\left(X\right)$. For $T\in ℬ\left(X\right)$, let $L_{J,T}$ and $R_{J,T}\in ℬ\left(J\right)$ denote the left and right multiplication defined by $L_{J,T}\left(A\right)=TA$ and $R_{J,T}\left(A\right)=AT$, respectively. In this paper, we study the transmission of some concepts related to recurrent operators between $T\in ℬ\left(X\right)$, and their elementary operators $L_{J,T}$ and $R_{J,T}$. In particular, we give necessary and sufficient conditions for $L_{J,T}$ and $R_{J,T}$ to be sequentially recurrent. Furthermore, we prove that $L_{J,T}$ is recurrent...

Our main purpose is to establish the existence of weak solutions of second order quasilinear elliptic systems ⎧ $-{\Delta }_{p}u+{\left|u\right|}^{p-2}u=f_{1\lambda ₁}{\left(x\right)\left|u\right|}^{q-2}u+2\alpha /(\alpha +\beta )g_{\mu }{\left|u\right|}^{\alpha -2}u{\left|v\right|}^{\beta }$, x ∈ Ω, ⎨ $-{\Delta }_{p}v+{\left|v\right|}^{p-2}v=f_{2\lambda ₂}{\left(x\right)\left|v\right|}^{q-2}v+2\beta /(\alpha +\beta )g_{\mu }{\left|u\right|}^{\alpha }{\left|v\right|}^{\beta -2}v$, x ∈ Ω, ⎩ u = v = 0, x∈ ∂Ω, where 1 < q < p < N and $\Omega \subset {ℝ}^N$ is an open bounded smooth domain. Here λ₁, λ₂, μ ≥ 0 and $f_{i{\lambda }_i}\left(x\right)={\lambda }_i{f}_{i+}\left(x\right)+f_{i-}\left(x\right)$ (i = 1,2) are sign-changing functions, where $f_{i\pm }\left(x\right)=max\pm f_i\left(x\right),0$, $g_{\mu }\left(x\right)=a\left(x\right)+\mu b\left(x\right)$, and ${\Delta }_{p}u={div\left(\right|\nabla u|}^{p-2}\nabla u)$ denotes the p-Laplace operator. We use variational methods.

The harmonic Cesàro operator is defined for a function f in $L^p\left(ℝ\right)$ for some 1 ≤ p < ∞ by setting $\left(f\right)\left(x\right):={\int }_x^{\infty }(f\left(u\right)/u)du$ for x > 0 and $\left(f\right)\left(x\right):=-{\int }_{-\infty }^x(f\left(u\right)/u)du$ for x < 0; the harmonic Copson operator ℂ* is defined for a function f in $L{¹}_{loc}\left(ℝ\right)$ by setting $*\left(f\right)\left(x\right):=(1/x){\int }^{x₀}f\left(u\right)du$ for x ≠ 0. The notation indicates that ℂ and ℂ* are adjoint operators in a certain sense. We present rigorous proofs of the following two commuting relations: (i) If $f\in L^p\left(ℝ\right)$ for some 1 ≤ p ≤ 2, then ${\left(\left(f\right)\right)}^{\wedge }\left(t\right)=*\left(f̂\right)\left(t\right)$ a.e., where f̂ denotes the Fourier transform of f. (ii) If $f\in L^p\left(ℝ\right)$ for some 1 < p ≤ 2, then...

In this paper, we obtain new sufficient conditions for the operators $F_{{\alpha }_1,{\alpha }_2,...,{\alpha }_n,\beta }\left(z\right)$ and $G_{{\alpha }_1,{\alpha }_2,...,{\alpha }_n,\beta }\left(z\right)$ to be univalent in the open unit disc $𝒰$, where the functions $f_1,f_2,...,f_n$ belong to the classes $S^{*}(a,b)$ and $𝒦(a,b)$. The order of convexity for the operators $F_{{\alpha }_1,{\alpha }_2,...,{\alpha }_n,\beta }\left(z\right)$ and $G_{{\alpha }_1,{\alpha }_2,...,{\alpha }_n,\beta }\left(z\right)$ is also determined. Furthermore, and for $\beta =1$, we obtain sufficient conditions for the operators $F_n\left(z\right)$ and $G_n\left(z\right)$ to be in the class $𝒦(a,b)$. Several corollaries and consequences of the main results are also considered.