We find an exact asymptotic formula for the singular values of the integral operator of the form ${\int }_{\Omega }T(x,y)k(x-y)·\mathrm{d}y{L}^2\left(\Omega \right)\to L^2\left(\Omega \right)$ ($\Omega \subset {ℝ}^m$, a Jordan measurable set) where $k\left(t\right)=k_0\left({(t_1^2+t_2^2+...t_m^2)}^{\frac{m}{2}}\right)$, $k_0\left(x\right)=x^{\alpha -1}L\left(\frac{1}{x}\right)$, $\frac{1}{2}-\frac{1}{2m}<\alpha <\frac{1}{2}$ and $L$ is slowly varying function with some additional properties. The formula is an explicit expression in terms of $L$ and $T$.

The purpose of this note is to present several criteria for essential self-adjointness. The method is based on ideas due to Shubin. This note is divided into two parts. The first part deals with symmetric first order systems on the line in the most general setting. Such a symmetric first order system of differential equations gives rise naturally to a symmetric linear relation in a Hilbert space. In this case even regularity is nontrivial. We will announce a regularity result and discuss...

In this paper we study integral operators of the form $$Tf\left(x\right)=\int |x-a_1{y|}^{-{\alpha }_1}\cdots {|x-a_{m}y|}^{-{\alpha }_m}f\left(y\right)\mathrm{d}y,$$ ${\alpha }_1+\cdots +{\alpha }_m=n$. We obtain the $L^p\left(w\right)$ boundedness for them, and a weighted $(1,1)$ inequality for weights $w$ in $A_p$ satisfying that there exists $c\ge 1$ such that $w\left(a_{i}x\right)\le cw\left(x\right)$ for a.e. $x\in {ℝ}^n$, $1\le i\le m$. Moreover, we prove ${\parallel Tf\parallel }_{\mathrm{B}MO}\le c{\parallel f\parallel }_{\infty }$ for a wide family of functions $f\in L^{\infty }\left({ℝ}^n\right)$.

The paper is devoted to the estimate u(x,k)Kk{capp,w(F)pw(B(x,))} 1p-1, $2p<n$ for a solution of a degenerate nonlinear elliptic equation in a domain $B(x_0,1)\setminus F$, $F\subset B(x_0,d)=\{x\in {ℝ}^n|x_0-x|<d\}$, $d<\frac{1}{2}$, under the boundary-value conditions $u(x,k)=k$ for $x\in \partial F$, $u(x,k)=0$ for $x\in \partial B(x_0,1)$ and where $0<\rho \le dist(x,F)$, $w\left(x\right)$ is a weighted function from some Muckenhoupt class, and $ca{p}_{p,w}\left(F\right)$, $w\left(B\right(x,\rho \left)\right)$ are weighted capacity and measure of the corresponding sets.

À une famille $\Omega$ de fonctions positives $\omega$ sur un groupe $G$ abélien localement compact, on associe l’ensemble $A^2$ des fonctions qui sont de carré sommable sur $G$ par rapport à $dx/\omega \left(x\right)$ pour au moins un $\omega \in \Omega$. Sous certaines conditions simples, portant sur $\Omega$, c’est une algèbre de convolution, contenue dans $L^1\left(G\right)$. On étudie particulièrement le cas $G=$ droite réelle, $\Omega =$ ensemble des fonctions paires, sommables, décroissantes sur $]0,\infty [$. Une caractérisation explicite de $\tilde{A}$ (ensemble des transformées de Fourier des $f\in A$)...