ﻻ يوجد ملخص باللغة العربية
This note is devoted to the study of Hyt{o}nens extrapolation theorem of compactness on weighted Lebesgue spaces. Two criteria of compactness of linear operators in the two-weight setting are obtained. As applications, we obtain two-weight compactness of commutators of Calder{o}n--Zygmund operators, fractional integrals and bilinear Calder{o}n--Zygmund operators.
The paper provides an extension, to fractional order Sobolev spaces, of the classical result of Murat and Brezis which states that the positive cone of elements in $H^{-1}(Omega)$ compactly embeds in $W^{-1,q}(Omega)$, for every $q < 2$ and for any o
We give a soft proof of Albertis Luzin-type theorem in [1] (G. Alberti, A Lusintype theorem for gradients, J. Funct. Anal. 100 (1991)), using elementary geometric measure theory and topology. Applications to the $C^2$-rectifiability problem are also discussed.
We give blow-up behavior for solutions to an elliptic system with Dirichlet condition, and, weight and boundary singularity. Also, we have a compactness result for this elliptic system with regular H{o}lderian weight and boundary singularity and Lipschitz condition.
In this note, we establish a strong form of the quantitive Sobolev inequality in Euclidean space for $p in (1,n)$. Given any function $u in dot W^{1,p}(mathbb{R}^n)$, the gap in the Sobolev inequality controls $| abla u - abla v|_{p}$, where $v$ is an extremal function for the Sobolev inequality.
We study the energy transfer in the linear system $$ begin{cases} ddot u+u+dot u=bdot v ddot v+v-epsilon dot v=-bdot u end{cases} $$ made by two coupled differential equations, the first one dissipative and the second one antidissipative. We see how