Osaka Journal of Mathematics

On the equivalence between trace and capacitary inequalities for the abstract contractive space of Bessel potentials

Ali Ben Amor

Full-text: Open access

Abstract

Let $F_{r,p}=V_{r,p}(L^p(X,m))$ be the abstract space of Bessel potentials and $\mu$ a positive smooth Radon measure on $X$. For $2\leq p\leq q < \infty$, we give necessary and sufficient criteria for the boundedness of $V_{r,p}$ from $L^p(X,m)$ into $L^p(X,\mu)$, provided $F_{r,p}$ is contractive. Among others, we shall prove that the boundedness is equivalent to a capacitary type inequality. Further we give necessary and sufficient conditions for $F_{r,p}$ to be compactly embedded in $L^q(\mu)$. Our method relies essentially on establishing a \textit{capacitary strong type inequality}.

Article information

Source
Osaka J. Math., Volume 42, Number 1 (2005), 11-26.

Dates
First available in Project Euclid: 21 July 2006

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1153494312

Mathematical Reviews number (MathSciNet)
MR2130960

Zentralblatt MATH identifier
1071.31003

Citation

Ben Amor, Ali. On the equivalence between trace and capacitary inequalities for the abstract contractive space of Bessel potentials. Osaka J. Math. 42 (2005), no. 1, 11--26. https://projecteuclid.org/euclid.ojm/1153494312


Export citation