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TRACE INEQUALITIES OF SOBOLEV TYPE IN THE UPPER TRIANGLE CASE

Published online by Cambridge University Press:  01 March 2000

CARME CASCANTE
Affiliation:
Departament de Matematica Aplicada i Analisi, Facultat de Matemàtiques, Universitat de Barcelona, Gran Via 585, 08071 Barcelona, Spaincascante@mat.ub.esortega@mat.ub.es
JOAQUÍN M. ORTEGA
Affiliation:
Departament de Matematica Aplicada i Analisi, Facultat de Matemàtiques, Universitat de Barcelona, Gran Via 585, 08071 Barcelona, Spaincascante@mat.ub.esortega@mat.ub.es
IGOR E. VERBITSKY
Affiliation:
Department of Mathematics, University of Missouri, Columbia, MO 65211, USAigor@math.missouri.edu
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Abstract

We give a new non-capacitary characterization of positive Borel measures $\mu$ on ${\Bbb R}^n$ such that the potential space $I_\alpha*L^p$ is imbedded in $L^q(d\mu)$ for $1< q < p<+\infty$, that is, the trace inequality $\|I_\alpha f\|_{L^q(d\mu)} \le C \, \|f\|_{L^p(d x)}$ holds, for Riesz potentials $I_\alpha = (- \Delta)^{-\alpha/2}$. A weak-type trace inequality is also characterized. A non-isotropic version on the unit sphere ${\Bbb S}^n$ is studied, as well as the holomorphic case for Hardy--Sobolev spaces $H_\alpha^p$ in the ball. 1991 Mathematics Subject Classification: primary 31C15, 42B20; secondary 32A35.

Type
Research Article
Copyright
2000 London Mathematical Society

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