Optimal domains for the kernel operator associated with Sobolev's inequality

Guillermo P. Curbera¹ ; Werner J. Ricker²

¹Facultad de Matemáticas Universidad de Sevilla Aptdo. 1160, Sevilla 41080, Spain
²Math.-Geogr. Fakultät Katholische Universität Eichstätt-Ingolstadt D-85072 Eichstätt, Germany

Studia Mathematica, Tome 158 (2003) no. 2, pp. 131-152

Résumé

Refinements of the classical Sobolev inequality lead to optimal domain problems in a natural way. This is made precise in recent work of Edmunds, Kerman and Pick; the fundamental technique is to prove that the (generalized) Sobolev inequality is equivalent to the boundedness of an associated kernel operator on $[0,1]$. We make a detailed study of both the optimal domain, providing various characterizations of it, and of properties of the kernel operator when it is extended to act in its optimal domain. Several results are devoted to identifying the maximal rearrangement invariant space inside the optimal domain. The methods and techniques used involve interpolation theory, Banach function spaces and vector integration.

Détail

DOI : 10.4064/sm158-2-3

Keywords: refinements classical sobolev inequality lead optimal domain problems natural made precise recent work edmunds kerman pick fundamental technique prove generalized sobolev inequality equivalent boundedness associated kernel operator make detailed study optimal domain providing various characterizations properties kernel operator extended act its optimal domain several results devoted identifying maximal rearrangement invariant space inside optimal domain methods techniques involve interpolation theory banach function spaces vector integration