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

Guillermo P. Curbera, and Werner J. Ricker. "Optimal domains for the kernel operator associated with Sobolev's inequality." Studia Mathematica 158.2 (2003): 131-152. <http://eudml.org/doc/285149>.

@article{GuillermoP2003,
abstract = {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.},
author = {Guillermo P. Curbera, Werner J. Ricker},
journal = {Studia Mathematica},
keywords = {Sobolev imbeddings; optimal domains; vector measure},
language = {eng},
number = {2},
pages = {131-152},
title = {Optimal domains for the kernel operator associated with Sobolev's inequality},
url = {http://eudml.org/doc/285149},
volume = {158},
year = {2003},
}

TY - JOUR
AU - Guillermo P. Curbera
AU - Werner J. Ricker
TI - Optimal domains for the kernel operator associated with Sobolev's inequality
JO - Studia Mathematica
PY - 2003
VL - 158
IS - 2
SP - 131
EP - 152
AB - 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.
LA - eng
KW - Sobolev imbeddings; optimal domains; vector measure
UR - http://eudml.org/doc/285149
ER -