In this Note we prove a two-weight Sobolev-Poincaré inequality for the function spaces associated with a Grushin type operator. Conditions on the weights are formulated in terms of a strong weight with respect to the metric associated with the operator. Roughly speaking, the strong condition provides relationships between line and solid integrals of the weight. Then, this result is applied in order to prove Harnack's inequality for positive weak solutions of some degenerate elliptic equations.
In questa Nota proviamo una disuguaglianza di Sobolev-Poincaré con due pesi per gli spazi funzionali associati ad un operatore tipo Grushin. Le condizioni sui pesi sono formulate in termini di un dato peso fortemente rispetto a una metrica naturale per l'operatore, dove la condizione -forte richiede opportune relazioni tra gli integrali di linea e gli integrali solidi del peso. Successivamente, questo risultato è applicato per provare la disuguaglianza di Harnack per le soluzioni deboli positive di certe equazioni ellittiche degeneri.
@article{RLIN_1994_9_5_2_167_0,
author = {Bruno Franchi and Cristian E. Guti\'errez and Richard L. Wheeden},
title = {Two-weight Sobolev-Poincar\'e inequalities and Harnack inequality for a class of degenerate elliptic operators},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
volume = {5},
year = {1994},
pages = {167-175},
zbl = {0811.46023},
mrnumber = {1292572},
language = {en},
url = {http://dml.mathdoc.fr/item/RLIN_1994_9_5_2_167_0}
}
Franchi, Bruno; Gutiérrez, Cristian E.; Wheeden, Richard L. Two-weight Sobolev-Poincaré inequalities and Harnack inequality for a class of degenerate elliptic operators. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 5 (1994) pp. 167-175. http://gdmltest.u-ga.fr/item/RLIN_1994_9_5_2_167_0/
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