Franchi, Gutiérrez, Wheeden: Two-weight Sobolev-Poincaré inequalities and Harnack inequality for a class of degenerate elliptic operators

Franchi, Bruno and Gutiérrez, Cristian E. and Wheeden, Richard L.:
Two-weight Sobolev-Poincaré inequalities and Harnack inequality for a class of degenerate elliptic operators (Disuguaglianze di Sobolev-Poincaré con due pesi e disuguaglianza di Hamack per una classe di operatori ellittici degeneri)
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni Serie 9 5 (1994), fasc. n.2, p. 167-175, (English)
pdf (1.21 MB), djvu (269 Kb). | MR1292572 | Zbl 0811.46023

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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 \( A_{\infty} \)» rispetto a una metrica naturale per l'operatore, dove la condizione \( A_{\infty} \)»-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.

Referenze Bibliografiche

[1] B. BOJARSKE, Remarks on Sobolev imbedding inequalities. Complex Analysis, Lecture Notes in Mathematics 1351, Springer, 1989, 52-68. | fulltext (doi) | MR 982072 | Zbl 0662.46037

[2] A. P. CALDERON, Inequalities for the maximal function relative to a metric. Studia Math., 57, 1976, 297-306. | fulltext mini-dml | MR 442579 | Zbl 0341.44007

[3] L. CAPOGNA - D. DANIELLI - N. GAROFALO, Embedding theorems and the Harnack inequality for solutions of nonlinear subelliptic equations. C.R. Acad. Sci. Paris, 316, 1993, 809-814. | MR 1218266 | Zbl 0797.35063

[4] S. CHANILLO - R. L. WHEEDEN, Weighted Poincaré and Sobolev inequalities and estimates for the Peano maximal function. Amer. J. Math., 107, 1985, 1191-1226. | fulltext (doi) | MR 805809 | Zbl 0575.42026

[5] S. CHANILLO - R. L. WHEEDEN, Harnack's inequality and mean-value inequalities for solutions of degenerate elliptic equations. Comm. Partial Differential Equations, 11, 1986, 1111-1134. | fulltext (doi) | MR 847996 | Zbl 0634.35035

[6] S. CHUA, Weighted Sobolev inequality on domains satisfying chain conditions. Proc. Amer. Math. Soc., to appear. | Zbl 0812.46020

[7] G. DAVID - S. SEMMES, Strong \( A_{\infty} \) weights, Sobolev inequalities and quasiconformal mappings In: C. SADOSKY (éd.), Analysis and Partial Differential Equations. Marcel Dekker, 1990, 101-111. | MR 1044784 | Zbl 0752.46014

[8] E. FABES - C. KENIG - R. SERAPIONI, The local regularity of solutions of degenerate elliptic equations. Comm. Partial Differential Equations, 7, 1982, 77-116. | fulltext (doi) | MR 643158 | Zbl 0498.35042

[9] J. C. FERNANDES, Mean value and Harnack inequalities for a certain class of degenerate parabolic equations. Revista Matematica Iberoamericana, 7, 1991, 247-286. | fulltext EuDML | fulltext (doi) | MR 1165071 | Zbl 0761.35050

[10] C. FEFFERMAN - D. H. PHONG, Subelliptic eigenvalue problems. In: W. BECKNER et al (eds.), Conference on Harmonic Analysis (Chicago 1980). Wadsworth 1981, 590-606. | MR 730094 | Zbl 0503.35071

[11] B. FRANCHI, Weighted Sobolev-Poincaré inequalities and pointwise estimates for a class of degenerate elliptic equations. Trans. Amer. Math. Soc., 327, 1991, 125-158. | fulltext (doi) | MR 1040042 | Zbl 0751.46023

[12] B. FRANCHI, Inégalités de Sobolev pour des champs de vecteurs lipschitziens. C.R. Acad. Sci. Paris, 311, 1990, 329-332. | MR 1071637 | Zbl 0734.46023

[13] B. FRANCHI - E. LANCONELLI, Hölder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients. Ann. Scuola Norm. Sup. Pisa, 10, (4), 1983, 523-541. | fulltext EuDML | fulltext mini-dml | MR 753153 | Zbl 0552.35032

[14] B. FRANCHI - R. SERAPIONI, Pointwise estimates for a class of strongly degenerate elliptic operators: a geometrical approach. Ann. Scuola Norm. Sup. Pisa, 14, (4), 1987, 527-568. | fulltext EuDML | fulltext mini-dml | MR 963489 | Zbl 0685.35046

[15] B. FRANCHI - C. GUTIERREZ - R. L. WHEEDEN, Weighted Sobolev-Poincaré inequalities for Grushin type operators. Comm. Partial Differential Equations, to appear. | fulltext (doi) | MR 1265808 | Zbl 0822.46032

[16] B. FRANCHI - S. GALLOT - R. L. WHEEDEN, Sobolev and isoperimetric inequalities for degenerate metrics. Math. Ann., to appear. | fulltext EuDML | fulltext (doi) | MR 1314734 | Zbl 0830.46027

[17] B. FRANCHI - S. GALLOT - R. L. WHEEDEN, Inégalités isopérimetriques pour des métriques dégénérées. C.R. Acad. Sci. Paris, Ser. A, 317, 1993, 651-654. | MR 1245092 | Zbl 0794.51011

[18] F. W. GEHRING, The \( L^{p} \)-integrability of the partial derivatives of a quasi conformai mapping. Acta Math., 130, 1973, 265-277. | MR 402038 | Zbl 0258.30021

[19] C. GUTIERREZ - R. L. WHEEDEN, Mean-value and Harnack inequalities for degenerate parabolic equations. Colloq. Math., 60/61, 1990, 157-194. | MR 1096367 | Zbl 0785.35057

[20] T. IWANIEK - C. A. NOLDER, Hardy-Littlewood inequality for quasiregular mappings in certain domains in \( \mathbb{R}^{n} \). Ann. Acad. Sci. Fenn., Series A I Math. 10, 1985, 267-282. | MR 802488 | Zbl 0588.30023

[21] D. JERISON, The Poincaré inequality for vector fields satisfying Hörmander condition. Duke Math. J., 53, 1986, 503-523. | fulltext mini-dml | fulltext (doi) | MR 850547 | Zbl 0614.35066

[22] G. LU, The sharp Poincaré inequality for free vector fields: an endpoint result. Revista Matématica Iberoamericana, to appear. | fulltext EuDML | fulltext (doi) | MR 1286482 | Zbl 0860.35006

[23] A. NAGEL - E. M. STEIN - S. WAINGER, Balls and metrics defined by vector fields I: basic properties. Acta Math., 155, 1985, 103-147. | fulltext (doi) | MR 793239 | Zbl 0578.32044

[24] O. SALINAS, Harnack inequality and Green function for a certain class of degenerate elliptic differential operators. Revista Matematica Iberoamericana, 7, 1991, 313-349. | fulltext EuDML | fulltext (doi) | MR 1165073 | Zbl 0770.35023

[25] E. SAWYER - R. L. WHEEDEN, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces. Amer. J. Math., 114, 1992, 813-874. | fulltext (doi) | MR 1175693 | Zbl 0783.42011

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Referenze Bibliografiche