A degenerate Sobolev inequality for a large open set in a homogeneous space

Author:
Scott Rodney

Journal:
Trans. Amer. Math. Soc. **362** (2010), 673-685

MSC (2000):
Primary 35Hxx

Published electronically:
September 15, 2009

MathSciNet review:
2551502

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In current literature, existence results for degenerate elliptic equations with rough coefficients on a large open set of a homogeneous space have been demonstrated; see the paper by Gutierrez and Lanconelli (2003). These results require the assumption of a Sobolev inequality on of the form

holding for and some . However, it is unclear when such an inequality is valid, as techniques often yield only a local version of (1):(2)

holding for , with as above. The main result of this work shows that the global Sobolev inequality (1) can be obtained from the local Sobolev inequality (2) provided standard regularity hypotheses are assumed with minimal restrictions on the quadratic form . This is achieved via a new technique involving existence of weak solutions, with global estimates, to a 1-parameter family of Dirichlet problems on and a maximum principle.

**1.**Eugene B. Fabes, Carlos E. Kenig, and Raul P. Serapioni,*The local regularity of solutions of degenerate elliptic equations*, Comm. Partial Differential Equations**7**(1982), no. 1, 77–116. MR**643158**, 10.1080/03605308208820218**2.**Eugene B. Fabes, Carlos E. Kenig, and Raul P. Serapioni,*The local regularity of solutions of degenerate elliptic equations*, Comm. Partial Differential Equations**7**(1982), no. 1, 77–116. MR**643158**, 10.1080/03605308208820218**3.**Bruno Franchi, Raul Serapioni, and Francesco Serra Cassano,*Approximation and imbedding theorems for weighted Sobolev spaces associated with Lipschitz continuous vector fields*, Boll. Un. Mat. Ital. B (7)**11**(1997), no. 1, 83–117 (English, with Italian summary). MR**1448000****4.**David Gilbarg and Neil S. Trudinger,*Elliptic partial differential equations of second order*, Springer, 1998.**5.**Cristian E. Gutiérrez and Ermanno Lanconelli,*Maximum principle, nonhomogeneous Harnack inequality, and Liouville theorems for 𝑋-elliptic operators*, Comm. Partial Differential Equations**28**(2003), no. 11-12, 1833–1862. MR**2015404**, 10.1081/PDE-120025487**6.**Peter D. Lax,*Functional analysis*, Pure and Applied Mathematics (New York), Wiley-Interscience [John Wiley & Sons], New York, 2002. MR**1892228****7.**Scott Rodney,*Existence of weak solutions to subelliptic partial differential equations in divergence form and the necessity of the Sobolev and Poincaré inequalities*, Ph.D. thesis, McMaster University, Hamilton, Ontario, Canada, 2007.**8.**H. L. Royden,*Real analysis*, 3rd ed., Macmillan Publishing Company, New York, 1988. MR**1013117****9.**Walter Rudin,*Principles of mathematical analysis*, 3rd ed., McGraw-Hill Book Co., New York-Auckland-Düsseldorf, 1976. International Series in Pure and Applied Mathematics. MR**0385023****10.**Walter Rudin,*Real and complex analysis*, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR**924157****11.**E. Sawyer and R. L. Wheeden,*Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces*, Amer. J. Math.**114**(1992), no. 4, 813–874. MR**1175693**, 10.2307/2374799**12.**Eric T. Sawyer and Richard L. Wheeden,*Hölder continuity of weak solutions to subelliptic equations with rough coefficients*, Mem. Amer. Math. Soc.**180**(2006), no. 847, x+157. MR**2204824**, 10.1090/memo/0847**13.**Eric T. Sawyer and Richard L. Wheeden,*Degenerate Sobolev spaces and weak solutions*, Trans. Amer. Math. Soc., to appear.**14.**Leon Simon,*Lectures on geometric measure theory*, Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, Australian National University, Centre for Mathematical Analysis, Canberra, 1983. MR**756417****15.**Elias M. Stein,*Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals*, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR**1232192**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
35Hxx

Retrieve articles in all journals with MSC (2000): 35Hxx

Additional Information

**Scott Rodney**

Affiliation:
Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854

DOI:
https://doi.org/10.1090/S0002-9947-09-04809-0

Received by editor(s):
October 1, 2007

Published electronically:
September 15, 2009

Article copyright:
© Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.