Subelliptic Cordes estimates

Authors:
András Domokos and Juan J. Manfredi

Journal:
Proc. Amer. Math. Soc. **133** (2005), 1047-1056

MSC (2000):
Primary 35H20, 35J70

DOI:
https://doi.org/10.1090/S0002-9939-04-07819-0

Published electronically:
November 19, 2004

MathSciNet review:
2117205

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove Cordes type estimates for subelliptic linear partial differential operators in non-divergence form with measurable coefficients in the Heisenberg group. As an application we establish interior horizontal $W^{2,2}$-regularity for p-harmonic functions in the Heisenberg group ${\mathbb H}^1$ for the range $\frac {\sqrt {17}-1}{2} \leq p < \frac {5+\sqrt {5}}{2}$.

- Luca Capogna, Donatella Danielli, and Nicola Garofalo,
*An embedding theorem and the Harnack inequality for nonlinear subelliptic equations*, Comm. Partial Differential Equations**18**(1993), no. 9-10, 1765–1794. MR**1239930**, DOI https://doi.org/10.1080/03605309308820992 - H. O. Cordes,
*Zero order a priori estimates for solutions of elliptic differential equations*, Proc. Sympos. Pure Math., Vol. IV, American Mathematical Society, Providence, R.I., 1961, pp. 157–166. MR**0146511** - A. Domokos,
*Differentiability of solutions for the non-degenerate $p$-Laplacian in the Heisenberg group*, J. Differential Equations 204(2004), 439-470. - A. Domokos and Juan J. Manfredi, $C^{1,\alpha }$-regularity for p-harmonic functions in the Heisenberg group for $p$ near $2$, to appear in
*The $p$-harmonic equation and recent advances in analysis*, Contemporary Mathematics, editor Pietro Poggi-Corradini. - David Gilbarg and Neil S. Trudinger,
*Elliptic partial differential equations of second order*, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR**737190** - Guozhen Lu,
*Polynomials, higher order Sobolev extension theorems and interpolation inequalities on weighted Folland-Stein spaces on stratified groups*, Acta Math. Sin. (Engl. Ser.)**16**(2000), no. 3, 405–444. MR**1787096**, DOI https://doi.org/10.1007/PL00011552 - S. Marchi,
*$C^{1,\alpha }$ local regularity for the solutions of the $p$-Laplacian on the Heisenberg group for $2\leq p<1+\sqrt 5$*, Z. Anal. Anwendungen**20**(2001), no. 3, 617–636. MR**1863937**, DOI https://doi.org/10.4171/ZAA/1035 - Silvana Marchi,
*$C^{1,\alpha }$ local regularity for the solutions of the $p$-Laplacian on the Heisenberg group. The case $1+\frac 1{\sqrt 5}<p\leq 2$*, Comment. Math. Univ. Carolin.**44**(2003), no. 1, 33–56. MR**2045844** - Duy-Minh Nhieu,
*Extension of Sobolev spaces on the Heisenberg group*, C. R. Acad. Sci. Paris Sér. I Math.**321**(1995), no. 12, 1559–1564 (English, with English and French summaries). MR**1367807** - Robert S. Strichartz,
*$L^p$ harmonic analysis and Radon transforms on the Heisenberg group*, J. Funct. Anal.**96**(1991), no. 2, 350–406. MR**1101262**, DOI https://doi.org/10.1016/0022-1236%2891%2990066-E - Giorgio Talenti,
*Sopra una classe di equazioni ellittiche a coefficienti misurabili*, Ann. Mat. Pura Appl. (4)**69**(1965), 285–304 (Italian). MR**201816**, DOI https://doi.org/10.1007/BF02414375

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Additional Information

**András Domokos**

Affiliation:
Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, Pennsylvania 15260

Address at time of publication:
Department of Mathematics and Statistics, California State University Sacramento, 6000 J Street, Sacramento, California 95819

Email:
domokos@csus.edu

**Juan J. Manfredi**

Affiliation:
Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, Pennsylvania 15260

MR Author ID:
205679

Email:
manfredi@pitt.edu

Keywords:
Cordes conditions,
subelliptic equations,
p-Laplacian

Received by editor(s):
August 13, 2003

Published electronically:
November 19, 2004

Additional Notes:
The authors were partially supported by NSF award DMS-0100107

Communicated by:
David S. Tartakoff

Article copyright:
© Copyright 2004
American Mathematical Society

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