# Hölder Continuity of Weak Solutions to Subelliptic Equations with Rough Coefficients

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*Eric T. Sawyer; Richard L. Wheeden*

We study interior regularity of weak solutions of second order linear divergence form equations with degenerate ellipticity and rough coefficients. In particular, we show that solutions of large classes of subelliptic equations with bounded measurable coefficients are Hölder continuous. We present two types of results dealing with such equations. The first type generalizes the celebrated Fefferman-Phong geometric characterization of subellipticity in the smooth case. We introduce a notion of \(L^q\)-subellipticity for the rough case and develop an axiomatic method which provides a near characterization of the notion of \(L^q\)-subellipticity. The second type deals with generalizing a case of Hörmanders's celebrated algebraic characterization of subellipticity for sums of squares of real analytic vector fields. In this case, we introduce a “flag condition” as a substitute for the Hörmander commutator condition which turns out to be equivalent to it in the smooth case. The question of regularity for quasilinear subelliptic equations with smooth coefficients provides motivation for our study, and we briefly indicate some applications in this direction, including degenerate Monge-Ampère equations.

#### Table of Contents

# Table of Contents

## Holder Continuity of Weak Solutions to Subelliptic Equations with Rough Coefficients

- Contents v6 free
- Overview ix10 free
- Chapter 1. Introduction 112 free
- Chapter 2. Comparisons of conditions 2940
- Chapter 3. Proof of the general subellipticity theorem 5364
- Chapter 4. Reduction of the proofs of the rough diagonal extensions of Hörmander's theorem 8798
- Chapter 5. Homogeneous spaces and subrepresentation inequalities 117128
- Chapter 6. Appendix 133144
- 1. Necessity of the Fefferman-Phong condition 133144
- 2. Necessity of the Sobolev and Poincaré inequalities 136147
- 3. The noninterference balls A (x,r) and the reverse Hölder condition 138149
- 4. Reverse Hölder examples 140151
- 5. Product reverse Hölder 143154
- 6. The noninterference conditions 147158
- 7. Other notions of weak solution 150161
- 8. Alternate methods of proof 151162

- Bibliography 153164