Regularity for Quasilinear Second-Order Subelliptic Equations

In this paper, we study the regularity of solutions of the quasilinear equation where X = ( X , ; . . , X , , , ) is a system of real smooth vector fields, A i j , B E Cw(Q x R m + l ) . Assume that X satisfies the Hormander condition and ( A , , ( x , z , c ) ) is positive definite. We prove that if u E S2@(Q) (see Section 2) is a solution of the above equation, then u E Cw(Q). Introduction In this work, we study the regularity of solutions of the following quasilinear second-order degenerate elliptic equation: where X = (X,,...,X,) is a system of real smooth vector fields defined in an open domain M of R", n 2 3, and Ajj, B E C"(R x Rrn+'), i, j = 1,. . , rn. We assume X satisfies the Hormander condition (see Section 1) and ( A j j ( X , z , < ) ) is positive definite on M x BBrn+'. Equation (*) is degenerate elliptic in general. But this degeneracy is described by a system of vector fields and it is independent of solutions. So we call equation (*) subelliptic as in the linear case. The Hormander condition permits us to define a metric p ( x , y ) associated with X on SZ cc M (see Section 1). In the induced geometry, the Hormander operator H = Cy==, X2 + c(x) possesses properties similar to those of the Laplacian. For examp[e, one has the precise estimation of Green's kernel, the Poincare and Harnack inequalities, etc. (See [l], [2], [7], [8], [ 101.) We use these properties to construct a Schauder type estimate for the operator H in the associate function space Sk@(SZ) (see Section 2). As for the elliptic equations, we prove that the Schauder estimate implies the regularity of solutions for quasilinear equations, i.e., if u E S2*a(SZ), Communications on Pure and Applied Mathematics, Vol. XLV, 77-96 (1992)