And Pointwise Estimates for a Class of Degenerate Elliptic Equations

In this paper we prove a Sobolev-Poincaré inequality for a class of function spaces associated with some degenerate elliptic equations. These estimates provide us with the basic tool to prove an invariant Harnack inequality for weak positive solutions. In addition, Holder regularity of the weak solutions follows in a standard way. 1 Let Sf = YJl ,=i 9j(ajjdj) be a second-order degenerate elliptic operator in divergence form with measurable coefficients. In this paper we shall obtain pointwise estimates for the weak solutions of Sfu — 0 (Holder continuity of the weak solutions and Harnack inequality for nonnegative solutions). Let us recall that the original results for elliptic operators were obtained by De Giorgi, Nash, and Moser. An extensive bibliography about the degenerate case can be found in [FL1, FL2, FS]. To introduce the results of the present paper, let us recall some recent results. In [FL1, FL2] a suitable metric d is associated with the differential operator Sf in such a way that we obtain a new geometry which is natural for the degenerate operator as the Euclidean geometry is natural for the Laplace operator (or, more precisely, as a suitable Riemannian geometry is natural for a secondorder elliptic operator). In the smooth case, this idea is contained in many papers: we refer to [FP, NSW]. The basic results in [FL1, FL2] are obtained via a precise description of this geometry under suitable technical hypotheses on the coefficients whose aim is to give a nonsmooth formulation of the Hörmander hypoellipticity condition for sum-of-squares operators. We note that the same idea is used in [NSW, S, J, V] to obtain pointwise estimates for sum-of-squares operators. On the other hand, a different class of degenerate elliptic operators is considered in [FKS]: instead of a geometrical degeneracy, a measure degeneracy is allowed. A typical example of this class is given by Sfu div(œ(x)Vu), where co is a weight function belonging to the ^2-class of Muckenhoupt. Unified results for a class containing both the operators in [FL1] and in [FKS] have Received by the editors August 1, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 46E35, 35J70. Partially supported by G.N.A.F.A. of C.N.R. and M.U.R.S.T., Italy. ©1991 American Mathematical Society 0002-9947/91 $1.00 + 5.25 per page