Revista de I Union Matematica Agentina Volumen 37, 1991. POINTWISE ESTIMATES FOR SOLUTIONS OF DEGENERATE PARABOLIC EQUATIONS CRISTIAN E. GUTIERREZ 261 The results I shall describe in this note are joint work with Richard L. Wheeden. They concern the regularity propert ies of weak solutions of a certain class of degenerate parabolic equations, the validity of a Harnack principl e for non-negative weak solutions and es timates for the fundamental solution. The proofs of the resul ts can be found in references [G-WI], [G-W2], [G- W3] and [G-W 4] . In order to place the results in proper perspective we recall some results in partial diferential equations. In the late 50's and early 60 's a theory for the following class of equations with non smooth coefcients was developed. Let n be a domain in Rn, Q = n x (a, b) and consider the operator in divergence form n Lu = 1 t - 2: (a ;j (x,t ) uxJXj ; ,j=1 where the coefcient matrix A(x,t) = (a;j (x,t)) is measurable, real, symmetric and there are two posi tive constants /\, A such that for every � E Rn, ( , ) being the Euclidean inner product. A function u E L 2 (Q) is a weak solution of Eu = 0 if \ xU E L 2 (Q) and I� {- u ' t + ( A ( x, t )\u , \') } dx dt = 0 for every ' E CJ(Q). The reason to assume only measurabi lity of the coefcients is be cause of the applications to non-linear equations. Nash [Na] proved that weak solutions are Holder continuous. De Giorgi [DeG] proved this result in the elliptic case. Moser [Mol], [Mo2] establi shed a parabolic Harnack principle for non-negative solutions and de rived from it the Holder continuity. This Harnack principle has a diference with the elliptic one, this is: values of a non-negative solution u at a given time tl are only comparable to values of u at a later time i 2 > i 1 . From the Harnack princi ple Aronson [Ar] proved that the fundamental solution of L behaves like the heat kernel. . ' The study of linear degenerate elliptic equations in divergence form began in the late 60's and early 70's with the work of Murthy ad St ampacchia [M- S ] , and Trudinger [ T I LT 2] . The author was partia.lly supported by NSF Grant #DMS-90-03 095 .