On the differentiability of the solutions of quasilinear partial differential equations

Abstract

is, for every x0, hypoelliptic and stronger than each Mj(D). (2) Each Nk(D) is strictly weaker than all Mj(D) together (cf. [2, p. 116, dl). (3) The functions aj(x, ti, * , t,) and g(x, t1, * -, t,) are infinitely differentiable. In this paper we want to outline a different proof which is more elementary than the original one, in the sense that it utilizes neither the Sobolev nor the Gagliardo-Nirenberg estimates but only rather straightforward estimates obtained from the Fourier transformation in L2. A severe disadvantage, on the other hand, is the fact that it requires rather strong a priori differentiability of the solutions. 1. The Schauder algebra. Let s be any real number. Put