Pointwise Schauder estimates of parabolic equations in Carnot groups

Heather Price

Published 2016-01-13Version 1

Schauder estimates were a historical stepping stone for establishing uniqueness and smoothness of solutions for certain classes of partial differential equations. Since that time, they have remained an essential tool in the field. Roughly speaking, the estimates state that the H\"older continuity of the coefficient functions and inhomogeneous term implies the H\"older continuity of the solution and its derivatives. This document establishes pointwise Schauder estimates for second order "parabolic" equations of the form $$\partial_{t}u(x,t)-\sum_{i,j=1}^{m_1} a_{ij}(x,t)X_iX_ju(x,t)=f(x,t)$$ where $X_{1},\ldots ,X_{m_1}$ generate the first layer of the Lie algebra stratification for a Carnot group. The Schauder estimates are shown by means of Campanato spaces. These spaces make the pointwise nature of the estimates possible by comparing solutions to their Taylor polynomials. As a prerequisite device, a version of both the mean value theorem and Taylor inequality are established with the parabolic distance incorporated.