  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.
