Pointwise Schauder estimates of parabolic equations in Carnot groups

TL;DR

This paper develops pointwise Schauder estimates for second-order parabolic equations on Carnot groups, using Campanato spaces and tailored mean value and Taylor inequalities to handle the sub-Riemannian structure.

Contribution

It introduces novel pointwise Schauder estimates for parabolic equations in Carnot groups, extending classical results to a sub-Riemannian setting with new analytical tools.

Findings

Established pointwise Schauder estimates using Campanato spaces.

Developed a version of mean value theorem and Taylor inequality for Carnot groups.

Provided a framework for regularity analysis of parabolic equations in sub-Riemannian geometry.

Abstract

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.