# Solutions to a system of equations for $C_{m}$ functions

### Charles Fefferman

Princeton University, USA### Garving K. Luli

University of California at Davis, USA

## Abstract

Fix $m≥0$, and let $A=(A_{ij}(x))_{1≤i≤N,1≤j≤M}$ be a matrix of semialgebraic functions on $R_{n}$ or on a compact subset $E⊂R_{n}$. Given $f=(f_{1},…,f_{N})∈C_{∞}(R_{n},R_{N})$, we consider the following system of equations:

$j=1∑M A_{ij}(x)F_{j}(x)=f_{i}(x)fori=1,…,N.$

In this paper, we give algorithms for computing a finite list of linear partial differential operators such that $AF=f$ admits a $C_{m}(R_{n},R_{M})$ solution $F=(F_{1},…,F_{M})$ if and only if $f=(f_{1},…,f_{N})$ is annihilated by the linear partial differential operators.

## Cite this article

Charles Fefferman, Garving K. Luli, Solutions to a system of equations for $C_{m}$ functions. Rev. Mat. Iberoam. 37 (2021), no. 3, pp. 911–963

DOI 10.4171/RMI/1217