JournalsrmiVol. 37, No. 3pp. 911–963

Solutions to a system of equations for CmC^m functions

  • Charles Fefferman

    Princeton University, USA
  • Garving K. Luli

    University of California at Davis, USA
Solutions to a system of equations for $C^m$ functions cover
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Abstract

Fix m0m\geq 0, and let A=(Aij(x))1iN,1jMA=(A_{ij}(x))_{1 \leq i \leq N, 1\leq j \leq M} be a matrix of semialgebraic functions on Rn\mathbb{R}^n or on a compact subset ERnE \subset \mathbb{R}^n. Given f=(f1,,fN)C(Rn,RN)f=(f_1,\ldots,f_N) \in C^\infty(\mathbb{R}^n, \mathbb{R}^N), we consider the following system of equations:

j=1MAij(x)Fj(x)=fi(x)for i=1,,N.\sum_{j=1}^M A_{ij} (x) F_j (x) = f_i (x) \quad\text{for } i =1,\ldots, N.

In this paper, we give algorithms for computing a finite list of linear partial differential operators such that AF=fAF=f admits a Cm(Rn,RM)C^m(\mathbb{R}^n,\mathbb{R}^M) solution F=(F1,,FM)F=(F_1,\ldots,F_M) if and only if f=(f1,,fN)f=(f_1,\ldots,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 CmC^m functions. Rev. Mat. Iberoam. 37 (2021), no. 3, pp. 911–963

DOI 10.4171/RMI/1217