On Local Continuous Solvability of Equations Associated to Elliptic and Canceling Linear Differential Operators

Abstract

Consider $A(x,D):C^{\infty }(\Omega ,E)\to C^{\infty }(\Omega ,F)$ an elliptic and canceling linear differential operator of order $\nu$ with smooth complex coefficients in $\Omega \subset {\mathbb{R}}^N$ from a finite dimension complex vector space $E$ to a finite dimension complex vector space $F$ and $A^{\ast }(x,D)$ {its} adjoint. In this work we characterize the (local) continuous solvability of the partial differential equation $A^{\ast }(x,D)v=f$ (in the distribution sense) for a given distribution $f$; more precisely we show that any $x_0\in \Omega$ is contained in a neighborhood $U\subset \Omega$ in which its continuous solvability is characterized by the following condition on $f$: for every $\epsilon >0$ and any compact set $K\subset \subset U$, there exists $\theta =\theta (K,\epsilon )>0$ such that the following holds for all smooth function $\phi$ supported in $K$: \begin{equation}\nonumber \left| f(φ) \right| \leq θ\|φ\|_{W^{ν-1,1}} + ε\|A(x,D) φ\|_{L^{1}}, \end{equation} where $W^{\nu -1,1}$ stands for the homogenous Sobolev space of all $L^1$ functions whose derivatives of order $\nu -1$ belongs to $L^1(U)$. This characterization implies and extends results obtained before for operators associated to elliptic complex of vector fields (see \cite{MP}); we also provide local analogues, for a large range of differential operators, to global results obtained for the classical divergence operator in [4] and [9].

arXiv admin note: substantial text overlap with arXiv:1701.02889