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

Abstract

Consider $A(x,D):C^{\infty}(\Omega,E) \rightarrow C^\infty(\Omega,F)$ 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 $A^{*}(x,D)$ {its} adjoint. In this work we characterize the (local) continuous solvability of the partial differential equation $A^{*}(x,D)v=f$ (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 $\varphi$ supported in $K$: \begin{equation}\nonumber \left| f(\varphi) \right| \leq \theta\|\varphi\|_{W^{\nu-1,1}} + \epsilon\|A(x,D) \varphi\|_{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].