Consider $A(x,D):C^{\infty }(\mathrm{\Omega },E)\to C^{\infty }(\mathrm{\Omega },F)$ an elliptic and canceling linear differential operator of order ν with smooth complex coefficients in $\mathrm{\Omega }\subset {\mathbb{R}}^N$ 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 $x_0\in \mathrm{\Omega }$ is contained in a neighborhood $U\subset \mathrm{\Omega }$ 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 φ supported in K:$|f(\phi )|⩽\theta {\Vert \phi \Vert }_{W^{\nu -1,1}}+\epsilon {\Vert A(x,D)\phi \Vert }_{L^1},$ 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 complexes of vector fields (see [1]); we also provide local analogues, for a large range of differential operators, to global results obtained for the classical divergence operator by Bourgain and Brezis in [2] and by De Pauw and Pfeffer in [3].

考虑 $\mathrm{一个}（X，d）：C^{\infty }（\mathrm{\Omega }，E）\to C^{\infty }（\mathrm{\Omega }，F）$具有光滑复系数的ν阶椭圆抵消线性微分算子。$\mathrm{\Omega }\subset {\mathbb{[R}}^{ñ}$从有限维复矢量空间E到有限维复矢量空间F和${\mathrm{一个}}^{⁎}（X，d）$它的伴随。在这项工作中，我们描述了偏微分方程的（局部）连续可解性${\mathrm{一个}}^{⁎}（X，d）v=F$（在分布意义上）对于给定的分布f；更确切地说，我们表明$X_0\in \mathrm{\Omega }$ 包含在附近 $ü\subset \mathrm{\Omega }$其中其连续可溶性的特征在于f上的以下条件：对于每个$\epsilon >0$ 和任何紧凑套装 $ķ\subset \subset ü$， 那里存在 $\theta =\theta （ķ，\epsilon ）>0$使得以下条件适用于K支持的所有平滑函数φ：$|F（\phi ）|⩽\theta {\Vert \phi \Vert }_{{\mathrm{w\; ^}}^{\nu -\mathrm{1个}，\mathrm{1个}}}+\epsilon {\Vert \mathrm{一个}（X，d）\phi \Vert }_{{\mathrm{大号}}^{\mathrm{1个}}}，$ 在哪里 ${\mathrm{w\; ^}}^{\nu -\mathrm{1个}，\mathrm{1个}}$ 代表所有人的同质Sobolev空间 ${\mathrm{大号}}^{\mathrm{1个}}$ 函数的阶导数 $\nu -\mathrm{1个}$ 属于 ${\mathrm{大号}}^{\mathrm{1个}}（ü）$。

这种表征暗示并扩展了先前获得的与矢量场的椭圆形复数相关的运算符的结果（请参见[1]）。我们还为大范围的微分算子提供了本地类似物，以类似于Bourgain和Brezis在[2]中以及De Pauw和Pfeffer在[3]中获得的经典散度算子的全局结果。