The structure of ${\cal A}$-free measures with uniformly singular part

Abstract

We prove that a singular part $\mu_s$ of a measure $\mu$ satisfying ${\cal A}\mu =0$ for a linear partial differential operator ${\cal A}$ defined on $R^d$ has the range in the intersection of kernels of the principal symbol of ${\cal A}$ if the singular part is singular with respect to all the variables (uniformly singular) i.e. it is such that for $\mu_s$-almost every $x\in R^d$ there exist positive functions $\alpha(\epsilon), \beta(\epsilon)$, $\epsilon \in R$, satisfying $\frac{\alpha(\epsilon)}{\epsilon}\to 0$, $ \frac{\epsilon}{\beta(\epsilon)}\to 0$ and a set $E_\epsilon\subset B(\mx,\alpha(\epsilon))$ such that $\lim_{\epsilon\to 0}\frac{\mu_s(B(x,\beta(\epsilon)) / E_\epsilon)}{|\mu_s|(E_\epsilon)}=0$.