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

Abstract

We prove that a singular part ${\mu }_s$ of a measure $\mu$ satisfying $\mathcal{A}\mu =0$ for a linear partial differential operator $\mathcal{A}$ defined on $R^d$ has the range in the intersection of kernels of the principal symbol of $\mathcal{A}$ 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 $E_ε\subset B(\mx,α(ε))$ such that ${\lim }_{\epsilon \to 0}\frac{{\mu }_s(B(x,\beta (\epsilon ))/E_{\epsilon })}{|{\mu }_s|(E_{\epsilon })}=0$.