The structure of ${\cal A}$-free measures with uniformly singular part
Darko Mitrović · arXiv (Cornell University) · 2016
We prove that a singular part $μ_s$ of a measure $μ$ satisfying ${\cal A}μ=0$ 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 $μ_s$-almost every $x\in R^d$ there exist positive functions $α(ε), β(ε)$, $ε\in R$, satisfying $\frac{α(ε)}ε\to 0$, $ \fracε{β(ε)}\to 0$ and a set $E_ε\subset B(\mx,α(ε))$ such that $\lim_{ε\to 0}\frac{μ_s(B(x,β(ε)) / E_ε)}{|μ_s|(E_ε)}=0$.