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

2016/12/31 by Mitrovic, Darko
#35D30 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1701.00078

Abstract

We prove that a singular part μ_s of a measure μ satisfying \cal Aμ=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∈ R^d there exist positive functions α(ε), β(ε), ε∈ R, satisfying \fracα(ε)ε→ 0, \fracεβ(ε)→ 0 and a set E_ε⊂ B(\mx,α(ε)) such that lim_{ε→ 0}(μ_s(B(x,β(ε)) / E_ε))/(|μ_s|(E_ε))=0.

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