The structure of 𝓐-free measures revisited

Abstract

We refine a recent result on the structure of measures satisfying a linear partial differential equation 𝓐μ = σ, μ, σ are Radon measures, considering the measure μ(x) = g(x)dx + μus(x̃)(μs(x̄) + dx̄) where x = (x̃,x̄) ∈ ℝk × ℝd−k, μus is a uniformly singular measure in x̃0 and the measure μs is a singular measure. We proved that for μus-a.e. x̃0 the range of the Radon-Nykodim derivative f~(x~0)=dμusd|μus|(x~0)$\begin{array}{} \tilde{f}(\tilde{{\bf x}}_0) = \frac{d \mu_{us}}{d | \mu_{us}|}(\tilde{{\bf x}}_0) \end{array}$ is in the set ∩ξ̃∈P̃𝓚erAP̃(ξ) and, if μs is different to zero, for μs-a.e. x̄0 the range of the Radon-Nykodim derivative f¯(x¯0)=dμsd|μs|(x¯0)$\begin{array}{} \bar{f}(\bar{{\bf x}}_0) = \frac{d \mu_{s}}{d | \mu_{s}|}(\bar{{\bf x}}_0) \end{array}$ is in the set ∪ξ̄∈P̄ 𝓚erAP̄(ξ) where P̃ × P̄ = P is a manifold determined by the main symbol AP = AP̃ ⋅ AP̄ of the operator 𝓐.

Keywords

• structure of measures
• 35d30