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.

中文翻译：

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重新审视无𝓐措施的结构

摘要 我们改进了最近关于满足线性偏微分方程的测度结构的结果 𝓐μ = σ, μ, σ 是氡测度，考虑到测度 μ(x) = g(x)dx + μus(x̃)(μs(x̄) ) + dx̄) 其中 x = (x̃,x̄) ∈ ℝk × ℝd−k，μus 是 x̃0 中的一致奇异测度，而测度 μs 是奇异测度。我们证明了对于 μus-ae x̃0，Radon-Nykodim 导数的范围 f~(x~0)=dμusd|μus|(x~0) $\begin{array}{} \tilde{f}(\tilde{ {\bf x}}_0) = \frac{d \mu_{us}}{d | \mu_{us}|}(\tilde{{\bf x}}_0) \end{array}$ 在集合∩ξ̃∈P̃𝓚erAP̃(ξ) 中，如果 μs 不为零，则对于 μs-ae x̄0 Radon-Nykodim 导数的范围 f¯(x¯0)=dμsd|μs|(x¯0) $\begin{array}{} \bar{f}(\bar{{\bf x}}_0 ) = \frac{d \mu_{s}}{d | \mu_{s}|}(\bar{{\bf x}}_0) \end{array}$ 在集合 ∪ξ̄∈P̄ 𝓚erAP̄(ξ) 中 P̃ × P̄ = P 是一个流形算符 𝓐 的符号 AP = AP̃ ⋅ AP̄。