2024/07/17 by Silvano Delladio, Delladio, Silvano
Mathematics · #28A05 #28A75 #31C40 #Analytic and geometric function theory #FOS: Mathematics #Functional Analysis (math.FA) #Functional Equations Stability Results #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2407.12343
openalex publication_date 2024/07/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Some properties of m-density points and density-degree functions are studied. Moreover the following main results are provided: \vskip2mm \beginitemize \item \it Let λ be a continuous differential form of degree h in \mathbf Rn (with h≥ 0) having the following property: There exists a continuous differential form Δ of degree h+1 in \rnn such that ∫_\mathbf RnΔ\wedgeω=∫_\mathbf Rnλ\wedge dω, for every C^∞c differential form ω of degree n-h-1 in \mathbf Rn. Moreover let μ be a C1 differential form of degree h+1 in \mathbf Rn and set E:=\y∈ \mathbf Rn \vert Δ(y)=μ(y)\. Then dμ(x) = 0 whenever x is a (n+1)-density point of E. \vskip2mm \item \it Let f:\mathbf Rn→ \mathbf R be a measurable function such that f(x)∈ \0\∪ [n,+∞] for a.e. x∈ \mathbf Rn. Then there exists a countable family \Fk\k=1^∞ of closed subsets of \mathbf Rn such that the corresponding sequence of density-degree functions \dFk\k=1^∞ converges almost everywhere to f.