2020/09/20 by Deb Kumar Giri, Giri, Deb Kumar
Engineering · Mathematics · #37A45 #42B10 #Algebraic and Geometric Analysis #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #Elasticity and Wave Propagation #FOS: Mathematics #Mathematical Analysis and Transform Methods #Primary 42A10 #Secondary 35L10
paper · pdf · doi:10.48550/arxiv.2009.09516
openalex publication_date 2020/09/20 · openalex created_date 2020/09/25 · openalex updated_date 2026/07/28
Let Γ be a smooth curve or finite disjoint union of smooth curves in the plane and Λ be any subset of the plane. Let \mathcal X(Γ) be the space of all finite complex-valued Borel measures in the plane which are supported on Γ and are absolutely continuous with respect to the arc length measure on Γ. Let AC(Γ,Λ)=\μ∈ X(Γ) : μ|Λ=0\, then we prove the following results: \beginenumerate[(a)] \item For a rational perturbation of Λβ namely, Λβθ=((\mathbb Z+\θ\)×\0\)∪(\0\×β\mathbb Z), where θ=1/p,~for some~p∈\mathbb N, and β is a positive real, AC(Γ,Λβθ) is infinite-dimensional whenever β>p. \smallskip \item For a rational perturbation of Λγ namely, Λγθ=((2\mathbb Z+\2θ\)×\0\)∪(\0\ ×2γ\mathbb Z), where θ=1/q,~for some~q∈\mathbb N, and γ is a positive real, AC(Γ+,Λγθ) is infinite-dimensional whenever γ>q. \endenumerate