2023/07/20 by Aparajita Dasgupta, Vishvesh Kumar, Dasgupta, Aparajita +5
Computer Science · Mathematics · #47G30 #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #FOS: Mathematics #Functional Analysis (math.FA) #Primary 35S05 #Secondary 43A85 #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2307.10825
openalex publication_date 2023/07/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this article, we introduce and study M-elliptic pseudo-differential operators in the framework of non-harmonic analysis of boundary value problems on a manifold Ω with boundary ∂ Ω, introduced by Ruzhansky and Tokmagambetov ( Int. Math. Res. Not. IMRN, (12), 3548-3615, 2016) in terms of a model operator \mathfrakL. More precisely, we consider a weighted \mathfrakL-symbol class Mρ, 0, Λm, m∈ ℝ, associated to a suitable weight function Λ on a countable set I and study elements of the symbolic calculus for pseudo-differential operators associated with \mathfrakL-symbol class Mρ, 0, Λm, by deriving formulae for the composition, adjoint, and transpose. Using the notion of M-ellipticity for symbols belonging to \mathfrakL-symbol class Mρ, 0, Λm, we construct the parametrix of M-elliptic pseudo-differential operators. Further, we investigate the minimal and maximal extensions for M-elliptic pseudo-differential operators and show that they coincide when the symbol σ∈ Mρ, 0, Λm, is M-elliptic. We provide a necessary and sufficient condition to ensure that the pseudo-differential operators Tσ with symbol in the \mathfrakL-symbol class Mρ, 0,Λ0 is a compact operator in L2(Ω) or a Riesz operator in Lp(Ω). Finally, we prove Gärding's inequality for pseudo-differential operators associated with symbol from Mρ, 0,Λ0 in the setting of non-harmonic analysis.