Boundedness of Operators Related to a Degenerate Schrödinger Semigroup

• Published in: Potential analysis 2022, Vol.57 (3), p.401-431
• Main Authors: Harboure, E., Salinas, O., Viviani, B.
• Format: Article
• Language: English
• Subjects: Functional Analysis; Geometry; Mathematics; Mathematics and Statistics; Operators (mathematics); Potential Theory; Probability Theory and Stochastic Processes; Semigroups
• ISSN: 0926-2601; 1572-929X
• DOI: 10.1007/s11118-021-09921-4

In this work we search for boundedness results for operators related to the semigroup generated by the degenerate Schrödinger operator L u = − 1 ω div A ⋅ ∇ u + V u , where ω is a weight, A is a matrix depending on x and satisfying λ ω ( x )| ξ | 2 ≤ A ( x ) ξ ⋅ ξ ≤Λ ω ( x )| ξ | 2 for some positive constants λ , Λ and all x , ξ in ℝ d , assuming further suitable properties on the weight ω and on the non-negative potential V . In particular, we analyze the behaviour of T ∗ , the maximal semigroup operator, L − α / 2 , the negative powers of L , and the mixed operators L − α / 2 V σ / 2 with 0 < σ ≤ α on appropriate functions spaces measuring size and regularity. As in the non degenerate case, i.e. ω ≡ 1, we achieve these results by first studying the case V = 0, obtaining also some boundedness properties in this context that we believe are new.