Boundedness of Operators Related to a Degenerate Schrödinger Semigroup

Abstract

In this work we search for boundedness results for operators related to the semigroup generated by the degenerate Schrödinger operator Lu=−1ωdivA⋅∇u+Vu, where ω is a weight, A is a matrix depending on x and satisfying λω(x)|ξ|2 ≤ A(x)ξ ⋅ ξ ≤Λω(x)|ξ|2 for some positive constants λ, Λ and all x, ξ in ℝd, 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/2Vσ/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.