In this work we search for boundedness results for operators related to the semigroup generated by the degenerate Schrödinger operator \({{\mathscr{L}}} u = -\frac {1}{\omega } \text {div} A\cdot \nabla u +V u\), 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 \(\mathbb {R}^{d}\), 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, \({{\mathscr{L}}}^{-\alpha /2}\), the negative powers of \({{\mathscr{L}}}\), and the mixed operators \({{\mathscr{L}}}^{-\alpha /2}V^{\sigma /2}\) 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.

在这项工作中，我们搜索与退化Schrödinger运算符\（{{\ mathscr {L}}} u =-\ frac {1} {\ omega} \ text {div} A \ CDOT \ nablaÚ+ V U \） ，其中ω是一个重量，阿是一个矩阵取决于X和满足λ ω（X）| ξ | ² ≤甲（X）ξ＆CenterDot;＆ξ ≤λ ω（X）| ξ | ²对于一些正常数λ，Λ和所有x，假设\（\ mathbb {R} ^ {d} \）中的ξ，在权重ω和非负电势V上假设其他合适的性质。特别是，我们分析T ^∗的行为，最大半群算子\（{{\ mathscr {L}}} ^ {-\ alpha / 2} \），\（{{\ mathscr {L }}} \） ，并将该混合运营商\（{{\ mathscr {L}}} ^ { - \的α/ 2}伏^ {\西格玛/ 2} \）其中0 < σ ≤ α上的适当功能的空间测量大小和规律性。如在非简并情况下，即ω＆equiv; 1，我们通过研究首先的情况下实现这些结果V = 0，在这种情况下，我们还获得了一些我们认为是新的有界属性。