Sharp resolvent estimates outside of the uniform boundedness range (Q1984831)

The authors of this paper consider the resolvent estimate for the Laplacian in Euclidean spaces. Precisely, it is of the form \[ \|(-\Delta -z)^{-1}f\|_{L^q(\mathbb{R}^d)}\leq C\|f\|_{L^p(\mathbb{R}^d)}\,,\qquad \forall\;z\in\mathbb{C}\backslash [0,\infty). \] They provide a complete characterization of sharp \(L^p\)-\(L^q\) resolvent estimates which could depend on \(z\). Additionally, they consider sharp resolvent estimates for the fractional Laplacians. Further, the obtain new results for the Bochner-Riesz operators of negative index.