Uniform Resolvent Estimates for Laplace–Beltrami Operator on the Flat Euclidean Cone

We study the \(L^p\rightarrow L^q\)-type uniform resolvent estimate for Laplace –Beltrami operator on the flat Euclidean cone \(C(\mathbb {S}_{\sigma }^1)\triangleq \mathbb {R}_{+}\times (\mathbb {R}/2\pi \sigma \mathbb {Z})\) equipped with the metric \(g(r,\theta )=dr^2+r^2d\theta ^2\) where the circle of radius \(\sigma >0\). The key ingredient is the resolvent kernel constructed by Zhang in (J Funct Anal 282(3):109311, 2022) and the Young inequality holds under the monotonicity assumption on the flat Euclidean cone.