Decay estimates for Schrödinger heat semigroup with inverse square potential in Lorentz spaces II
Let $H:=-Δ+V$ be a nonnegative Schrödinger operator on $L^2({\bf R}^N)$, where $N\ge 2$ and $V$ is a radially symmetric inverse square potential. Let $\|\nabla^αe^{-tH}\|_{(L^{p,σ}\to L^{q,θ})}$ be the operator norm of $\nabla^αe^{-tH}$ from the Lorentz space $L^{p,σ}({\bf R}^N)$ to $L^{q,θ}({\bf R}^N)$, where $α\in\{0,1,2,\dots\}$. We establish both of upper and lower decay estimates of $\|\nabla^αe^{-tH}\|_{(L^{p,σ}\to L^{q,θ})}$ and study sharp decay estimates of $\|\nabla^αe^{-tH}\|_{(L^{p,σ}\to L^{q,θ})}$. Furthermore, we characterize the Laplace operator $-Δ$ from the view point of the decay of $\|\nabla^αe^{-tH}\|_{(L^{p,σ}\to L^{q,θ})}$.
math.AP↗