Decay estimates for Schrödinger heat semigroup with inverse square potential in Lorentz spaces II

Abstract

Let $H:=-\Delta +V$ be a nonnegative Schrödinger operator on $L^2({\mathbf{R}}^N)$, where $N\ge 2$ and $V$ is a radially symmetric inverse square potential. Let $\Vert {\nabla }^{\alpha }{e}^{-tH}{\Vert }_{(L^{p,\sigma }\to L^{q,\theta })}$ be the operator norm of ${\nabla }^{\alpha }{e}^{-tH}$ from the Lorentz space $L^{p,\sigma }({\mathbf{R}}^N)$ to $L^{q,\theta }({\mathbf{R}}^N)$, where $\alpha \in \{0,1,2,\dots \}$. We establish both of upper and lower decay estimates of $\Vert {\nabla }^{\alpha }{e}^{-tH}{\Vert }_{(L^{p,\sigma }\to L^{q,\theta })}$ and study sharp decay estimates of $\Vert {\nabla }^{\alpha }{e}^{-tH}{\Vert }_{(L^{p,\sigma }\to L^{q,\theta })}$. Furthermore, we characterize the Laplace operator $-\Delta$ from the view point of the decay of $\Vert {\nabla }^{\alpha }{e}^{-tH}{\Vert }_{(L^{p,\sigma }\to L^{q,\theta })}$.