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Let $H:=-\Delta+V$ be a nonnegative Schr\"odinger operator on $L^2({\bfR}^N)$, where $N\ge 2$ and $V$ is a radially symmetric inverse squarepotential. Let $\|\nabla^\alpha e^{-tH}\|_{(L^{p,\sigma}\to L^{q,\theta})}$ bethe operator norm of $\nabla^\alpha e^{-tH}$ from the Lorentz space$L^{p,\sigma}({\bf R}^N)$ to $L^{q,\theta}({\bf R}^N)$, where$\alpha\in\{0,1,2,\dots\}$. We establish both of upper and lower decayestimates of $\|\nabla^\alpha e^{-tH}\|_{(L^{p,\sigma}\to L^{q,\theta})}$ andstudy sharp decay estimates of $\|\nabla^\alpha e^{-tH}\|_{(L^{p,\sigma}\toL^{q,\theta})}$. Furthermore, we characterize the Laplace operator $-\Delta$from the view point of the decay of $\|\nabla^\alphae^{-tH}\|_{(L^{p,\sigma}\to L^{q,\theta})}$.

Decay estimates for Schr\"odinger heat semigroup with inverse square potential in Lorentz spaces II