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

Let $H:=-\mathrm{\Delta }+V$ be a nonnegative Schr"odinger operator on $L^2({\bf R}^N)$,where$N\ge 2$and$V$ is a radially symmetric inverse square potential. Let $\mathrm{\parallel }{\mathrm{\nabla }}^{\alpha }{e}^{-tH}{\mathrm{\parallel }}_{(L^{p,\sigma }\to L^{q,\theta })}$ be the operator norm of ${\mathrm{\nabla }}^{\alpha }{e}^{-tH}$ 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 $\mathrm{\parallel }{\mathrm{\nabla }}^{\alpha }{e}^{-tH}{\mathrm{\parallel }}_{(L^{p,\sigma }\to L^{q,\theta })}$ and study sharp decay estimates of $|\nabla^\alpha e^{-tH}|{(L^{p,\sigma}\to L^{q,\theta})}$\mathrm{.}Furthermore,wecharacterizetheLaplaceoperator$-\Delta$fromtheviewpointofthedecayof$|\nabla^\alpha e^{-tH}|{(L^{p,\sigma}\to L^{q,\theta})}$.