论文标题
Lorentz空间II中具有逆平方电位的Schrödinger热半群的衰减估计值II
Decay estimates for Schrödinger heat semigroup with inverse square potential in Lorentz spaces II
论文作者
论文摘要
令$ h：= - δ+v $是$ l^2（{\ bf r}^n）$上的非负Schrödinger运算符，其中$ n \ ge 2 $和$ v $是径向对称的倒数平方电位。令$ \ | \ nabla^αe^{ - th} \ | _ {（l^{p，σ} \ to lorentz space $ l^nabla^a^{ - to p， $ l^{q，θ}（{\ bf r}^n）$，其中$α\ in \ {0,1,2，\ dots \} $。我们建立了$ \ | \ | \ nabla^αe^{ - th} \ | _ {（l^{p，σ} \ to l^{q，θ}）} $的上层和下衰减估计值和研究敏锐的衰减估计值$ \ | \ | \ | \ | \ nabla^αe^al^{ - l^{q，θ}）} $。此外，我们从$ \ | \ | \ nabla^αe^a^{ - th} \ | _ {（l^{p，σ} \ to l^{q，θ}}} $的衰减中表征了拉普拉斯操作员$-Δ$。
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 $\|\nabla^αe^{-tH}\|_{(L^{p,σ}\to L^{q,θ})}$.
