标题： 梯度估计在诺伊曼半群及其应用 显示英文标题

标题： Gradient Estimate on the Neumann Semigroup and Applications

摘要： 我们证明了在边界为$C^2$-光滑或凸的$d$-维紧致区域$\OO$上，Neumann半群$P_t$的梯度具有以下精确上界：$$\|\nn P_t\|_{1\to \infty}\le \ff{c}{t^{(d+1)/2}},\ \ t>0,$$，其中$c>0$是依赖于区域的常数，$\|\cdot\|_{1\to\infty}$是从$L^1(\OO)$到$L^\infty(\OO)$的算子范数。 此估计意味着对Neumann热核的梯度给出了高斯类型的逐点上界，这被应用于紧凸域上非齐次Neumann问题解的正则性、Hardy空间和Riesz变换的研究。 显示英文摘要

摘要： We prove the following sharp upper bound for the gradient of the Neumann semigroup $P_t$ on a $d$-dimensional compact domain $\OO$ with boundary either $C^2$-smooth or convex: $$\|\nn P_t\|_{1\to \infty}\le \ff{c}{t^{(d+1)/2}},\ \ t>0,$$ where $c>0$ is a constant depending on the domain and $\|\cdot\|_{1\to\infty}$ is the operator norm from $L^1(\OO)$ to $L^\infty(\OO)$. This estimate implies a Gaussian type point-wise upper bound for the gradient of the Neumann heat kernel, which is applied to the study of the Hardy spaces, Riesz transforms, and regularity of solutions to the inhomogeneous Neumann problem on compact convex domains.