论文标题
$ l^{p} $梯度估计和calderón-齐格蒙德不平等现象
$L^{p}$ gradient estimates and Calderón--Zygmund inequalities under Ricci lower bounds
论文作者
论文摘要
在本文中，我们研究了泊松方程解决方案的第一和第二阶$ l^{p} $估计的有效性，具体取决于基础歧管的几何形状。我们首先在假设RICCI张量以局部积分意义上降低限制并构造第一个反例子的假设，表明它们是错误的，通常是错误的，没有曲率限制。 Next, we obtain $L^p$ estimates for the second order Riesz transform (or, equivalently, the validity of $L^{p}$ Calderón--Zygmund inequalities) on the whole scale $1<p<+\infty$ by assuming that the injectivity radius is positive and that the Ricci tensor is either pointwise lower bounded or non-negative in a global integral sense.当$ 1 <p \ leq 2 $时，如果还可以控制RICCI的衍生产品，则可以在高阶Riesz变换上获得类似的$ l^p $界限。在$ p $的相同范围内，对于具有较低RICCI边界的流形和频谱的正底部的流形，我们表明Laplacian的$ l^{p} $ NORM控制着整个$ W^{2，P} $ - 在紧凑型功能上。
In this paper we investigate the validity of first and second order $L^{p}$ estimates for the solutions of the Poisson equation depending on the geometry of the underlying manifold. We first present $L^{p}$ estimates of the gradient under the assumption that the Ricci tensor is lower bounded in a local integral sense and construct the first counterexample showing that they are false, in general, without curvature restrictions. Next, we obtain $L^p$ estimates for the second order Riesz transform (or, equivalently, the validity of $L^{p}$ Calderón--Zygmund inequalities) on the whole scale $1<p<+\infty$ by assuming that the injectivity radius is positive and that the Ricci tensor is either pointwise lower bounded or non-negative in a global integral sense. When $1<p \leq 2$, analogous $L^p$ bounds on even higher order Riesz transforms are obtained provided that also the derivatives of Ricci are controlled up to a suitable order. In the same range of values of $p$, for manifolds with lower Ricci bounds and positive bottom of the spectrum, we show that the $L^{p}$ norm of the Laplacian controls the whole $W^{2,p}$-norm on compactly supported functions.
