加权范数不等式、非对角估计与椭圆算子。部分IV：流形和权上的里斯变换

Reverse Poincare inequalities, Isoperimetry, and Riesz transforms in Carnot groups

We prove an optimal reverse Poincaré inequality for the heat semigroup generated by the sub-Laplacian on a Carnot group of any step. As an application we give new proofs of the isoperimetric inequality and of the boundedness of the Riesz transform in Carnot groups.

Sobolev不等式、里斯变换和里奇流

Sobolev Inequalities, Riesz Transforms and the Ricci Flow

本文研究了由一个Sobolev不等式进一步导出Sobolev不等式的问题。我们使用几种不同的方法，包括贝塞尔势和里斯变换。将所得结果应用于里奇流，将作者关于沿里奇流的$W^{1,2}$ Sobolev不等式的结果推广到一般p的$W^{1,p}$和$W^{2,p}$ Sobolev不等式.

流形上的里斯变换与Poincaré不等式

Riesz transform on manifolds and Poincaré inequalities

研究了完备黎曼流形上的里斯变换的$L^p$不等式在二重性质和Poincaré不等式下的有效性，当$p>2$和$p<2$在完备黎曼流形上时$L^p$不等式的逆不等式的有效性.

$L^{p}$ gradient estimates and Calderón--Zygmund inequalities under Ricci lower bounds

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<+…

Hardy spaces and heat kernel regularity

In this paper, we show the equivalence between the boundedness of the Riesz transform $dΔ^{-1/2}$ on $L^p$, $p\in (2,p_0)$, and the equality $H^p=L^p$, $p\in(2,p_0)$, in the class of manifold whose measure is doubling and for which the scaled Poincaré inequalities hold. Here, $H^p$ is a Hardy space of exact $1-$forms, naturally associated with the Riesz transform.

加权范数不等式、非对角估计与椭圆算子

Weighted norm inequalities, off-diagonal estimates and elliptic operators

我们给出了“非积分”奇异算子的广义Calderón-Zygmund理论的概述，即没有核边界但有适当的非对角估计的算子。这个理论是强大的，足以获得加权估计，这样的运营商和他们的消除与$\BMO$功能。当$p\le q$起重要作用时，我们给出了$L^p-L^q$非对角估计。它们特别适用于二阶椭圆算子生成的半群，并且指数范围$(p,q)$对于由该半群及其梯度构造的许多算子都规定了$L^p$理论.这些应用程序进行了总结。