Weighted norm inequalities, off-diagonal estimates and elliptic operators
Pascal Auscher, José Maria Martell
备注：
survey for the El Escorial 2008 proceedings
发表信息：
Harmonic Analysis and Partial Differential Equations. Proceedings of the 8th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial, June 16-20, 2008, Contemporary Mathematics 505 (2010), 61--83
We give an overview of the generalized Calderón-Zygmund theory for "non-integral" singular operators, that is, operators without kernels bounds but appropriate off-diagonal estimates. This theory is powerful enough to obtain weighted estimates for such operators and their commutators with $\BMO$ functions. $L^p-L^q$ off-diagonal estimates when $p\le q$ play an important role and we present them. They are particularly well suited to the semigroups generated by second order elliptic operators and the range of exponents $(p,q)$ rules the $L^p$ theory for many operators constructed from the semigroup and its gradient. Such applications are summarized.
Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: General operator theory and weights
这是一系列四篇文章的第一部分。在这项工作中，我们感兴趣的加权范数估计。我们把重点放在两个不同性质的结果：一个是基于一个好的$λ$不等式与两个参数和其他使用Calderón-Zygmund分解。这些结果很好地适用于奇异的“非积分”算子及其具有有界平均振荡函数的算子。奇异意味着它们是0阶的，“非整数”意味着它们没有一个带有大小估计的核的整表示，甚至是粗略的，所以它们可能不在$1 < p < \infty$的所有$L^p$空间上有界。然后用适当的局部$L^p-L^q$估计值替换逐点估计值。我们获得加权$L^p$估计的$p$的范围是不同的$(1,\infty)$和隔离的权利类的权重。特别是，我们证明了这样一个类的外推定理' à la Rubio de Francia'，从而向量值估计。
