论文标题
沿变量的奇异积分一个子空间
Singular integrals along variable codimension one subspaces
论文作者
论文摘要
本文通过$ {\ Mathbb r}^n $在$ {\ mathbb r}^n $上与最大运算符进行了处理，该$是通过将tensor产品的任意旋转量的tensor旋转组成的，$ d $维的hörmander--mihlin乘数与$ n-d $坐标的身份，在特定的condimension 1案例1案例1 $ d = n-1 $中。这些最大运算符自然与分化问题和最大调制的奇异积分有关，例如Sjölin对Carleson最大运算符的概括。我们的主要结果是一个弱型$ l^{2}（{\ Mathbb r}^n）$ - 在带限制功能上估计，导致了几种成果。第一个是最大运算符的尖锐$ l^2（{\ Mathbb r}^n）$估算，仅限于有限集的基础性方面，仅限于有限的旋转。第二个是Carleson-Sjölin定理的版本。此外，我们可以在BESOV空间中获得该功能$ b_ {p，1}^0（{\ Mathbb r}^n）$，$ 2 \ le p <\ p <\ infty $，可以从他们的平均值中恢复其平均值，沿其可测量选择的Codimension $ 1 $ $ 1 $子领域，Zygmund $ n $ n $ n的形式。
This article deals with maximal operators on ${\mathbb R}^n$ formed by taking arbitrary rotations of tensor products of a $d$-dimensional Hörmander--Mihlin multiplier with the identity in $n-d$ coordinates, in the particular codimension 1 case $d=n-1$. These maximal operators are naturally connected to differentiation problems and maximally modulated singular integrals such as Sjölin's generalization of Carleson's maximal operator. Our main result, a weak-type $L^{2}({\mathbb R}^n)$-estimate on band-limited functions, leads to several corollaries. The first is a sharp $L^2({\mathbb R}^n)$ estimate for the maximal operator restricted to a finite set of rotations in terms of the cardinality of the finite set. The second is a version of the Carleson--Sjölin theorem. In addition, we obtain that functions in the Besov space $B_{p,1}^0({\mathbb R}^n)$, $2\le p <\infty$, may be recovered from their averages along a measurable choice of codimension $1$ subspaces, a form of Zygmund's conjecture in general dimension $n$.
