Regularizations of general singular integral operators.
Regularizations of general singular integral operators.
复制标题
一般奇异积分算子的正则化。
DOI：
10.4171/rmi/712
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发表时间：
2010
影响因子：
1.2
通讯作者：
S. Treil
中科院分区：
文献类型：
--
作者：
C. Liaw;S. Treil
在奇异积分算子的理论中，通常需要大量的努力来严格定义这样的算子。这是因为这些算子的核在对角线上不是局部可积的，所以即使对于好的函数，形式上定义算子或其双线性形式的积分也没有很好的定义（被积函数不在L^1中）。然而，由于核函数只在对角线上有奇点，双线性形式是很好的定义，比如对于有界紧支持的分离支持函数。
In the theory of singular integral operators significant effort is often required to rigorously define such an operator. This is due to the fact that the kernels of such operators are not locally integrable on the diagonal, so the integral formally defining the operator or its bilinear form is not well defined (the integrand is not in L^1) even for nice functions. However, since the kernel only has singularities on the diagonal, the bilinear form is well defined say for bounded compactly supported functions with separated supports.
One of the standard ways to interpret the boundedness of a singular integral operators is to consider regularized kernels, where the cut-off function is zero in a neighborhood of the origin, so the corresponding regularized operators with kernel are well defined (at least on a dense set). Then one can ask about uniform boundedness of the regularized operators. For the standard regularizations one usually considers truncated operators.
The main result of the paper is that for a wide class of singular integral operators (including the classical Calderon-Zygmund operators in non-homogeneous two weight settings), the L^p boundedness of the bilinear form on the compactly supported functions with separated supports (the so-called restricted L^p boundedness) implies the uniform L^p-boundedness of regularized operators for any reasonable choice of a smooth cut-off of the kernel. If the kernel satisfies some additional assumptions (which are satisfied for classical singular integral operators like Hilbert Transform, Cauchy Transform, Ahlfors--Beurling Transform, Generalized Riesz Transforms), then the restricted L^p boundedness also implies the uniform L^p boundedness of the classical truncated operators.