Estimates for the maximal singular integral in terms of the singular integral: the case of even kernels
Estimates for the maximal singular integral in terms of the singular integral: the case of even kernels
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根据奇异积分估计最大奇异积分：偶核的情况
DOI：
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
Juan Verdera Melenchón
中科院分区：
文献类型：
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作者：
Joan Mateu Bennassar;Joan Orobitg i Huguet;Juan Verdera Melenchón
设T是Rn中光滑齐次Calderon-Zygmund奇异积分算子.在本文中，我们研究的问题，控制最大奇异积分T？f的奇异积分Tf。最基本的控制形式之一，可以考虑的是估计的L2（Rn）规范的T？f乘以Tf的L2（Rn）范数。我们表明，如果T是一个更高的顺序Riesz变换，那么有更强的逐点不等式T？f（x）CM（Tf）（x），其中C是常数，M是Hardy-Littlewood极大算子.我们证明，L 2估计T？的T是等价的，甚至光滑齐次Calderon-Zygmund算子，T？和M（T）。我们的主要结果用一个代数条件刻画了L2不等式和点态不等式，该代数条件用T的核（x）jxjn表示，其中
Let T be a smooth homogeneous Calder on-Zygmund singular integral operator in R n . In this paper we study the problem of controlling the maximal singular integral T ? f by the singular integral Tf. The most basic form of control one may consider is the estimate of the L 2 (R n ) norm of T ? f by a constant times the L 2 (R n ) norm of Tf. We show that if T is an even higher order Riesz transform, then one has the stronger pointwise inequality T ? f(x) C M(Tf)(x), where C is a constant and M is the Hardy-Littlewood maximal operator. We prove that the L 2 estimate of T ? by T is equivalent, for even smooth homogeneous Calder on-Zygmund operators, to the pointwise inequality between T ? and M(T ). Our main result characterizes the L 2 and pointwise inequalities in terms of an algebraic condition expressed in terms of the kernel ( x) jxjn of T , where is an