Banach Center Publications, Volume ** Institute of Mathematics Polish Academy of Sciences Warszawa 201* Problems on Averages and Lacunary Maximal Functions
Banach Center Publications, Volume ** Institute of Mathematics Polish Academy of Sciences Warszawa 201* Problems on Averages and Lacunary Maximal Functions
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巴纳赫中心出版物，卷 ** 波兰科学院数学研究所 华沙 201* 平均值和空隙极大函数问题
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通讯作者：
James Wright
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作者：
A. Seeger;James Wright
我们证明了三个结果卷积算子和缺极大函数相关的扩张措施。首先，在维数的假设和Lp正则性界的假设下，得到了缺项极大算子的H1到L1，∞界.其次，得到了平面凸曲线上与平均值相关的缺项极大算子的L2有界性的一个充要条件。最后证明了这种平均值的Lp正则性结果.我们提出了各种开放的问题。1.导论.考虑一个紧支撑的有限Borel测度μ，定义它的并矢扩张为μ k，f = μ，f（2 k ·）.本文的主要对象是卷积f → f * µ k和缺极大函数，
We prove three results concerning convolution operators and lacunary maximal functions associated to dilates of measures. First we obtain an H 1 to L 1,∞ bound for lacunary maximal operators under a dimensional assumption on the underlying measure and an assumption on an L p regularity bound for some p > 1. Secondly, we obtain a necessary and sufficient condition for L 2 boundedness of lacunary maximal operator associated to averages over convex curves in the plane. Finally we prove an L p regularity result for such averages. We formulate various open problems. 1. Introduction. We consider a compactly supported finite Borel measure µ and define its dyadic dilates by µ k , f = µ, f (2 k ·). The main objects of this paper are the convolutions f → f * µ k and the lacunary maximal function given by