A note on weighted norm inequalities for the Hardy-Littlewood maximal operator
A note on weighted norm inequalities for the Hardy-Littlewood maximal operator
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关于 Hardy-Littlewood 极大算子的加权范数不等式的注释
DOI：
10.1090/s0002-9939-1983-0684636-9
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发表时间：
1983
影响因子：
2.5
通讯作者：
R. Fefferman
中科院分区：
文献类型：
--
作者：
M. Christ;R. Fefferman
本文给出了R”中Hardy-Littlewood极大算子的权模不等式的一个极其简单的证明。本文给出了极大算子11 Mf 11 Lr（w）Cp 1f 1 Lr（w）在权wEAp，p > 1时有界性的一个极初等的证明.结果是众所周知的，是由于Muckenhoupt [1]。在R. Coifman和C. February man [21]，但仍然依赖于一个“反向保持器”不等式，并且不是完全初等的。而最近索耶在[3]中解决了极大算子的两权问题，在一权情形下得到了一个简单的新证明，但其中涉及一个与Ap不明显等价的权条件。在[4]中，Hunt、Kurtz和Neugebauer证明了这个等价性，并且他们的证明避免了反向的保持器不等式。最后，我们的证明避免了反向的保持器不等式和Ap条件的重新表述。证明是非常相似的索耶的[3]和基督的替代证明索耶的[5]。我们假定读者熟悉加权不等式的基本定义，如[2]中所示。然后，令f C LP（w），其中w C Ap（R '），p > 1。我们Calder 6 n-Zygmund在高度C 'k处分解f，其中k是整数，其中Cn是仅取决于维数n的某个大常数，并且我们稍后指定。设Qk j 2 1是高度Cnk处的Calder 6 n-Zygmund立方体。设Efk = Q-kUQIcQkQ '。那我们有fRn吗B k，（I）< Pw（E k）（此后我们将JEq写成q（E），其中q在Lloi（Rn）中，qE = q（E）/ E？，按照索耶的方法，设置w /（P -）w（Et）IQ| ? B~[（'）f（~fa I）o E k）I（J）（Q）-）I编辑1982年6月15日收到。1980年数学学科分类第42 B25章
In.this note we give an extremely simple proof of the weight norm inequalities for the Hardy-Littlewood maximal operator in R". The purpose of this note is to provide an extremely elementary proof of the boundedness of the maximal operator, 11 Mf 11 Lr(w) Cp 1 f 1 Lr(w) when the weight w E Ap, p > 1. The result is well known and is due to Muckenhoupt [1]. The proof was simplified in R. Coifman and C. Fefferman [21, but still depended on a "reverse Holder" inequality and was not totally elementary. Rather recently Sawyer in [3] solved the two weight problem for the maximal operator, which yielded a simple new proof in the one weight case, but which involved a condition on the weight not obviously equivalent to Ap. In [4] Hunt, Kurtz, and Neugebauer proved this equivalence and their proof avoids the reverse Holder inequality. Finally, our proof avoids both the reverse Holder inequality and also the reformulation of the Ap condition. The proof is very similar to Sawyer's [3] and Christ's alternate proof to Sawyer's [5]. We shall assume the reader is familiar with the basic definitions from weighted inequalities as appear in, say, [2]. Then, let f C LP(w) where w C Ap(R'), p > 1. We Calder6n-Zygmund decompose f at heights C'k for k an integer, where Cn is some large constant depending only on the dimension n, and which we specify later. Let Qk j 2 1, be the Calder6n-Zygmund cubes at height Cnk. Let Efk = Q-k UQIcQkQ' . Then we have fRn ? B k,, ( I < Pw(E k (henceforth we write q(E) for JEq where q is in Lloi (Rn), qE = q(E)/ E ?, and following Sawyer, set a w /(P -) w(Et) IQ | ?B~~[(')f( ~fa I)o E k) I(J) (Q)-)I Received by the editors June 15, 1982. 1980 Mathematics Subject Classification. Primary 42B25.