WEIGHTED POINCARE AND SOBOLEV INEQUALITIES AND ESTIMATES FOR WEIGHTED PEANO MAXIMAL FUNCTIONS
WEIGHTED POINCARE AND SOBOLEV INEQUALITIES AND ESTIMATES FOR WEIGHTED PEANO MAXIMAL FUNCTIONS
复制标题
加权Poincare和Sobolev不等式以及加权PEANO极大值函数的估计
DOI：
10.2307/2374351
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发表时间：
1985
影响因子：
1.7
通讯作者：
R. Wheeden
中科院分区：
文献类型：
--
作者：
Sagun Chanillo;R. Wheeden
1.导论.在本文中，我们得到了涉及两个权函数的局部Poincare和Sobolev不等式。我们得到的估计包含了Fabes，Kenig和Serapioni在[8]中对单权函数情形证明的估计。我们还考虑了加权Peano导数的概念，并表明，与一个给定的权重函数形成的Peano导数可以估计的最大函数涉及第二个权重函数。在一般Lebesgue测度的特殊情况下，这类结果导出了A. P. Calderon和A. Zygmund关于Sobolev空间中函数的局部可微性和Tonelli意义下有界变差的函数的局部可微性（见[3]和[4]）。我们所有的结果都是基于[5]中基本估计的变体，但我们不预先假设[5]中的任何事实。设v（x）是R”上的权重函数;即，设v（x）是关于Lebesgue测度非负局部可积的.我们将使用符号v（E）= SE v（x）dx;普通勒贝格测度将被表示为IE l。如果Bh（x）是以x为中心，半径为h的开球，我们说v满足加倍条件（关于勒贝格测度），如果
1. Introduction. In this paper, we derive local Poincare and Sobolev inequalities involving two weight functions. The estimates we obtain include those proved by Fabes, Kenig and Serapioni in [8] for the one weight function case. We also consider the notion of weighted Peano derivatives and show that the Peano derivative formed with a given weight function can be estimated in terms of a maximal function involving a second weight function. In the special case of ordinary Lebesgue measure, results of this kind lead to the theorems of A. P. Calderon and A. Zygmund about local differentiability of functions in Sobolev spaces and of functions which are of bounded variation in the sense of Tonelli (see [3] and [4]). All our results are based on variants of the basic estimate in [5], but we do not presuppose any facts from [5]. Let v(x) be a weight function on R"; i.e., let v(x) be nonnegative and locally integrable with respect to Lebesgue measure. We shall use the notation v(E) = SE v(x) dx; ordinary Lebesgue measure will be denoted IE l. If Bh(x) is the open ball with center x and radius h, we say v satisfies the doubling condition (with respect to Lebesgue measure) if