Cases of equality in the riesz rearrangement inequality
Cases of equality in the riesz rearrangement inequality
复制标题
riesz 重排不等式中的平等案例
DOI：
10.2307/2118534
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发表时间：
1996
影响因子：
4.9
通讯作者：
Almut Burchard
中科院分区：
文献类型：
--
作者：
Almut Burchard
确定了Riesz重排不等式zzf(Y)g(x？Y)h(X)dydx ZZ f(Y)g(x？Y)h(X)dydx其中f，g和h是函数f，g和h在Rn上的球面递减重排。我们将我们的结果应用于弱Young不等式。Riesz重排不等式指出，在球面重排下，泛函I(f；g；h)：=Z f ghdx=Zz f(Y)g(x？y)h(X)dydx(1：1)永远不会减小，即对于Rn上右端被取值的任何三个非负可测函数(f；g；h)，1。非负可测函数f的球减重排f是与f相等的球减函数.我们将它定义为f(X)=sup n S>；0 j(N S(F))！N jxj n o；其中N S(F)：=n x 2 R n j f(X)>；S o是f在高度S的水平集，且！N表示单位球在Rn中的测度。也就是说，f的水平集是与f的相应水平集相同的中心测度球。如果f的正值对应的所有水平集都有nite测度，例如，f对于某个p<；1在L p中。本文确定了文(1.2)中的等价性。满足(1.2)式的函数的三元组将被称为不等式的优化三元组或优化器。(1.2)有许多优化器。一个原因是I在一大组Aane变换下是不变的：对于任何线性映射，L，行列式1，向量a，b，c=a+b，在Rn中，我们有g？表示由g？定义的函数？(X)：=g(？x)。显然，在这些对称性下，等同于球面递减函数三元组的任何三元组函数都是优化器。还有第二个理由期待许多优化器。考虑f和g具有紧支承时的情况。那么卷积f_g也有紧支承。如果h是包含f g的支集的特征函数，则f；g；h在(1.2)中产生等式，而不考虑…
We determine the cases of equality in the Riesz rearrangement inequality ZZ f (y)g(x ? y)h(x) dydx ZZ f (y)g (x ? y)h (x) dydx where f , g , and h are the spherically decreasing rearrangements of the functions f , g, and h on R n. We apply our results to the weak Young inequality. The Riesz rearrangement inequality states that the functional I(f; g; h) := Z f gh dx = ZZ f(y)g(x?y)h(x) dydx (1:1) never decreases under spherical rearrangement, that is, 1 for any triple (f; g; h) of nonnegative measurable functions on R n for which the right hand side is deened. The spherically decreasing rearrangement, f , of a nonnegative measurable function f is the spherically decreasing function equimeasurable to f. We will deene it by f (x) = sup n s > 0 j (N s (f)) ! n jxj n o ; where N s (f) := n x 2 R n j f(x) > s o is the level set of f at height s, and ! n denotes the measure of the unit ball in R n. That is, the level sets of f are the centered balls of equal measure as the corresponding level sets of f. This deenition makes sense if all level sets corresponding to positive values of f have nite measure, for example, if f is in L p for some p < 1. In this paper, we determine the cases of equality in (1.2). A triple of functions that satisses (1.2) with equality will be called an optimizing triple, or optimizer, of the inequality. There are many optimizers of (1.2). One reason is that I is invariant under a large group of aane transformations: For any linear map, L, of determinant 1, and vectors a, b, and c = a + b in R n , we have where g ? denotes the function deened by g ? (x) := g(?x). Clearly, any triple of functions that is equivalent to a triple of spherically decreasing functions under these symmetries is an optimizer. There is a second reason to expect many optimizers. Consider the case when f and g have compact support. Then also the convolution f g has compact support. If h is the characteristic function of a set that contains the support of f g, then f; g; h produce equality in (1.2) regardless …