Boundedness of commutators of fractional and singular integrals for the extreme values of $p$
Boundedness of commutators of fractional and singular integrals for the extreme values of $p$
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DOI：
10.1215/ijm/1256068988
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发表时间：
1997-12
影响因子：
0.6
通讯作者：
E. Harboure;C. Segovia;J. Torrea
中科院分区：
文献类型：
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作者：
E. Harboure;C. Segovia;J. Torrea
众所周知，只要B是BMO函数[C-R-W]，奇异积分与可测函数B（x）相乘的算子是Lp，< p <0 0上的有界算子.此外，如果所有Riesz变换的换位子对于某些p有界，1 < p < oo，则函数B必然属于BMO。对于分数次积分算子I，后来，Segovia和Torrea [S-T]在向量值算子的更一般的上下文中考虑了这个问题，包括在这种方法中，例如与极大函数相关的算子。本文在函数B上找到了H L和Ln/ BMO有界的充分条件。在大多数情况下，给定的条件也是必要的。关于B是BMO函数的讨论，见[P]。我们已经选择了工作在一般情况下的向量值算子的奇异积分型，以包括一个更大的类的算子。沿着这条线，我们首先证明两个一般定理（定理A和B在第2节）表示的条件B的核给定的操作。然后，在第三节中，我们将我们的定理应用到一些特殊的情况，如希尔伯特变换，分数次积分和极大算子的光滑逼近的身份。作为一个例子，希尔伯特变换的分解子仅在B等于常数的平凡情况下从H到L是有界的;这也是另一个极端的情况，L到BMO。对于分数次积分也证明了类似的结果，因此，由于常数函数对应于BMO中的零函数，因此对于非零BMO函数，希尔伯特变换或分数次积分的换位子在极端情况下是没有界的;见定理（3.1）和（3.10）。在周期性的情况下，图像得到改善。事实上，我们证明了具有共轭函数的算子从L到BMO是有界的当且仅当B属于比BMO更严格的一类，即空间BMO，p（t）log t1-1
It is well known that commutators of singular integrals with multiplication by a measurable function b(x) are bounded operators on Lp, < p < oo, as long as b is a BMO function [C-R-W]. Moreover if the commutator of all Riesz transforms are bounded for some p, 1 < p < oo, the function b must necessarily belong to BMO. Similar results are also known for the fractional integral operators I,, in connection with the boundedness from Lp into Lq, < p < n/u, 1/q 1/p/n [Ch]. Later on, Segovia and Torrea [S-T] have considered this problem in the more general context of vector valued operators including in this approach, commutators associated for example to maximal functions. It this paper we find sufficient conditions on the function b in order to obtain H L and Ln/ BMO boundedness of such commutators. In most of the cases the given conditions will be also necessary. See [P] for a discussion in the case b is aBMO function. We have chosen to work in the general context of vector valued operators of singular integral type as to include a larger class of commutators. Following this line, we first prove two general theorems (Theorems A and B in Section 2) expressing the conditions on b in terms of the kernel of the given operator. Afterwards, in Section 3, we apply our theorems to some particular cases like the Hilbert transform, fractional integrals and maximal operators of smooth approximations to the identity. As an example, commutators with the Hilbert transform are bounded from H into L only in the trivial case that b equals a constant; this is also the case in the other extreme, L into BMO. Similar results are proven for the fractional integral, therefore since a constant function corresponds to the zero function in BMO we have that for non-zero BMO functions the commutator with the Hilbert transform or fractional integral is not bounded in the extreme cases; see Theorems (3.1) and (3.10). The picture improves in the periodic case. In fact we prove that commutators with the conjugate function are bounded from L into BMO if and only if b belongs to a class a little bit more restricted than BMO, the space BMO for p(t) log t1-1