Hardy-Littlewood, Hausdorff-Young-Paley inequalities, and L-L Fourier multipliers on compact homogeneous manifolds
Hardy-Littlewood, Hausdorff-Young-Paley inequalities, and L-L Fourier multipliers on compact homogeneous manifolds
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DOI：
10.1016/j.jmaa.2019.07.010
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发表时间：
2015-04
影响因子：
1.3
通讯作者：
R. Akylzhanov;E. Nursultanov;Michael Ruzhansky
中科院分区：
文献类型：
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作者：
R. Akylzhanov;E. Nursultanov;Michael Ruzhansky
本文证明了描述紧致齐次流形上函数的“大小”与其傅立叶系数的“大小”之间关系的新的不等式。这些不等式可以看作是Hardy和Littlewood[17]在圆周上得到的Hardy-Littlewood不等式的非对易形式。对于群SU(2)的例子，我们证明了所得到的Hardy-Littlewood不等式是尖锐的，从而给出了一个关于函数的傅里叶系数为L p(SU(2))的判据。我们还在一般紧致齐次流形上建立了Paley和Hausdorff-Young-Paley不等式。利用后者得到了紧齐次流形上1<；p≤2≤q<；∞的傅立叶乘子的L p-L q有界性的条件以及一般(非不变)算子在紧Lie群上的L p-L q有界性的条件.我们还记录了全序离散集上Marcinkiewicz插值定理的一个抽象形式，用于证明酉对偶上具有不同Plcherel测度的情形。
In this paper we prove new inequalities describing the relationship between the “size” of a function on a compact homogeneous manifold and the “size” of its Fourier coefficients. These inequalities can be viewed as noncommutative versions of the Hardy-Littlewood inequalities obtained by Hardy and Littlewood [17] on the circle. For the example case of the group SU (2) we show that the obtained Hardy-Littlewood inequalities are sharp, yielding a criterion for a function to be in L p (SU (2)) in terms of its Fourier coefficients. We also establish Paley and Hausdorff-Young-Paley inequalities on general compact homogeneous manifolds. The latter is applied to obtain conditions for the L p-L q boundedness of Fourier multipliers for 1< p≤ 2≤ q<∞ on compact homogeneous manifolds as well as the L p-L q boundedness of general (non-invariant) operators on compact Lie groups. We also record an abstract version of the Marcinkiewicz interpolation theorem on totally ordered discrete sets, to be used in the proofs with different Plancherel measures on the unitary duals.