各向异性Musielak-Orlicz函数空间的实变理论

The theory of Musielak-Orlicz function spaces plays an important role in various fields of harmonic analysis,elasticity,fluid dynamics and nonlinear PDEs.The anisotropic weighted Hardy spaces are very geneal and keep the core natures of the classic Hardy spaces,which include the corresponding classical、parabolic、weighted、Orlicz and Musielak-Orlicz Hardy spaces. The applicant and his collaborators have partly established some real-variable theories of weighted anisotropic/product/Musielak-Orlicz Hardy spaces. This project will go on establishing some real variable theory of anisotropic Musielak-Orlicz (localized/product) Hardy spaces including the Littlewood-Paley theory, atomic/ molecular decompositions, different kinds of maximal function characterizations, the duality between anisotropic Musielak-Orlicz (localized) BMO spaces and anisotropic Musielak-Orlicz (localized) Hardy spaces, the Carleson measure characterization of anisotropic Musielak-Orlicz BMO spaces, which will be applied to study the boundedness of some kinds of (sub-)linear operators, including (fractional) singular integral operators, pointwise multipliers, Riesz transform of uniformly eliptical operators related to some potential functions, multilinear Hausdorff operators and so on.

Musielak-Orlicz(M-O)型函数空间理论在调和分析、弹力学、流体力学、非线性PDEs等问题上都有重要应用.本项目所研究的各向异性M-O型Hardy空间包括了经典的、抛物的、加权的、Orlicz型及经典M-O型的Hardy空间,具有广泛的一般性,且保持了Hardy空间的核心性质.申请人及合作者已部分发展了加权各向异性/乘积/M-O型Hardy空间的实变理论.本项目拟进一步建立各向异性M-O型(局部/乘积)Hardy空间的Littlewood-Paley理论,原子分解,分子分解,各种极大函数特征,各向异性M-O型的(局部)BMO空间与各向异性M-O型Hardy的对偶关系及其Carleson测度特征等实变理论,并将其应用于各类(次)线性算子在上述空间中有界性的研究中,包括(分数次)奇异积分算子、点态乘子、具有势函数V的一致抛物型算子的Riesz变换,多线性Hausdorff算子等.

本项目研究了强\弱各向异性Musielak-Orlicz(M-O)型Hardy空间的原子特征, 分子特征, 极大函数特征等实变特征以及各向异性BMO函数空间的实变理论。作为应用, 本项目进一步研究了各向异性奇异积分算子, 各向异性分数次积分算子， Marcinkiewicz算子, Bochner-Riesz平均算子, 各向异性多线性算子, 交换子等算子在强\弱各向异性M-O型Hardy空间上的有界性。此研究工作将经典各向齐性的 抛物型的，加权型的，Orlicz型的Hardy空间理论都统一在M-O型各向异性Hardy空间理论下进行研究. 此函数空间, 即具有广泛的一般性,又包含了相应经典函数空间的核心性质. 另外, 本项目的很多结论即使在经典的各向齐性情形下也是新的. 项目成果为各向异性的小波理论提供了更多的等价范数刻画和不等式估计.