变指数空间上Littlewood-Paley算子及相关算子的研究与应用

Littlewood-Paley 算子及相关算子在各类变指数函数空间上的有界性和加权估计, 变指数函数空间的Littlewood-Paley刻画, 以及分数阶不可压缩 Navier-Stokes 程在此类函数空间中解的适定性是本项目研究的核心内容. 具体将研究下面几个问题. (1) Littlewood-Paley算子在变指数函数空间上的有界性和加权估计; (2) 参数型粗糙核 Littlewood-Paley 算子在变指数空间上的有界性和加权估计; (3) Marcinkiewicz积分在变指数空间上的有界性和加权估计; (4) Littlewood-Paley交换子在变指数函数空间上的有界性; (5) 非齐型空间上变指数函数空间的若干问题; (6) 变指数 Lebesgue 空间上最优界及最值函数问题; (7) 分数阶不可压缩Navier-Stokes方程在变指数空间上解的适定性.

The core of the project is to study the the boundedness and weighted estimates of Littlewood-Paley operators and related operators on spaces with variable exponent, as well as the Littlewood-Paley characterizations of these spaces with variable exponent and their applications to the well-posedness of the solutions for the fractional order incompressible Navier-Stokes equations. Specifically, we will study the following questions in the project. (1) The boundedness and weighted estimates of Littlewood-Paley operators on spaces with variable exponent; (2) the boundedness and weighted estimates of parameterized Littlewood-Paley operators with rough kernel on spaces with variable exponent; (3) the boundedness and weighted estimates of Marcinkiewicz integrals on spaces with variable exponent; (4) The boundedness for the commutators of Littlewood-Paley operators on spaces with variable exponent; (5) Several problems on spaces with variable exponent in nonhomogeneous space; (6) Sharp bounds and functions on Lebesgue spaces with variable exponent; (7) The well-posedness of the solutions for the fractional order incompressible Navier-Stokes equations on spaces with variable exponent.