Hermite函数和Laguerre函数在许多数学领域中发挥着重要作用，也是欧氏空间上的调和分析中的重要工具和研究对象。本项目研究Hardy空间中的函数关于广义Hermite函数系、多元广义Hermite函数系和多元Laguerre函数系展开的Hardy-Littlewood型定理和一般系数乘子问题，分别引入合理的系数条件，确定尽可能大的指标范围，建立具有深刻意义的各种乘子定理。这些问题的研究将利用调和分析实方法和泛函分析的手段，并且由于基函数的复杂性，需要采用一些新方法或新途径。

The Hermite functions and the Laguerre functions play an important role in many fields of mathematics, and are also important tools and research subjects in harmonic analysis on Euclidean spaces. This project is to study the Hardy-Littlewood type theorems and the problems on coefficient multipliers of functions in Hardy spaces associated with expansions in terms of the generalized Hermite functions, the multivariate generalized Hermite functions and the Laguerre functions. Some conditions on multipliers will be introduced in each case separately, the range of indexes will be determined as large as possible, and various theorems on coefficient multipliers will be proved. The real-variable methods in harmonic analysis and some theory in functional analysis will be applied in study of these problems, and because of complexity of these basis functions, some new methods or tools will be used.