Course Category： Professional Basic Course
Prerequisites：Mathematical analysis, Real variable functions, Functional analysis, General topology.
Aims & Requirements：
The curriculum is a core course for both the postgraduates and the doctorates from all fields of mathematics and also an elective course for graduates who major in theoretical physics. The main contents of the course include basic operator theory, Fourier analysis on Euclidean spaces, the generalized function theory, Hardy spaces Hp and BMO spaces. Through learning the course, the students can master the basic concepts, methods, skills of modern real analysis and lay foundation for further studying modern mathematics and doing further research work in mathematics.
Chapter 1 Basic operator theory
Convolution operators; Hardy－Littlewood maximal operators; Identity approximation operators; Interpolation theorem.
Chapter 2 Fourier analysis on Euclidean spaces
The basic theory of the Fourier transform; The theory of the Fourier transform.
Chapter 3 Generalized function theory
The Schwartz function spaces; Generalized function spaces.
Chapter 4 Hardy spaces Hp and BMO spaces
Hardy spaces Hp and BMO spaces; Calderon-Zygmund theory.
1. Walter Rudin, Real and complex Analysis, Third Edition.
2. Anthony W. Knapp, Advanced Real Analysis, Birkhauser Boston Basel Berlin, 2005
1. E. M. Stein, R. Shakarchi, Real Analysis World Scientific, 2006.
2. R. L. Wheeden and A. Zygmund, Measure and Integral, 1977.
Author：：Dunyan Yan（Mathematical Science, GUCAS）
Date： November, 2011.