The Application Of Riesz-thorin Interpolation Theorem And The Promotion Of The Clarkson Inequality

In the Fourier analysis,Riesz-Thorin interpolation theorem is very important.The application of the theorem include the Young inequality,the Hausdorff-Young theorem and the Clarkson inequality ,etc.The Clarkson inequality is about the p-th power of the complex absolute, and obtained by Clarkson in researching the uniform convexity of the LP space.In this paper, the Clarkson inequality will be investigated in two sections. One section is to use Hilbert space to replace the complex space; the other is to extend the parallelogram rule to high dimension. In the promotion of Clarkson inequality the Riesz-Thorin interpo-lation theorem was used, and the results obtained to a large extent to expand its scope of application.