Interpolation of Operators

Abstract

In this chapter we overview the technique of interpolation of operators, which is widely used in harmonic analysis in connection with Lebesgue spaces. The underlying idea is to obtain boundedness of an operator based on the available information in the endpoints. In the first section we will deal with the Riesz-Thorin interpolation theorem, also known as the complex method, and give some applications, viz. Hausdorff-Young inequality and Young’s inequality for convolution. In the second section we prove the Marcinkiewicz interpolation theorem in Lebesgue spaces and also mention the theorem in its natural environment, namely in the framework of Lorentz spaces. We end the chapter with the Young’s convolution inequality in Lorentz spaces.