Schatten Classes, Interpolation Spaces, Singular Integral Operators and their Commutators

The treatment of most of the content in this thesis was motivated by the study of the article "Schatten Classes and Commutators of Singular Integral Operators" by S. Janson and T. H. Wolff (1982). To this end, the first chapter introduces the Schatten classes S^p, which are specific classes of compact operators characterized by the property that the sequence of singular values of these operators is p-summable. In the second chapter, a detailed discussion of interpolation spaces is provided. A rigorous definition of these spaces is given, followed by a focus on interpolation using the real method, with particular attention to Lorentz spaces and Schatten-Lorentz classes (of which L^p spaces and S^p classes are special cases). The third chapter addresses homogeneous singular integral operators of convolution type. Specifically, after a brief general discussion necessary for understanding the work of S. Janson and T. H. Wolff, the focus shifts to the case of the Hilbert transform on the line, the circle, and a general Lipschitz curve. The Hilbert transform H on the line and the circle share several analogies, such as boundedness as operators on L^p (for 1 < p < \infty) and from L^1 to weak-L^1, as well as the existence of an integral operator that, under suitable assumptions on the real-valued function f to which it is applied, takes boundary values equal to f + iHf. For the Hilbert transform on Lipschitz curves, the boundedness as an operator on L^2 is presented. In the fourth chapter, the content of the article by S. Janson and T. H. Wolff is presented. It establishes a necessary and sufficient condition for the commutator between a singular integral operator and a pointwise multiplier by a function f to belong to the Schatten class S^p. This condition is that f must belong to an appropriate Besov space if p > n, or that f must be constant if 0<p≤n.