Lp;r spaces: Cauchy Singular Integral, Hardy Classes and Riemann-Hilbert Problem in this Framework

In the present work the space Lp;r which is continuously embedded into Lp is introduced. The corresponding Hardy spaces of analytic functions are defined as well. Some properties of the functions from these spaces are studied. The analogs of some results in the classical theory of Hardy spaces are proved for the new spaces. It is shown that the Cauchy singular integral operator is bounded in Lp;r. The problem of basisness of the system {A(t)e^int;B(t)e^−int}n∈Z+, is also considered. It is shown that under an additional condition this system forms a basis in Lp;r if and only if the Riemann-Hilbert problem has a unique solution in corresponding Hardy class H^+p;r×H^+p;r.