$L_{p;r} $ spaces: Cauchy Singular Integral, Hardy Classes and Riemann-Hilbert Problem in this Framework

In the present work the space  $L_{p;r} $ which is continuously embedded into $L_{p} $  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 $L_{p;r} $. The problem of basisness of the system  $left{Aleft(tright)e^{{mathop{rm int}} }; Bleft(tright)e^{-{mathop{rm int}} } right}_{nin Z_{+} }, $  is also considered. It is shown that under an additional condition this system forms a basis in $L_{p;r} $  if and only if the Riemann-Hilbert problem has a unique solution in corresponding Hardy class ${  H}_{p;r}^{+} times {  H}_{p;r}^{+} $.