General Fractional Integrals and Derivatives with the Sonine Kernels

In this paper, we address the general fractional integrals and derivatives with the Sonine kernels on the spaces of functions with an integrable singularity at the point zero. First, the Sonine kernels and their important special classes and particular cases are discussed. In particular, we introduce a class of the Sonine kernels that possess an integrable singularity of power function type at the point zero. For the general fractional integrals and derivatives with the Sonine kernels from this class, two fundamental theorems of fractional calculus are proved. Then, we construct the n-fold general fractional integrals and derivatives and study their properties.

Automated summary

The paper discusses the general fractional integrals and derivatives with Sonine kernels on function spaces with integrable singularities at the point zero, introducing a special class of Sonine kernels with integrable singularities of power function type at zero. Two fundamental theorems of fractional calculus are proved, n-fold general fractional integrals and derivatives are constructed, and their properties are studied.