Fractional Operators in Generalized Hölder Spaces

This monograph is devoted to the study of the mapping properties of mixed Riemann-Liouville fractional integrals and integral operators of the Volterra convolution type in generalized Hölder spaces of functions of several variables. The study examines issues of boundedness, continuity, and isomorphism of these operators in classical, weighted, and generalized Hölder spaces defined by mixed continuity modules.Basic information on Hölder spaces, fractional integral and differential operators is provided, along with an overview of recent results in fractional analysis. Conditions for boundedness and theorems on the mapping properties of mixed Riemann-Liouville fractional integrals in Hölder spaces of functions of several variables are obtained.For integral operators of the Volterra convolution type, estimates for the mixed continuity moduli of the images of functions are established, theorems of the Sigmund type are proven, and results characterizing the change in the smoothness properties of functions under the action of operators are obtained. The results obtained can be used in the study of integral and fractional differential equations, as well as in problems of functional analysis and mathematical physics.