On the fractional Laplacian of a function with respect to another function

Abstract

The theories of fractional Laplacians and of fractional calculus with respect to functions are combined to produce, for the first time, the concept of a fractional Laplacian with respect to a bijective function. The theory is developed both in the 1-dimensional setting and in the general (Formula presented.) -dimensional setting. Fourier transforms with respect to functions are also defined, and the relationships between Fourier transforms, fractional Laplacians, and Marchaud-type derivatives are explored. Function spaces for these operators are carefully defined, including weighted (Formula presented.) spaces and a new type of Schwartz space. The theory developed is then applied to construct solutions to some partial differential equations involving both fractional time derivatives and fractional Laplacians with respect to functions, with illustrative examples.