Difference between Riesz derivative and fractional Laplacian on the proper subset of ℝ

In general, the Riesz derivative and the fractional Laplacian are equivalent on ℝ. But they generally are not equivalent with each other on any proper subset of ℝ. In this paper, we focus on the difference between them on the proper subset of ℝ.

On the Generalized Riesz Derivative

The goal of this paper is to construct an integral representation for the generalized Riesz derivative R Z D x 2 s u ( x ) for k < s < k + 1 with k = 0 , 1 , ⋯ , which is proved to be a one-to-one and linearly continuous mapping from the normed space W k + 1 ( R ) to the Banach space C ( R ) . In addition, we show that R Z D x 2 s u ( x ) is continuous at the end points and well defined for s = 1 2 + k . Furthermore, we extend the generalized Riesz derivative R Z D x 2 s u ( x ) to the space C k ( R n ) , where k is an n-tuple of nonnegative integers, based on the normalization of distribution and surface integrals over the unit sphere. Finally, several examples are presented to demonstrate computations for obtaining the generalized Riesz derivatives.

On Riesz derivative

This paper focuses on studying Riesz derivative. An interesting investigation on properties of Riesz derivative in one dimension indicates that it is distinct from other fractional derivatives such as Riemann-Liouville derivative and Caputo derivative. In the existing literatures, Riesz derivative is commonly considered as a proxy for fractional Laplacian on ℝ. We show the equivalence between Riesz derivative and fractional Laplacian on ℝn with n ≥ 1 in details.