On m-order logarithmic Laplacians and the applications

In this paper, we study [Formula: see text]-order logarithmic Laplacian [Formula: see text], which is a singular integro-differential operator with symbol [Formula: see text] by the Fourier transform. With help of these logarithmic Laplacians, we build the [Formula: see text]th order Taylor expansion for fractional Laplacian with respect to the order and the Riesz operators: for [Formula: see text] and [Formula: see text], [Formula: see text] and [Formula: see text] where [Formula: see text] is the [Formula: see text]-fractional Laplacian, [Formula: see text] is [Formula: see text]-order of Riesz operator with the form [Formula: see text] in [Formula: see text]. Moreover, we analyze qualitative properties of these operators based on the order [Formula: see text], such as basic regularity and the Dirichlet eigenvalues.