On m-order logarithmic Laplacians and related propeties

In this article, we study $m$-order logarithmic Laplacian $\mathcal{L}_m$, which is a singular integro-differential operator with symbol $\big(2\ln |\cdot|\big)^m$ by the Fourier transform. With help of these logarithmic Laplacians, we build the $n$-th order Taylor expansion for fractional Laplacian with respect to the order and the Riesz operators: for $u \in C^\infty_c(\mathbb{R}^N)$ and $x \in \mathbb{R}^N$, $$ (-\Delta)^s u (x) = u(x) + \sum^{n}_{m=1} \frac{s^m}{m!}\mathcal{L}_mu(x) + o(s^n) \quad {\rm as}\ \, s\to 0^+ $$ and $$ \big(\Phi_s\ast u\big)(x) = u(x) + \sum^{n}_{m=1}(-1)^m\frac{s^m}{m!}\mathcal{L}_mu(x) + o(s^n) \quad {\rm as}\ \, s\to 0^+, $$ where $ (-\Delta)^s$ is the $s$-fractional Laplacian, $\Phi_s\ast u$ is $s$-order of Riesz operator with the form $\Phi_s(x)=\kappa_{N,s}|x|^{2s-N}$ in $\mathbb{R}^N\setminus\{0\}$. Moreover, we analyze qualitative properties of these operators based on the order $m$, such as basic regularity and the Dirichlet eigenvalues.