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 $(2\ln \mathrm{\mid }\cdot \mathrm{\mid }{)}^m$ 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 $u\in C_c^{\mathrm{\infty }}({\mathbb{R}}^N)$ and $x\in {\mathbb{R}}^N$, $$(-\mathrm{\Delta }{)}^{s}u(x)=u(x)+\sum _{m=1}^n\frac{s^m}{m!}{\mathcal{L}}_{m}u(x)+o(s^n)\mathrm{a}\mathrm{s}s\to 0^{+}$$ and $$({\mathrm{\Phi }}_s\ast u)(x)=u(x)+\sum _{m=1}^n(-1{)}^m\frac{s^m}{m!}{\mathcal{L}}_{m}u(x)+o(s^n)\mathrm{a}\mathrm{s}s\to 0^{+},$$ where $(-\mathrm{\Delta }{)}^s$ is the $s$-fractional Laplacian, ${\mathrm{\Phi }}_s\ast u$ is $s$-order of Riesz operator with the form ${\mathrm{\Phi }}_s(x)={\kappa }_{N,s}\mathrm{\mid }x{\mathrm{\mid }}^{2s-N}$ 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.