On m-order logarithmic Laplacians and related propeties

TL;DR

This paper introduces m-order logarithmic Laplacians, explores their properties, and uses them to develop Taylor expansions for fractional Laplacians and Riesz operators, revealing new insights into their regularity and eigenvalues.

Contribution

It constructs the m-order logarithmic Laplacian operators and applies them to derive Taylor expansions for fractional Laplacians and Riesz operators, advancing understanding of their properties.

Findings

Established Taylor expansions for fractional Laplacian and Riesz operators using logarithmic Laplacians.

Analyzed qualitative properties such as regularity and Dirichlet eigenvalues of these operators.

Provided new tools for studying fractional and nonlocal operators in analysis.

Abstract

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.