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| Format: | Preprint |
| Veröffentlicht: |
2023
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| Online-Zugang: | https://arxiv.org/abs/2307.06198 |
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| _version_ | 1866913451083825152 |
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| author | Chen, Huyuan |
| author_facet | Chen, Huyuan |
| contents | 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 $x \in \mathbb{R}^N$, $$
(-Δ)^s u (x) = u(x) + \sum^{n}_{m=1} \frac{s^m}{m!}\mathcal{L}_mu(x) + o(s^n) \quad {\rm as}\ \, s\to 0^+ $$ and $$ \big(Φ_s\ast u\big)(x) = u(x) + \sum^{n}_{m=1}(-1)^m\frac{s^m}{m!}\mathcal{L}_mu(x) + o(s^n) \quad {\rm as}\ \, s\to 0^+, $$ where $ (-Δ)^s$ is the $s$-fractional Laplacian, $Φ_s\ast u$ is $s$-order of Riesz operator with the form $Φ_s(x)=κ_{N,s}|x|^{2s-N}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_06198 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On m-order logarithmic Laplacians and related propeties Chen, Huyuan Analysis of PDEs 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 $x \in \mathbb{R}^N$, $$ (-Δ)^s u (x) = u(x) + \sum^{n}_{m=1} \frac{s^m}{m!}\mathcal{L}_mu(x) + o(s^n) \quad {\rm as}\ \, s\to 0^+ $$ and $$ \big(Φ_s\ast u\big)(x) = u(x) + \sum^{n}_{m=1}(-1)^m\frac{s^m}{m!}\mathcal{L}_mu(x) + o(s^n) \quad {\rm as}\ \, s\to 0^+, $$ where $ (-Δ)^s$ is the $s$-fractional Laplacian, $Φ_s\ast u$ is $s$-order of Riesz operator with the form $Φ_s(x)=κ_{N,s}|x|^{2s-N}$ 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. |
| title | On m-order logarithmic Laplacians and related propeties |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2307.06198 |