The Fractional-Logarithmic Laplacian:Fundamental Properties and Eigenvalues
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| Formato: | Preprint |
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2026
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| author | Chen, Huyuan Chen, Rui Hauer, Daniel |
| author_facet | Chen, Huyuan Chen, Rui Hauer, Daniel |
| contents | In this paper, we introduce, for the first time, the fractional--logarithmic Laplacian \( (-Δ)^{s+\log} \), defined as the derivative of the fractional Laplacian \( (-Δ)^t \) at \( t=s \). It is a singular integral operator with Fourier symbol \( |ξ|^{2s}(2\ln|ξ|) \), and we prove the pointwise integral representation \[ (-Δ)^{s+\log}u(x) = c_{n,s}\,\mathrm{PV}\!\int_{\mathbb{R}^n} \frac{u(x)-u(y)}{|x-y|^{n+2s}}\bigl(-2\ln|x-y|\bigr)\,dy + b_{n,s}(-Δ)^s u(x), \] where \( c_{n,s} \) is the normalization constant of the fractional Laplacian and \( b_{n,s}:=\frac{d}{ds}c_{n,s}.\) We also establish several equivalent formulations of \( (-Δ)^{s+\log} \), including the singular-integral representation, the Fourier-multiplier representation, the spectral-calculus definition, and an extension characterization.
We develop the associated functional framework on both \( \mathbb{R}^n \) and bounded Lipschitz domains, introducing the natural energy spaces and proving embedding results. In particular, we obtain a compact embedding at the critical exponent \( 2_s^*=\frac{2n}{n-2s},\) a phenomenon that differs from the classical Sobolev and fractional Sobolev settings. We further study the Poisson problem, proving existence and \( L^\infty \)-regularity results. We then investigate the Dirichlet eigenvalue problem and establish qualitative spectral properties. Finally, we derive a Weyl-type asymptotic law for the eigenvalue counting function and for the \( k \)-th Dirichlet eigenvalue, showing that the high-frequency behavior combines the fractional Weyl scaling with a logarithmic growth factor, thereby interpolating between the fractional Laplacian and the logarithmic Laplacian. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_06581 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Fractional-Logarithmic Laplacian:Fundamental Properties and Eigenvalues Chen, Huyuan Chen, Rui Hauer, Daniel Analysis of PDEs 35B65, 47G20, 35A01, 35P20, 35R11 In this paper, we introduce, for the first time, the fractional--logarithmic Laplacian \( (-Δ)^{s+\log} \), defined as the derivative of the fractional Laplacian \( (-Δ)^t \) at \( t=s \). It is a singular integral operator with Fourier symbol \( |ξ|^{2s}(2\ln|ξ|) \), and we prove the pointwise integral representation \[ (-Δ)^{s+\log}u(x) = c_{n,s}\,\mathrm{PV}\!\int_{\mathbb{R}^n} \frac{u(x)-u(y)}{|x-y|^{n+2s}}\bigl(-2\ln|x-y|\bigr)\,dy + b_{n,s}(-Δ)^s u(x), \] where \( c_{n,s} \) is the normalization constant of the fractional Laplacian and \( b_{n,s}:=\frac{d}{ds}c_{n,s}.\) We also establish several equivalent formulations of \( (-Δ)^{s+\log} \), including the singular-integral representation, the Fourier-multiplier representation, the spectral-calculus definition, and an extension characterization. We develop the associated functional framework on both \( \mathbb{R}^n \) and bounded Lipschitz domains, introducing the natural energy spaces and proving embedding results. In particular, we obtain a compact embedding at the critical exponent \( 2_s^*=\frac{2n}{n-2s},\) a phenomenon that differs from the classical Sobolev and fractional Sobolev settings. We further study the Poisson problem, proving existence and \( L^\infty \)-regularity results. We then investigate the Dirichlet eigenvalue problem and establish qualitative spectral properties. Finally, we derive a Weyl-type asymptotic law for the eigenvalue counting function and for the \( k \)-th Dirichlet eigenvalue, showing that the high-frequency behavior combines the fractional Weyl scaling with a logarithmic growth factor, thereby interpolating between the fractional Laplacian and the logarithmic Laplacian. |
| title | The Fractional-Logarithmic Laplacian:Fundamental Properties and Eigenvalues |
| topic | Analysis of PDEs 35B65, 47G20, 35A01, 35P20, 35R11 |
| url | https://arxiv.org/abs/2602.06581 |