Potential Theory of the Fractional-Logarithmic Laplacian: Global Regularity and Critical Compact Embeddings
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866917315378937856 |
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| author | Chen, Rui |
| author_facet | Chen, Rui |
| contents | We develop a potential-theoretic and functional framework for the fractional--logarithmic Laplacian $(-Δ)^{s+\ln}$ and its inhomogeneous counterpart $(λI-Δ)^{s+\ln}$ with $λ>1$. Their inverses yield logarithmic analogues of the classical Riesz and Bessel potentials. We introduce the logarithmic Bessel kernel $K_{s+\ln}^λ$, derive explicit representations (including a Gamma-mixture formula and a Fourier--Bessel representation), and compute sharp pointwise asymptotics as $|x|\to0$ and $|x|\to\infty$, with explicit leading constants; in particular, the far-field profile and its prefactor are independent of $s$. We also establish a measure-level bridge between the homogeneous and inhomogeneous symbols, which yields $L^p$ regularity for global solutions of $(λI-Δ)^{s+\ln}u=f$ and $(-Δ)^{s+\ln}u=f$ and motivates the logarithmic Bessel spaces $\mathcal L^{p}_{s+\ln,λ}$. We relate these spaces to the classical Bessel scale and prove their equivalence to the logarithmic Bessel potential spaces of Opic and Trebels. As applications, we obtain endpoint embeddings into H"older-type spaces at the critical line $n=2sp$ and a compactness theory exhibiting a strict logarithmic gain, including compactness at the critical Sobolev exponent $p^*=\frac{np}{n-2sp}$ in the subcritical regime. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_04879 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Potential Theory of the Fractional-Logarithmic Laplacian: Global Regularity and Critical Compact Embeddings Chen, Rui Analysis of PDEs 35S05, 46E35, 42B20, 35B65, 46E30 We develop a potential-theoretic and functional framework for the fractional--logarithmic Laplacian $(-Δ)^{s+\ln}$ and its inhomogeneous counterpart $(λI-Δ)^{s+\ln}$ with $λ>1$. Their inverses yield logarithmic analogues of the classical Riesz and Bessel potentials. We introduce the logarithmic Bessel kernel $K_{s+\ln}^λ$, derive explicit representations (including a Gamma-mixture formula and a Fourier--Bessel representation), and compute sharp pointwise asymptotics as $|x|\to0$ and $|x|\to\infty$, with explicit leading constants; in particular, the far-field profile and its prefactor are independent of $s$. We also establish a measure-level bridge between the homogeneous and inhomogeneous symbols, which yields $L^p$ regularity for global solutions of $(λI-Δ)^{s+\ln}u=f$ and $(-Δ)^{s+\ln}u=f$ and motivates the logarithmic Bessel spaces $\mathcal L^{p}_{s+\ln,λ}$. We relate these spaces to the classical Bessel scale and prove their equivalence to the logarithmic Bessel potential spaces of Opic and Trebels. As applications, we obtain endpoint embeddings into H"older-type spaces at the critical line $n=2sp$ and a compactness theory exhibiting a strict logarithmic gain, including compactness at the critical Sobolev exponent $p^*=\frac{np}{n-2sp}$ in the subcritical regime. |
| title | Potential Theory of the Fractional-Logarithmic Laplacian: Global Regularity and Critical Compact Embeddings |
| topic | Analysis of PDEs 35S05, 46E35, 42B20, 35B65, 46E30 |
| url | https://arxiv.org/abs/2603.04879 |