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arXiv 2603.04879math.AP
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分数-对数拉普拉斯算子：势、正则性及临界紧嵌入

The Fractional-Logarithmic Laplacian: Potentials, Regularity, and Critical Compact Embeddings

Rui Chen

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AI总结：

本文研究分数-对数拉普拉斯算子的势理论和正则性，探讨其临界紧嵌入性质，提出新的对数Bessel空间，并讨论其与经典Bessel空间的关系。

摘要：

我们为分数-对数拉普拉斯算子$(-Δ)^{s+\ln}$及其非齐次形式$(λI-Δ)^{s+\ln}$$(λ>1)开发了势理论和$L^p$正则性结果。这些算子产生经典Riesz和Bessel势的对数类比。对于相关的对数Bessel核$K_{s+\ln}^λ$，我们获得了表示公式和在原点和无穷远处的尖锐点wise渐进行为，包括显式的主要常数。关键成分是一个在测度层面连接齐次和非齐次符号的桥梁。这使得我们能够在方程$(λI-Δ)^{s+\ln}u=f$和$(-Δ)^{s+\ln}u=f$之间转换，并获得全局$L^p$估计、分布解的适定性，以及自然的对数Bessel空间$\mathcal L^p_{s+\ln,λ}$的尺度。我们还讨论了这些空间对$λ$的依赖性，以及它们与经典Bessel空间和Opic和Trebels引入的对数Bessel势空间的关系。作为应用，我们证明了端点嵌入和临界紧性结果。在临界线$n=2sp$上，我们获得具有对数模连续性的嵌入，得到在有界域上的局部紧性和在径向类中的全局紧性。在亚临界情况$n>2sp$下，我们证明在临界Sobolev指数$p^*=\frac{np}{n-2sp}$处的紧性，恢复临界Lebesgue阈值的紧性，这一现象在经典Sobolev和Bessel尺度中不存在。

英文摘要：

We develop potential-theoretic and \(L^p\)-regularity results for the fractional--logarithmic Laplacian \((-Δ)^{s+\ln}\) and its inhomogeneous counterpart \((λI-Δ)^{s+\ln}\), \(λ>1\). These operators lead to logarithmic analogues of the classical Riesz and Bessel potentials. For the associated logarithmic Bessel kernel \(K_{s+\ln}^λ\), we obtain representation formulas and sharp pointwise asymptotics at both the origin and infinity, including explicit leading constants. A key ingredient is a measure-level bridge between the homogeneous and inhomogeneous symbols. This allows us to pass between the equations $(λI-Δ)^{s+\ln}u=f$ and $(-Δ)^{s+\ln}u=f,$ and yields global \(L^p\) estimates, well-posedness for distributional solutions, and a natural scale of logarithmic Bessel spaces \(\mathcal L^p_{s+\ln,λ}\). We also discuss the dependence of these spaces on \(λ\), their relation to the classical Bessel spaces and with the logarithmic Bessel potential spaces introduced by Opic and Trebels. As applications, we prove endpoint embeddings and critical compactness results. On the critical line \(n=2sp\), we obtain embeddings with a logarithmic modulus of continuity, local compactness on bounded domains, and global compactness in the radial class. In the subcritical case \(n>2sp\), we prove compactness at the critical Sobolev exponent $p^*=\frac{np}{n-2sp},$ recovering compactness at the borderline Lebesgue threshold, a phenomenon absent from the classical Sobolev and Bessel scales.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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