Potential Theory of the Fractional-Logarithmic Laplacian: Global Regularity and Critical Compact Embeddings

TL;DR

This paper develops a potential-theoretic framework for the fractional-logarithmic Laplacian, deriving explicit kernels, asymptotics, and regularity results, and establishing new embeddings and compactness properties at critical exponents.

Contribution

It introduces the logarithmic Bessel potential spaces, explicit representations of the logarithmic Bessel kernel, and establishes regularity and compactness results for the fractional-logarithmic Laplacian.

Findings

Explicit formulas for the logarithmic Bessel kernel.

Sharp asymptotics of the kernel at zero and infinity.

New compactness results with logarithmic gain at critical exponents.

Abstract

We develop a potential-theoretic and functional framework for the fractional--logarithmic Laplacian $(-\Delta)^{s+\ln}$ and its inhomogeneous counterpart $(\lambda I-\Delta)^{s+\ln}$ with $\lambda>1$. Their inverses yield logarithmic analogues of the classical Riesz and Bessel potentials. We introduce the logarithmic Bessel kernel $K_{s+\ln}^{\lambda}$, 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 $(\lambda I-\Delta)^{s+\ln}u=f$ and $(-\Delta)^{s+\ln}u=f$ and motivates the logarithmic Bessel spaces $\mathcal L^{p}_{s+\ln,\lambda}$. 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.