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

Abstract

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.

Theorems & Definitions (52)

• Proposition 1.1
• Proposition 1.2
• Proposition 1.3
• Proposition 1.4
• Definition 1.1
• Proposition 1.5
• Proposition 1.6
• Theorem 1.1
• Theorem 1.2
• Proposition 1.7