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

Abstract

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 $(\hat{I}»I-\hat{I}”{)}^{s+\ln }u=f$ and $(-\hat{I}”{)}^{s+\ln }u=f,$ 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 $p^{\ast }=\frac{np}{n-2sp},$ recovering compactness at the borderline Lebesgue threshold, a phenomenon absent from the classical Sobolev and Bessel scales.