$L^p$ isomorphisms for the fractional Laplacian on $\mathbb{R}^n$ and their application

Abstract

A classical theory of Amrouche, Girault and Giroire (1994) resolves the Laplace equation on $\mathbb{R}^n$ through an isomorphism between weighted Sobolev spaces. Inspired by their framework, we develop the corresponding $L^p$ theory for the fractional Laplacian: we introduce weighted fractional Sobolev spaces $Λ^{s,p}(\mathbb{R}^n)$, gauged at top order by $\|(-\triangle)^{\frac{s}{2}}u\|_{L^p}$, and prove that $(-\triangle)^{\frac{s}{2}}:Λ^{s,p}(\mathbb{R}^n)/\mathcal P_{[s-n/p]}\to L^p(\mathbb{R}^n)$ is an isomorphism for all $s\in(0,2)$ and $p\in(1,\infty)$; thus $(-\triangle)^{\frac{s}{2}}u=f$ is solvable for every $f\in L^p(\mathbb{R}^n)$, uniquely modulo an explicit finite-dimensional space of polynomials. More generally, extending the scale by duality to negative orders, the fractional Laplacians of the appropriate orders map its spaces isomorphically onto one another and compose exactly. A key ingredient is a family of weighted Hardy and Poincaré inequalities adapted to $\|(-\triangle)^{\frac{s}{2}}u\|_{L^p}$, established here for $1\le s<2$ by a reduction to the gradient. Since these inequalities hold a priori only on a dense class, while the density of $C_{\mathrm{c}}^\infty(\mathbb{R}^n)$ in $Λ^{s,p}(\mathbb{R}^n)$ is itself nontrivial, we first prove the isomorphism on the closure of the Schwartz class, which coincides with $Λ^{s,p}(\mathbb{R}^n)$ since the difference of two solutions is an entire $s$-harmonic function, hence a polynomial; the density follows as a by-product. As an application, we obtain existence and uniqueness, with explicit kernels and compatibility conditions, for the fractional Stokes system in $\mathbb{R}^n$.