Fractional $p$-Laplacians via Neumann problems in unbounded metric measure spaces

We prove well-posedness, Harnack inequality and sharp regularity of solutions to a fractional $p$-Laplace non-homogeneous equation $(-Δ_p)^su =f$, with $0<s<1$, $1<p<\infty$, for data $f$ satisfying a weighted $L^{p’}$ condition in a doubling metric measure space $(Z,d_Z,ν)$ that is possibly unbounded. Our approach is inspired by the work of Caffarelli and Silvestre \cite{CS} (see also Mol{č}anov and Ostrovski{ĭ} \cite{MO}), and extends the techniques developed in \cite{CKKSS}, where the bounded case is studied. Unlike in \cite{EbGKSS}, we do not assume that $Z$ supports a Poincaré inequality. The proof is based on the well-posedness of the Neumann problem on a Gromov hyperbolic space $(X,d_X, μ)$ that arises as an hyperbolic filling of $Z$.

💡 Research Summary

This paper establishes a comprehensive analytical framework for the fractional (p)-Laplacian ((-\Delta_{p})^{s}) (with (0<s<1) and (1<p<\infty)) on possibly unbounded doubling metric measure spaces ((Z,d_{Z},\nu)). The authors extend the Caffarelli‑Silvestre extension method, previously applied mainly in Euclidean or bounded settings, to the general metric context without assuming a Poincaré inequality on (Z).

The central construction views (Z) as the boundary of a uniform domain (\Omega) whose interior ((\Omega,d,\mu)) is a hyperbolic filling of (Z). Under three structural hypotheses—(H0) (\Omega) is a uniform domain, (H1) ((\Omega,d,\mu)) is doubling and supports a (p)-Poincaré inequality, and (H2) the boundary (\partial\Omega) is complete, uniformly perfect, doubling, and equipped with a co‑dimensional Hausdorff measure (\nu) comparable to (\mu)—the authors define the homogeneous Sobolev space (D^{1,p}(\Omega)) and the homogeneous Besov space (HB^{\theta}_{p,p}(\partial\Omega)) with (\theta=1-\Theta/p).

The Neumann problem for the Cheeger (p)-Laplacian is posed with data (f) belonging to a novel weighted space (L^{p’}(\partial\Omega,\nu_{J})\cap L^{p’}(\partial\Omega,\nu)) where (\nu_{J}) incorporates a kernel (J(x,x_{0})) depending on a reference point (x_{0}\in\partial\Omega). The problem seeks (u\in D^{1,p}(\Omega)) satisfying
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