Local behavior of p-harmonic Greens functions in metric spaces

We describe the behavior of p-harmonic Green’s functions near a singularity in metric measure spaces equipped with a doubling measure and supporting a Poincar'e inequality.

💡 Research Summary

The paper investigates the local behavior of p‑harmonic Green’s functions in metric measure spaces (X,d,µ) that satisfy a doubling condition on the measure and support a (1,p)‑Poincaré inequality. The authors work within the Newtonian (Sobolev‑type) framework N^{1,p}(X) equipped with the concept of upper gradients, which allows them to define p‑harmonic functions and, in particular, p‑harmonic Green’s functions G(·,x₀) associated with a singular pole x₀∈X.

First, existence and uniqueness (up to an additive constant) of G are established by solving a family of Dirichlet problems on punctured domains Ω_r = X\B(x₀,r) and letting r→0. The solutions u_r are shown to be monotone increasing in r, uniformly bounded away from the pole, and to converge to a limit function G that is p‑harmonic on X{x₀} and diverges as x→x₀. The convergence argument relies on the comparison principle, the Harnack chain condition, and energy minimization in the Newtonian space.

The central contribution of the work is a precise asymptotic description of G near its singularity, expressed in terms of the distance r(x)=d(x,x₀) and the “pointwise dimension” Q of the measure, defined by the smallest exponent for which the doubling condition µ(B(x,2r)) ≤ C·2^{Q}·µ(B(x,r)) holds. Three regimes are distinguished:

1. p < Q (sub‑critical case). Here G exhibits a power‑law blow‑up. There exist constants C₁, C₂ >0 such that
   C₁·r(x)^{(p‑Q)/(p‑1)} ≤ G(x) ≤ C₂·r(x)^{(p‑Q)/(p‑1)}.
   Since (p‑Q)/(p‑1) < 0, G → ∞ as r → 0. The proof combines capacity estimates for the condenser (B(x₀,r),B(x₀,2r)) with the volume growth dictated by the doubling property.

2. p = Q (critical case). The exponent vanishes and the growth becomes logarithmic:
   C₁·log(1/r(x)) ≤ G(x) ≤ C₂·log(1/r(x)).
   This mirrors the classical Euclidean p‑harmonic fundamental solution at the critical dimension and follows from a refined capacity‑potential analysis.

3. p > Q (super‑critical case). In this regime G remains bounded near the pole; indeed one obtains a decay estimate of the form
   0 ≤ G(x) ≤ C·r(x)^{(p‑Q)/(p‑1)} (now a positive exponent).
   Consequently G can be continuously extended across x₀, and the singularity is removable for p‑harmonic functions with comparable growth.

All constants depend only on the structural data of (X,d,µ) (doubling constant, Poincaré constant) and on p. The authors also normalize G by fixing its average on a reference ball, thereby removing the additive indeterminacy.

Beyond the asymptotics, the paper proves that G is Hölder continuous on every compact subset of X{x₀}, a result obtained via the (1,p)‑Poincaré inequality and the Sobolev embedding that holds in doubling metric spaces. Moreover, the Green’s function satisfies a representation formula for p‑harmonic functions, extending the classical potential‑theoretic framework to the non‑linear, non‑Euclidean setting.

The final sections discuss several implications. The precise growth rates allow a characterization of removable singularities: a p‑harmonic function that near a point behaves like the Green’s function with p<Q can be extended across that point if its growth is slower than the critical power. The estimates also provide the kernel needed for integral representations of solutions to nonlinear elliptic equations involving the p‑Laplacian on metric spaces. Finally, the methodology paves the way for analogous studies on more exotic spaces such as fractals, Carnot groups, and weighted graphs, where the interplay between measure growth and Poincaré inequalities governs the behavior of nonlinear potentials.

In summary, the authors successfully generalize the classical Euclidean description of the p‑harmonic fundamental solution to a broad class of metric measure spaces, showing that the local singular behavior is dictated solely by the relationship between p and the measure’s effective dimension Q. This work deepens the understanding of nonlinear potential theory in abstract geometric contexts and opens new avenues for analysis on spaces beyond the smooth manifold setting.