Elliptic Harnack inequality and Poincaré inequality for $p$-energies on metric measure spaces

Technical Summary: Elliptic Harnack Inequality and Poincaré Inequality for p-energies on Metric Measure Spaces

Problem Statement
The paper addresses the structural relationship between functional inequalities in the context of nonlinear $p$-energies on metric measure spaces $(X, d, m)$. While the equivalence between the elliptic Harnack inequality (EHI), two-sided capacity bounds, the Poincaré inequality (PI), and the cutoff Sobolev inequality (CS) is well-established for linear Dirichlet forms ($p=2$), the nonlinear case ($p > 1$) lacks a linear semi-group or heat kernel, making such equivalences more difficult to prove.

Specifically, the author seeks to complete the nonlinear analogue of the classical equivalence:
$$\text{EHI} + \text{cap}(\Psi) \iff \text{PI}(\Psi) + \text{CS}(\Psi)$$
where $\Psi$ is a suitable scaling function. Previous work by the author [28] had already established the implications $\text{EHI} + \text{cap}(\Psi) \Rightarrow \text{CS}(\Psi)$ and $\text{PI}(\Psi) + \text{CS}(\Psi) \Rightarrow \text{EHI} + \text{cap}(\Psi)$. The remaining open step, which this paper resolves, is to derive the Poincaré inequality $\text{PI}(\Psi)$ from the elliptic Harnack inequality and two-sided capacity bounds.

Methodology
The proof strategy combines techniques from the author's previous work [28] with recent methods introduced in [11] regarding parabolic Harnack inequalities on spaces with upper gradient structures. The argument proceeds in two main steps:

1. Reduction to Faber–Krahn Inequality:
   Following the strategy in [11], the author reduces the proof of $\text{PI}(\Psi)$ to establishing a Faber–Krahn inequality, denoted $\text{FK}(\Psi)$. This inequality provides a lower bound for the first eigenvalue of the $p$-Laplacian on a domain in terms of the domain's measure and the scaling function $\Psi$.
   The author proves that $\text{PI}(\Psi)$ follows if one can establish an estimate of the form:
   $$\left( \fint_{B(x_0, R)} |f - c|^p dm \right)^{1/p} \lesssim \left( \frac{\Psi(R)}{V(x_0, R)} \int_{B(x_0, AR)} d\Gamma(f) \right)^{1/p} + \epsilon \fint_{B(x_0, R)} |f| dm$$
   for a suitable constant $\epsilon$. This is achieved by decomposing the function $f$ into a harmonic part and a remainder, utilizing the oscillation estimates derived from EHI.

2. Establishing Faber–Krahn via Wolff Potentials:
   To prove $\text{FK}(\Psi)$, the author employs a standard Maz'ya decomposition technique, which reduces the problem to estimating the measure of a set $K$ in terms of its relative capacity:
   $$m(K) \lesssim \Psi(R) \, \text{cap}(K, X \setminus B(x_0, AR)).$$
   This estimate is derived using Wolff potential estimates established in [28]. The proof involves:

   • Constructing a capacitary potential $u$ associated with the set $K$.
   • Utilizing the Riesz measure $\mu$ associated with the superharmonic function $u$.
   • Applying Wolff potential estimates to bound $u(x_0)$ from below, linking the potential to the measure $\mu$ and the capacity.
   • Using the volume doubling (VD) and capacity bounds to control the Wolff potential series.
   • Applying the 5B-covering lemma to sum the measures of disjoint balls covering $K$, thereby bounding $m(K)$ by the capacity.

Key Assumptions
The results hold under the following structural assumptions on the metric measure space $(X, d, m)$ and the $p$-energy $(\mathcal{E}, \mathcal{F})$:

• Geometric: Volume Doubling (VD) and the Chain Condition (CC).
• Analytic:
  • EHI: The Elliptic Harnack Inequality holds.
  • cap($\Psi$): Two-sided capacity bounds hold.
  • BSP: Bottom Spectrum Positivity (positivity of the first eigenvalue on bounded domains).
  • LSC: Lower Semi-Continuity condition for superharmonic functions.
  • Additional Condition: Either the Linearly Locally Connected (LLC) condition holds, or the Annulus Elliptic Harnack Inequality (EHI$_{\text{ann}}$) holds.

Main Results

1. Theorem 1.1: Under the assumptions of VD, LSC, BSP, EHI, and cap($\Psi$), if either LLC or EHI$_{\text{ann}}$ holds, then the Poincaré inequality $\text{PI}(\Psi)$ holds.
2. Corollaries 1.3 and 1.4: By combining Theorem 1.1 with previous results from [28], the paper establishes the full equivalence:
   $$\text{EHI} + \text{cap}(\Psi) \iff \text{PI}(\Psi) + \text{CS}(\Psi)$$
   This equivalence holds under the same structural assumptions (VD, LSC, BSP, and either LLC or EHI$_{\text{ann}}$).

Significance and Claims
The paper claims to provide the "remaining step" toward the full nonlinear analogue of the classical equivalence known in the linear setting ($p=2$).

• Structural Unification: The result demonstrates that the nonlinear theory of $p$-energies possesses the same structural form as the classical Dirichlet form theory at the elliptic level. Specifically, the "Harnack–capacity side" and the "Poincaré–cutoff side" are equivalent.
• Robustness: In the absence of heat kernels (which generally do not exist for nonlinear $p$-energies), this equivalence provides a robust substitute for the classical equivalences known in the linear setting.
• Technical Contribution: The work adapts arguments from [11] (originally for parabolic Harnack inequalities on spaces with upper gradients) to the more general setting of metric measure spaces, including fractals, by establishing the necessary Faber–Krahn inequality via Wolff potential estimates.

The author notes that for the specific case $p=2$, the implication $\text{EHI} + \text{cap}(\Psi) \Rightarrow \text{PI}(\Psi)$ is already known under VD alone. However, for general $p > 1$, the additional assumptions of LSC, BSP, and either LLC or EHI$_{\text{ann}}$ are required due to the nonlinearity and the limitations of the current techniques.