A simplified characterization of stable-like heat kernel estimates

We study heat kernel estimates for symmetric pure jump processes on general metric measure spaces. Building on recent progress in the local setting due to S.~Eriksson-Bique, we develop a non-local version of the Whitney blending technique and use it to relate stable-like heat kernel estimates to capacity upper bounds. Under two-sided stable-like bounds on the jump kernel, we show that a capacity upper bound across annuli implies a cutoff Sobolev inequality, and we obtain a characterization of stable-like heat kernel estimates in terms of these conditions. As a consequence, we give an affirmative answer to a conjecture of A. Grigor’yan, E. Hu, and J. Hu.

💡 Research Summary

The paper addresses the long‑standing problem of characterising stable‑like heat kernel estimates for symmetric pure‑jump Markov processes on general metric measure spaces. Classical characterisations for diffusions require three analytic‑geometric conditions: volume doubling (VD), a Poincaré inequality, and a cut‑off Sobolev inequality (CSJ). While CSJ is a powerful tool, verifying it in concrete non‑local settings (e.g., graphs with polynomial volume growth, fractal spaces, or mixed diffusion‑jump processes) is notoriously difficult. Grigor’yan, Hu, and Hu conjectured that a much simpler condition—an upper bound on capacity across annuli—should be sufficient to replace CSJ in the characterisation of stable‑like heat kernel bounds.

The author builds on the recent “Whitney blending” technique introduced by Eriksson‑Bique for local Dirichlet forms and develops a non‑local analogue suitable for pure‑jump Dirichlet forms. The main technical innovation is a three‑step construction:

1. Cut‑off Sobolev‑type inequality for equilibrium potentials. Lemma 2.6 shows that when the cut‑off function is taken as the equilibrium potential of a compact set K relative to an open set Ω, a Sobolev‑type inequality holds for functions vanishing near K or near Ω.

2. Whitney decomposition and function splitting. For any f∈F, a point x₀ and radius r, the author builds two auxiliary functions f₁, f₂ using a Whitney cover of the annulus B(x₀,2r)∖B(x₀,r). f₁ vanishes near B(x₀,r) and f₂ near B(x₀,2r). Controlling the Dirichlet energies of f₁ and f₂ in terms of E(f,f) requires sharp two‑sided estimates on the jump kernel J and the use of truncated Dirichlet forms. This is the most delicate part (Lemma 2.9).

3. Self‑improvement to full CSJ. The inequality obtained in step 2 is a non‑local analogue of the simplified cut‑off Sobolev inequality introduced in earlier work. By a self‑improvement argument (following Murugan 2024), the author upgrades it to the full CSJ ϕ inequality (Proposition 2.10, §2.6).

With CSJ ϕ established under the assumptions of volume doubling, two‑sided stable‑like bounds on the jump kernel (denoted J ϕ ≤ and J ϕ ≥), and a capacity upper bound (Cap ϕ ≤), the author proves Theorem 1.6: these three conditions imply the cut‑off Sobolev inequality for the non‑local Dirichlet form.

The second main result, Theorem 1.7, combines Theorem 1.6 with known equivalences between CSJ ϕ and stable‑like heat kernel estimates (HK ϕ) from the literature (e.g., CKW21, GHH18). It shows that, on a metric measure space satisfying VD and the weaker quasi‑reverse volume doubling (QR‑VD), the following are equivalent:

• (a) The jump kernel satisfies the two‑sided bound J ϕ and the capacity upper bound Cap ϕ ≤.
• (b) The heat kernel of the associated Markov semigroup satisfies the two‑sided stable‑like estimates HK ϕ (i.e., the on‑diagonal and off‑diagonal bounds (1.5) and (1.6)).

Thus the conjecture of Grigor’yan, Hu, and Hu is resolved affirmatively: a simple capacity upper bound across annuli is sufficient to guarantee stable‑like heat kernel estimates, without needing to verify CSJ directly.

The paper also discusses several important implications:

• Simplified verification: In many concrete models (e.g., reflected diffusions, boundary traces, or processes on Laakso spaces), establishing capacity bounds is far easier than proving exit‑time estimates or CSJ. This opens a practical pathway for checking heat kernel estimates in new settings.
• Generality: The results hold under QR‑VD, which is satisfied by many graphs and fractal‑like spaces that fail the stronger reverse volume doubling (R‑VD). Hence the theory applies to a broader class of spaces than previous works.
• Methodological novelty: Extending Whitney blending to non‑local forms required new tools, such as sharp two‑sided kernel estimates and the use of truncated forms, which may be of independent interest for future studies of non‑local Dirichlet forms.
• Future directions: The author points out that analogous capacity‑based characterisations remain open for settings involving sub‑Gaussian estimates, mixed diffusion‑jump processes, and other variants (items (iii), (v), (vi) in the introduction). Extending the non‑local Whitney blending technique to those contexts is highlighted as a promising research avenue.

In summary, the paper provides a clean, robust characterisation of stable‑like heat kernel estimates for symmetric pure‑jump processes: the combination of two‑sided jump kernel bounds and a capacity upper bound across annuli is both necessary and sufficient. This not only settles a notable conjecture but also offers a more accessible toolkit for analysts working with non‑local stochastic processes on irregular spaces.