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Local and non-local $p$ -energies on metric measure spaces

This paper establishes that on metric measure spaces, a local regular $p$ -energy satisfying specific Poincaré and cutoff Sobolev inequalities can be subordinated to a non-local $p$ -energy with a jumping kernel that scales strictly faster at small scales, thereby extending the classical subordination principle to a nonlinear framework.

Original authors: Meng Yang

Published 2026-06-26

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Meng Yang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to understand how heat, or information, or even a random walker moves across a strange, crumpled landscape. In mathematics, this landscape is called a "metric measure space" (think of it as a weirdly shaped city where the streets aren't straight lines, and the "population density" of the city varies wildly).

This paper, by Meng Yang, is about understanding the rules that govern how things move on this landscape, specifically comparing two different ways of moving: Local and Non-Local.

The Two Types of Movement

Local Movement (The Cautious Walker):
Imagine a person walking through a city who can only take tiny steps. They can only move from their current spot to the immediate next spot. They can't jump. In math, this is like a "local energy." It depends entirely on what is happening right next to you. If you want to know how much effort this walker expends, you only look at their immediate neighbors.
Non-Local Movement (The Teleporting Bird):
Now, imagine a bird flying over the same city. This bird can fly from one side of the city to the other in a single hop. It doesn't care about the streets in between; it just jumps from point A to point Z. In math, this is a "non-local energy." It depends on the distance between two points, even if they are far apart. The "jumping kernel" is the rulebook that says how likely the bird is to jump a certain distance.

The Big Question: Can We Predict the Bird from the Walker?

The author asks a fundamental question: If we know the rules for the cautious walker (Local), can we predict the rules for the teleporting bird (Non-Local)?

Specifically, the paper looks at a scenario where the bird's jumping rules are "heavier" or "stronger" than the walker's walking rules at small distances. The author wants to know: If the walker follows a specific set of safety rules (called Poincaré and Cutoff Sobolev inequalities), does the bird automatically follow a similar set of safety rules?

The Main Discovery: The "Subordination" Principle

The paper proves a "Yes." It shows that if the local walker is well-behaved (satisfies certain mathematical inequalities that keep things from exploding or behaving chaotically), then the non-local bird will also be well-behaved, provided the bird's jumping rules are "strictly above" the walker's walking rules at small scales.

Think of it like this:

The Walker's Rule: "I can only walk 1 meter at a time, and I have a specific energy cost for doing so."
The Bird's Rule: "I can jump 10 meters, 100 meters, or even 1,000 meters."
The Result: If the walker's 1-meter walk is stable and predictable, and the bird's jumps are "stronger" (meaning the bird is very active even at small distances), then the bird's entire flight pattern is also stable and predictable.

The paper provides a specific formula (a "scaling function") that translates the walker's rules into the bird's rules. It's like having a translator that takes the instructions for a slow, local process and automatically generates the correct instructions for a fast, global process.

Why is this a Big Deal?

It's a Generalization: Previous math mostly looked at "quadratic" cases (like standard heat diffusion, where $p=2$ ). This paper works for any power $p > 1$ . This is like upgrading from a simple straight-line ruler to a flexible, curved measuring tape that works for all kinds of shapes.
No "Heat Kernel" Needed: Usually, to prove these things, mathematicians need to know exactly how heat spreads over time (the "heat kernel"). This paper is clever because it proves the bird's rules without needing to know the heat kernel first. It builds the bridge directly from the local rules to the non-local rules using pure logic and geometry.
Fractals and Weird Spaces: The paper is designed to work on "fractals" (shapes that look the same no matter how much you zoom in, like a snowflake or a coastline). These are places where normal calculus (gradients and derivatives) doesn't work. This paper gives us a new way to do calculus on these weird shapes.

The "BBM" Connection

The paper also touches on a famous idea called the Bourgain-Brezis-Mironescu (BBM) theory.

The Analogy: Imagine you have a video of a bird flying. If you slow the video down enough (or look at it through a specific lens), the bird's flight starts to look exactly like the walker's steps.
The Paper's Contribution: The author shows how to set up the "lens" (the mathematical inequalities) so that you can see this connection clearly on these weird, fractal landscapes where no smooth roads exist.

Summary in One Sentence

This paper proves that if a "local" process (like a slow walker) follows specific stability rules on a strange geometric landscape, then a "non-local" process (like a teleporting bird) that jumps more aggressively will automatically follow similar stability rules, allowing mathematicians to predict complex global behavior based on simple local rules.

Technical Summary: Local and non-local p-energies on metric measure spaces

Problem Statement
This paper investigates the relationship between local and non-local regular $p$ -energies on metric measure spaces $(X, d, m)$ for $p > 1$ . Specifically, it addresses the "subordination" phenomenon: under what geometric and analytic conditions does a local regular $p$ -energy, satisfying specific functional inequalities, induce a non-local regular $p$ -energy via a jumping kernel?

The study is motivated by the classical Dirichlet form framework ( $p=2$ ), where subordination principles link Brownian motion (local) to stable-like processes (non-local). In the linear case, it is known that if a local Dirichlet form satisfies a Poincaré inequality and a cutoff Sobolev inequality, the associated non-local form (defined by a kernel with specific scaling) satisfies analogous non-local inequalities. The central challenge addressed here is extending these results to the nonlinear setting ( $p \neq 2$ ) on general metric measure spaces, particularly fractals where gradient structures may be absent.

The paper poses the following core questions:

Given a strongly local regular $p$ -energy satisfying a Poincaré inequality $PI(L)(\Psi)$ and a cutoff Sobolev inequality $CS(L)(\Psi)$ , for which scaling functions $\Upsilon$ does the induced non-local $p$ -form (defined by a jumping kernel $K^{(\Upsilon)}$ ) constitute a regular $p$ -energy?
Does this non-local form satisfy non-local functional inequalities ($PI(J)$ and $CS(J)$), and if so, what is the resulting scaling function $\Xi$ ?
Can these results be derived directly from the local functional inequalities without relying on heat kernel estimates?

Methodology
The author employs a purely analytic approach, avoiding reliance on heat kernel estimates or probabilistic representations. The methodology relies on the following key components:

Equivalence of Inequalities: The paper first establishes the equivalence between various versions of the cutoff energy inequality ($CE$) and the cutoff Sobolev inequality ($CS$) in both local and non-local settings. Crucially, it proves that the "weak" versions (without continuity assumptions on cutoff functions) imply the "strong" versions (where cutoff functions are Hölder continuous). This equivalence is vital because the existence of Hölder continuous cutoff functions is necessary to establish the regularity of the energy space.
Regularity of Solutions: To prove the strong versions of the inequalities, the author utilizes the interior and boundary regularity of solutions to $p$ -Laplace type equations ( $-\Delta_p u = f$ ). This involves establishing oscillation inequalities and weak Harnack inequalities. In the non-local setting, special care is taken to handle "tail terms" arising from the non-local nature of the operator.
Partition of Unity and Discrete Convolution: To derive non-local inequalities from local ones, the paper constructs partitions of unity with controlled energy. A novel analytic argument involving discrete convolution is used to transfer the local Poincaré inequality to the non-local setting.
Scaling Analysis: The relationship between the scaling functions of the local energy ( $\Psi$ ), the jumping kernel ( $\Upsilon$ ), and the resulting non-local energy ( $\Xi$ ) is analyzed through series convergence conditions and integral estimates.

Key Contributions and Results

Equivalence of Cutoff Conditions:
Theorem 2.1 and Theorem 2.2 establish that for both local and non-local regular $p$ -energies, the strong, continuous, and weak versions of the cutoff energy inequality ($CE$) and the cutoff Sobolev inequality ($CS$) are equivalent, provided the volume doubling condition (VD) holds. This resolves a technical gap regarding the regularity of cutoff functions in the nonlinear setting.
Regularity of Non-local $p$ -energies:
Proposition 2.6 demonstrates that a non-local $p$ -form satisfying the non-local Poincaré inequality and the continuous cutoff Sobolev inequality is indeed a regular $p$ -energy. This confirms that the functional inequalities are sufficient to ensure the space contains sufficiently many continuous functions for potential theory.
The Subordination Theorem (Main Result):
Theorem 2.9 provides the main "subordination" result. It states that if a strongly local regular $p$ -energy satisfies $PI(L)(\Psi)$ and $CE(L)_{strong}(\Psi)$ , and if the scaling function $\Upsilon$ of the jumping kernel satisfies a strict upper growth condition at small scales relative to $\Psi$ :
$\sum_{n=0}^{\infty} \frac{\Psi(2^{-n})}{\Upsilon(2^{-n})} < +\infty$
then the induced non-local $p$ -form is a regular $p$ -energy. Furthermore, it satisfies non-local Poincaré and cutoff Sobolev inequalities with a new scaling function $\Xi$ explicitly determined by:
$\Xi(r) = \Psi(r) \int_0^r \frac{d\Psi(t)}{\Upsilon(t)}$
The paper proves that this growth condition is not only sufficient but also necessary for the non-local space to contain non-constant functions.
Generalization of Classical Examples:
Corollary 2.11 applies these results to specific scaling functions (piecewise power functions), recovering known results for stable-like processes on fractals and extending them to cases with light polynomial tails at infinity. Corollary 2.12 establishes a BBM-type (Bourgain-Brezis-Mironescu) Poincaré inequality for non-local energies derived from local ones, providing a bridge between non-local and local $p$ -energies in the limit.

Significance and Claims
The paper claims to provide a nonlinear extension of the classical subordination principle beyond the Dirichlet form framework ( $p=2$ ). Its significance lies in:

Direct Derivation: It answers whether non-local functional inequalities can be derived directly from local ones without heat kernel estimates, affirming this is possible via the constructed analytic arguments.
Fractal Applicability: The results are formulated for general metric measure spaces, including fractals where no gradient or upper gradient structure is available, relying instead on the existence of local $p$ -energies and functional inequalities.
Technical Novelty: The proof of the implication from weak to strong cutoff inequalities via PDE regularity, and the use of discrete convolution to transfer local inequalities to the non-local setting, are presented as new analytic techniques, even in the linear case.

The author notes that these results cover examples where jumping kernels have light polynomial tails at infinity, a regime where previous methods relying on heat kernel estimates might be less direct. The work is positioned as a foundational step in understanding the interplay between local and non-local $p$ -energies on general spaces.