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Closing the gap: Maz'ya-Shaposhnikova and asymptotics of fractional perimeters

This paper generalizes the Maz'ya-Shaposhnikova formula for $p=2$ to functions that may not vanish at infinity by introducing a "mass at infinity" concept, thereby unifying the asymptotic behavior of Gagliardo seminorms and nonlocal perimeters within a single functional framework that extends to metric measure spaces.

Original authors: Elisa Davoli, Alberto Fanizza, Marco Picerni

Published 2026-05-06

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CC BY 4.0

Original authors: Elisa Davoli, Alberto Fanizza, Marco Picerni

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 standing in a room (let's call it $\Omega$ ) and you want to measure how much "energy" or "tension" exists in the air around you. In mathematics, this energy is often calculated by looking at how much a function (like a temperature map or a height map) changes between every possible pair of points.

Usually, mathematicians have two main ways to look at this energy:

The "Close-Up" View (s → 1): If you look at points very close to each other, this energy tells you about the smoothness of the surface (like the slope of a hill). This is a famous result called the Bourgain-Brezis-Mironescu (BBM) formula.
The "Far-Out" View (s → 0): If you look at points very far apart, the energy usually tells you about the total amount of "stuff" (mass) in the room. This is the Maz'ya-Shaposhnikova (MS) formula.

The Problem:
The classic "Far-Out" view (MS formula) has a strict rule: it only works if the "stuff" (the function) eventually disappears completely as you move far away from the room. It assumes the universe is empty at the edges.

But what if the "stuff" doesn't disappear? What if you have a pattern that repeats forever, or a value that stays constant at infinity? The old formula breaks down because it can't handle the "tail" of the function—the part that stretches out to infinity. It's like trying to measure the weight of a river by only looking at the water inside a bucket, ignoring the fact that the river flows forever.

The Solution: "Closing the Gap"
This paper introduces a new way to measure that energy for functions that don't vanish at infinity. The authors, Davoli, Fanizza, and Picerni, propose a new concept called "Mass at Infinity."

Here is the analogy they use:
Imagine the room is a house, and the rest of the world is the "outside."

Old Method: You only count the energy between points inside the house and points inside the house, or inside and outside, but you assume the outside eventually becomes empty.
New Method: They realize that as you zoom out (letting the parameter $s$ go to zero), the "kernel" (the mathematical tool that measures distance) starts to concentrate its attention on the horizon. It's as if the interaction between a point inside your house and a point "at infinity" becomes the most important thing.

They define a new quantity, $\alpha(u)$ , which measures the "mass" or "weight" of the function as it stretches out to the horizon. Think of it as measuring the average value of the function on a giant sphere that surrounds the universe, getting larger and larger until it represents "infinity."

The Big Discovery
The authors prove a new formula that acts as a universal translator between two different worlds:

For normal functions (that vanish at infinity): Their new formula shrinks down to the classic MS formula. It says, "If the outside is empty, the energy is just proportional to the total amount of stuff inside."
For weird functions (that don't vanish): Their new formula reveals that the energy is actually an interaction energy. It's not just about how much stuff is inside the room; it's about the "tension" between the stuff inside the room and the "stuff" waiting at the horizon.

They show that the limit of this energy is exactly the average of the squared difference between the value inside the room and the value at the horizon.

Metaphor: Imagine a rubber sheet stretched over a frame (the room). If the sheet is pulled tight by a constant force from the horizon, the energy isn't just about the sheet's weight; it's about how much the sheet inside the frame differs from the force pulling it from the edge.

Key Takeaways from the Paper:

The "Mass at Infinity": They created a mathematical definition for how much "weight" a function has at the very edge of the universe. If this number exists, you can calculate the energy limit.
Unifying Framework: This single new formula explains both the classic result (for functions that die out) and the behavior of "fractional perimeters" (which measure the boundary of shapes) for shapes that might have infinite tails.
Gamma-Convergence: They proved that this isn't just a point-by-point coincidence. If you have a sequence of these energy measurements, they will mathematically "settle down" into this new formula, even if you look at them through a slightly blurry lens (weak topology).
Beyond Flat Space: They showed this idea works not just in our flat, Euclidean space, but also in more exotic geometric worlds (like Carnot Groups, which are used to model complex movements in robotics or physics) and spaces with different rules for distance.

What They Did NOT Do:
The paper is purely theoretical mathematics. They did not apply this to:

Medical imaging or clinical uses.
Specific engineering designs.
Predicting future physical phenomena.

They strictly focused on proving that this new mathematical "bridge" exists, works for a wider class of functions, and unifies two previously separate ideas in the study of non-local interactions. They essentially fixed the "leak" in the old formula that only worked when the universe was empty at the edges.

Technical Summary: Closing the Gap: Maz'ya-Shaposhnikova and Asymptotics of Fractional Perimeters

Problem Statement
The study of the asymptotics of Gagliardo seminorms is central to nonlocal analysis, bridging the gap between fractional Sobolev spaces $W^{s,p}$ and classical spaces as the parameter $s$ approaches 0 or 1. The classical Maz'ya-Shaposhnikova (MS) formula [36] describes the limit as $s \to 0^+$ for functions $u \in L^p(\mathbb{R}^d)$ , stating that $s [u]_{W^{s,p}}^p$ converges to a multiple of the $L^p$ norm. However, this formula relies on the structural assumption that $u$ vanishes at infinity (i.e., $u \in L^p(\mathbb{R}^d)$ ). This restriction artificially suppresses the "tail behavior" of functions, which is precisely the regime where the nonlocal nature of the fractional kernel is most pronounced as $s \to 0^+$ .

Conversely, recent work on the asymptotics of fractional perimeters [20] for characteristic functions of sets $E$ revealed that if a set has non-vanishing mass at infinity, the limit involves a term $\alpha_1(E)$ representing the "nonlocal tail." This suggests that for general functions not in $L^p(\mathbb{R}^d)$ , the limit should not be a simple multiple of the local $L^p$ norm but rather an interaction energy between the function's values inside a bounded domain $\Omega$ and its behavior "at infinity."

Methodology
The authors focus on the Hilbertian case $p=2$ and introduce a localized Gagliardo seminorm on a bounded Lipschitz domain $\Omega$ :
$[u]_{W^{s,2}(Q\Omega)}^2 = \iint_{Q\Omega} \frac{|u(x) - u(y)|^2}{|x-y|^{d+2s}} \, dx \, dy,$
where $Q\Omega = (\mathbb{R}^d \times \mathbb{R}^d) \setminus (\Omega^c \times \Omega^c)$ . This functional accounts for interactions involving at least one point in $\Omega$ , isolating the energy from interior-interior, interior-exterior, and exterior-interior interactions while ignoring exterior-exterior contributions.

The core methodological innovation is the definition of the $p$ -mass at infinity for a measurable function $u$ :
$\alpha_p(u) := \lim_{s \to 0^+} s \int_{B_1^c(0)} \frac{u(y)}{|y|^{d+sp}} \, dy.$
The authors establish that this quantity is translation-invariant and well-defined for functions that may not be globally integrable. They decompose the localized seminorm into four terms:

Interior-Interior: Shown to vanish as $s \to 0^+$ (Lemma 2.5).
Interior-Exterior (Squared): Converges to a term involving the mass of the constant function 1 and the integral of $u^2$ over $\Omega$ .
Cross-Term: Involves the product $u(x)u(y)$ , which is expanded using the positive and negative parts of $u$ ( $u_+, u_-$ ) and analyzed via the mass at infinity.
Exterior-Interior (Squared): Converges to a term involving the mass of $u^2$ and the volume of $\Omega$ .

The proofs rely on a combination of direct estimates, the fractional Hardy inequality, and a generalized version of the MS formula in metric measure spaces (Proposition 2.3 from [28]), adapted to handle measures weighted by the function values themselves.

Key Contributions and Results

Generalized Maz'ya-Shaposhnikova Formula (Theorem 1.3):
The paper proves that for $u \in L^2_{\text{loc}}(\mathbb{R}^d)$ with finite mass at infinity (specifically $\alpha(u_+), \alpha(u_-), \alpha(u^2)$ exist), the limit of the localized seminorm is:
$\lim_{s \to 0^+} \frac{s}{2} [u]_{H^s(Q\Omega)}^2 = \sum_{k=0}^2 \binom{2}{k} (-1)^k \alpha(u^k) \int_\Omega u^{2-k}(x) \, dx.$
This formula unifies the classical MS formula (recovered when $u \in L^2(\mathbb{R}^d)$ so $\alpha(u^k)=0$ for $k \ge 1$ ) and the fractional perimeter asymptotics.
Geometric Interpretation (Theorem 2.11):
For functions $u$ that admit a radial limit $u_\infty(\theta)$ as $|x| \to \infty$ , the limiting functional is explicitly characterized as an interaction energy between the values of $u$ inside $\Omega$ and its limit at infinity:
$\lim_{s \to 0^+} \frac{s}{2} [u]_{H^s(Q\Omega)}^2 = \frac{1}{2} \int_\Omega \int_{S^{d-1}} |u(x) - u_\infty(\theta)|^2 \, d\mathcal{H}^{d-1}(\theta) \, dx.$
This clarifies that the limit is not a norm but an energy measuring the deviation of the function from its behavior at infinity.
Gamma-Convergence (Theorem 3.4):
The authors establish that the sequence of functionals $F_s(u, \Omega) = \frac{s}{2}[u]_{H^s(Q\Omega)}^2$ Gamma-converges with respect to the weak- $L^2(\Omega)$ topology to the functional identified in Theorem 1.3. This result implies that the limiting functional is the lower semicontinuous envelope of the set-functional describing the asymptotic fractional perimeter (Equation 1.6).
Extension to Metric Measure Spaces (Theorem 4.5):
The results are extended to general metric measure spaces $(M, d, m)$ under specific assumptions on the convolution kernels $\rho_n$ and the existence of mass at infinity. This includes applications to Carnot Groups.
Distinction from Gaussian Perimeters (Remark 4.9):
The paper notes that the derived asymptotics do not hold for the Gaussian fractional perimeter. In the Gaussian setting, the measure is finite, and the kernels do not concentrate mass at infinity in the same way; consequently, the limit depends on the total measure of sets rather than local boundary interactions.

Significance and Claims
The paper claims to "close the gap" between the classical Maz'ya-Shaposhnikova formula and the asymptotics of nonlocal perimeters by providing a unified framework that accommodates functions with non-vanishing mass at infinity.

Unification: The identified limiting functional reduces to the classical $L^2$ norm for globally integrable functions and recovers the pointwise limit of $s$ -fractional perimeters for characteristic functions of sets.
Structural Insight: By introducing the notion of "mass at infinity," the authors rigorously characterize the long-range interactions that dominate the energy as $s \to 0^+$ . They demonstrate that the limit is an interaction energy between the interior of the domain and the "horizon" (infinity), rather than a local norm.
Robustness: The results are shown to hold in the sense of Gamma-convergence, ensuring stability under weak convergence, and are generalized to the metric measure setting, covering spaces like Carnot Groups.

The authors explicitly state that the case for general $p \neq 2$ requires a different approach and is the subject of a companion paper [13], noting that for odd $p$ , a closed formula analogous to the binomial expansion used here may not be attainable.

Technical Summary: Closing the Gap: Maz'ya-Shaposhnikova and Asymptotics of Fractional Perimeters

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