Maz'ya-type bounds for sharp constants in fractional Poincaré-Sobolev inequalities

1. Problem Statement and Context

The paper addresses the problem of characterizing the sharp constants (denoted $\lambda^s_{p,q}(\Omega)$) in fractional Poincaré-Sobolev inequalities associated with an arbitrary open set $\Omega \subseteq \mathbb{R}^N$.

The quantity of interest is defined as:
$$\lambda^s_{p,q}(\Omega) := \inf_{u \in C^\infty_0(\Omega), \|u\|_{L^q}=1} [u]^p_{W^{s,p}(\mathbb{R}^N)}$$
where $[u]_{W^{s,p}}$ is the Gagliardo-Slobodeckĭ seminorm, $s \in (0,1)$ is the fractional order, and $1 \le p < \infty$.

The Core Challenge:
In the local case ($s=1$), Maz'ya and Shubin established that the positivity of the Poincaré constant $\lambda_p(\Omega)$ is equivalent to the finiteness of a specific geometric quantity called the interior capacitary radius ($R_{p,\gamma}(\Omega)$). This radius allows balls to "cross" the boundary of $\Omega$ slightly, provided the portion outside $\Omega$ has small capacity.

In the fractional (nonlocal) setting, a direct reversal of the standard inradius bound is impossible because the frequencies $\lambda^s_{p,q}(\Omega)$ are insensitive to "capacitary perturbations" (removing sets of zero fractional capacity). The authors aim to bridge this gap by introducing a fractional capacitary inradius and proving two-sided bounds (Maz'ya-type bounds) for $\lambda^s_{p,q}(\Omega)$ in terms of this new geometric quantity.

2. Methodology

The authors adapt the strategy used by the first author and Brasco in the local case [11], but face significant technical hurdles due to the nonlocal nature of the fractional Laplacian. Their methodology involves:

1. Definition of Fractional Capacitary Inradius:
   They define $R^s_{p,\gamma}(\Omega)$ as the supremum of radii $r$ such that there exists a ball $B_r(x_0)$ where the part of the ball outside $\Omega$ ($B_r(x_0) \setminus \Omega$) is $(s,p,\gamma)$-negligible. Negligibility is defined via the relative fractional capacity:
   $$\text{cap}^s_{p}(B_r(x_0) \setminus \Omega; B_{2r}(x_0)) \le \gamma \cdot \text{cap}^s_{p}(B_r(x_0); B_{2r}(x_0)).$$

2. Tiling and Covering Arguments:
   To prove the lower bound, they construct a tiling of $\mathbb{R}^N$ with balls of radius $r > R^s_{p,\gamma}(\Omega)$. By the definition of the inradius, every ball in the tiling contains a "significant" amount of the complement of $\Omega$ in terms of fractional capacity.

3. New Fractional Inequalities:
   A major methodological innovation is the derivation of new fractional Maz'ya-Poincaré inequalities.

   • The Asymmetric Seminorm: Standard fractional Poincaré inequalities on balls fail to capture the correct asymptotic behavior as $s \to 0$ or $s \to 1$. To fix this, the authors introduce an asymmetric seminorm on "strips" (integrating over $B_r \times \mathbb{R}^N$ rather than $B_r \times B_r$):
     $$\left( \iint_{B_r(x_0) \times \mathbb{R}^N} \frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}} \, dx dy \right)^{1/p}$$
   • Sharp Asymptotics: They prove that this specific seminorm behaves correctly in the limits: it converges to the $L^p$ norm as $s \to 0$ and to the gradient norm as $s \to 1$.

4. Capacity Approximation:
   For the upper bound, they use smooth approximations of the capacitary potential of the set $B_r \setminus \Omega$ to construct test functions that nearly minimize the Rayleigh quotient.

3. Key Contributions and Results

A. Main Theorems (Maz'ya-Type Bounds)

The paper establishes two-sided estimates for the sharp constant $\lambda^s_{p,q}(\Omega)$ in terms of the fractional capacitary inradius $R^s_{p,\gamma}(\Omega)$.

• Theorem 1.4 (Lower Bound):
  $$\gamma \sigma \left( \frac{1}{R^s_{p,\gamma}(\Omega)} \right)^{sp - N + \frac{Np}{q}} \le \lambda^s_{p,q}(\Omega)$$
  The constant $\sigma$ depends on $N, p, s, q$ and has sharp asymptotic behavior: $\sigma \sim 1/s$ as $s \to 0$ and $\sigma \sim 1/(1-s)$ as $s \to 1$.

• Theorem 1.5 (Upper Bound):
  $$\lambda^s_{p,q}(\Omega) \le C \left( \frac{1}{R^s_{p,\gamma}(\Omega)} \right)^{sp - N + \frac{Np}{q}}$$
  This holds for a sufficiently small $\gamma < \gamma_0(N, p, s)$. The constant $C$ also exhibits the correct asymptotic limits.

  • Note: For $p=1$, $\gamma_0 = 1$ (optimal). For $p > 1$, $\gamma_0 < 1$ is currently required, though the authors conjecture $\gamma_0=1$ might hold generally.

B. Characterization of Embedding Positivity

Corollary 1.7 provides a necessary and sufficient condition for the embedding $D^{s,p}_0(\Omega) \hookrightarrow L^q(\Omega)$:
$$D^{s,p}_0(\Omega) \hookrightarrow L^q(\Omega) \iff R^s_{p,\gamma}(\Omega) < \infty$$
for some $0 < \gamma < \gamma_0$. This generalizes the local result that the embedding holds if and only if the capacitary inradius is finite.

C. Fractional Cheeger's Constant

Corollary 1.8 applies the results to the case $p=q=1$, characterizing the positivity of the fractional Cheeger constant $h_s(\Omega)$:
$$h_s(\Omega) > 0 \iff R^s_{1,\gamma}(\Omega) < \infty$$
This is an optimal characterization valid for the full range of admissible $\gamma \in (0,1)$.

D. New Technical Inequalities

The paper introduces several independent results of interest:

• Fractional Maz'ya-Poincaré Inequalities (Lemma 4.11): Inequalities involving the asymmetric seminorm that are sharp as $s \to 0$ and $s \to 1$.
• Fractional Poincaré-Wirtinger Estimates: New bounds for functions with zero mean on balls.
• L1-W s,p Estimates: Explicit computation of sharp constants for $p=1$ on balls (Lemma 3.2).

4. Significance and Impact

1. Resolution of a Nonlocal Obstruction: The paper successfully overcomes the difficulty that fractional Poincaré constants are invariant under removal of sets with zero capacity. By using the capacitary inradius, they provide a geometric characterization that is sensitive to the "shape" of the domain in a nonlocal sense.
2. Sharp Asymptotics: Unlike previous attempts to generalize Maz'ya's bounds to the fractional setting, the constants derived here are sharp with respect to the fractional parameter $s$. They correctly recover the local behavior ($s \to 1$) and the $s \to 0$ limit (related to the $L^p$ norm), which is crucial for applications in spectral geometry and asymptotic analysis.
3. Unification of Local and Nonlocal: The results unify the theory of Poincaré inequalities, showing that the same geometric principle (capacitary inradius) governs the spectral properties of both local and nonlocal operators.
4. Applications: The bounds have direct implications for:
   • The first eigenvalue of the fractional $p$-Laplacian.
   • The existence and bounds of the fractional torsion function (Corollary 1.9).
   • Regularity theory and spectral geometry of nonlocal operators.

In summary, this work provides a rigorous and optimal geometric framework for understanding the spectral properties of fractional Sobolev spaces on general open sets, extending the classical legacy of Maz'ya into the nonlocal realm with precise control over the fractional parameter.