Robust interpolation inequalities via Chebyshev-type integral inequalities

Technical Summary: Robust Interpolation Inequalities via Chebyshev-Type Integral Inequalities

Problem Statement
The paper addresses the lack of robustness in classical log-convex interpolation inequalities within the scale of Gagliardo seminorms. Standard interpolation estimates, such as $|u|_{W^{\eta,p}} \leq C |u|_{W^{\sigma,p}}^\theta |u|_{W^{s,p}}^{1-\theta}$, fail to remain consistent as the fractional order $\eta$ approaches the endpoints $0$ or $1$. Specifically, the seminorms $|u|_{W^{\eta,p}}$ exhibit singular blow-up behavior as $\eta \to 0^+$ and $\eta \to 1^-$ for non-constant functions. While previous works (Brezis–Bourgain–Mironescu [BBM01] and Maz'ya–Shaposhnikova [MS02]) established asymptotic limits involving factors like $\eta(1-\eta)$ to annihilate these singularities, a unified, robust interpolation inequality that avoids these anomalies across the entire scale $s \in [0,1]$ had not been fully established with dimension-free constants.

The primary motivation for this work is the study of the asymptotic stability of weak solutions to the regional Dirichlet problem associated with the regional fractional $p$-Laplacian $(-\Delta)^s_{p,\Omega}$ as the fractional order $s$ approaches $1$. The authors aim to establish convergence results that are uniform with respect to $s$.

Methodology
The core methodology relies on a decomposition of the Gagliardo seminorm into short-range (local) and long-range (nonlocal) interactions, defined via the averaged $L^p$-modulus of continuity $U(t)$.

1. Decomposition: The Gagliardo seminorm is split into integrals over short distances ($0 < |x-y| \leq r$) and long distances ($|x-y| > r$). These correspond to the short-range interaction $L_p(s, u, r)$ and the long-range interaction (fractional tail) $T_p(s, u, r)$.
2. Chebyshev-Type Inequalities: A significant portion of the paper (Section 2) is dedicated to establishing new integral inequalities for general non-synchronous (and non-asynchronous) functions. The authors generalize the classical Chebyshev inequality to cases involving $(b,c)$-synchronicity and weak monotonicity. These inequalities are applied to singular integrals of the form $\int g(t) t^{z-1} dt$ to prove monotonicity properties of the interaction terms with respect to the fractional order $s$.
3. Monotonicity Analysis: The authors prove that the map $s \mapsto L_p(s, u, r)$ is increasing, while $s \mapsto T_p(s, u, r)$ is essentially decreasing (specifically, "2-almost decreasing" for $d \geq 2$). This monotonicity is the cornerstone for deriving the robust interpolation estimates.
4. Renormalization: The paper utilizes specific renormalized seminorms $[u]_{W^{s,p}}$ and $[u]^*_{W^{s,p}}$, incorporating normalization constants (e.g., $s(1-s)$ and constants related to the fractional Laplacian) to ensure consistent limits as $s \to 0$ and $s \to 1$.

Key Contributions and Results

• Robust Interpolation Inequalities (Theorem 1.1): The paper establishes robust log-convex interpolation inequalities for renormalized seminorms $[u]_{W^{\eta,p}}$ and Gagliardo seminorms $|u|_{W^{\eta,p}}$. These inequalities hold for all $0 \leq \sigma < \eta < s \leq 1$ with constants that remain bounded and well-defined as $\eta \to 0$ or $\eta \to 1$.

  • For the renormalized seminorm, the inequality takes the form $[u]_{W^{\eta,p}} \leq \kappa N_p(\sigma, \eta, s) [u]_{W^{s,p}}^\theta [u]_{W^{\sigma,p}}^{1-\theta}$, where the constant $N_p$ is dimension-free and bounded.
  • In the specific case $p=2$, the authors derive a sharper estimate with a constant of 1 for the renormalized Sobolev seminorms, utilizing the Fourier transform.

• Monotonicity of Interactions (Theorem 1.3): The authors prove that the short-range interaction $L_p(s, u, r)$ is continuous and increasing in $s$, while the long-range interaction $T_p(s, u, r)$ is continuous and essentially decreasing (up to a factor of 2 for $d \geq 2$). This confirms that $L_p$ captures the local regularity (converging to the gradient norm as $s \to 1$) and $T_p$ captures the nonlocal behavior (converging to the $L^p$ norm as $s \to 0$).

• Asymptotic Analysis (Section 4): The paper provides a relatively simple approach to the asymptotic limits of fractional seminorms as $s \to 0^+$ and $s \to 1^-$, recovering the results of Maz'ya–Shaposhnikova and Brezis–Bourgain–Mironescu. It also generalizes the Maz'ya–Shaposhnikova formula to arbitrary open sets $\Omega$, characterizing the limit via an "escape sector" $S_\Omega(x)$.

• Convergence of Regional Dirichlet Problems (Theorem 1.4): As a primary application, the paper proves the strong convergence of weak solutions $u_s$ to the regional fractional $p$-Laplacian Dirichlet problem to the weak solution $u_1$ of the local $p$-Laplacian problem. Specifically, if the data converges appropriately, then $\|u_s - u_1\|_{W^{\eta,p}(\Omega)} \to 0$ for all $0 \leq \eta < 1$ as $s \to 1^-$. This result relies on the robust interpolation inequalities and a new robust regional Poincaré–Friedrichs inequality.

• Open Problem Resolution: The robust interpolation inequality established here provides a positive answer to an open question posed in [BSY23, Question 1.10] regarding the robustness of interpolation in a general setting.

Significance and Claims
The paper claims that its primary contribution is the establishment of robust interpolation inequalities that avoid the singularities at the endpoints of the fractional scale. By decomposing the seminorm and utilizing generalized Chebyshev-type inequalities, the authors achieve estimates where the constants are independent of the fractional parameter's proximity to 0 or 1.

The significance of this work lies in its application to the stability of nonlocal-to-local limits. The authors demonstrate that these inequalities are sufficient to prove the strong convergence of solutions to the regional fractional $p$-Laplacian to the local $p$-Laplacian, a result that requires uniform control over the seminorms as $s \to 1$. The paper also highlights a gap in dimension $d=1$ regarding the monotonicity of the long-range interaction, formulating a conjecture for a constant $C_p$ in this case, while noting that the established results hold for $d \geq 2$.

The work is presented as a technical advancement in the analysis of fractional Sobolev spaces, providing tools (robust extension operators, monotonicity of singular integrals) that facilitate the study of asymptotic behaviors in nonlocal problems.