A fractional Hardy-Sobolev inequality of Michael-Simon type on convex hypersurfaces

1. Problem Statement

The paper addresses the extension of classical and fractional functional inequalities to the setting of hypersurfaces $M = \partial\Omega \subset \mathbb{R}^{n+1}$, where $\Omega$ is an open convex set. Specifically, the authors aim to establish a fractional Caffarelli-Kohn-Nirenberg (CKN) type interpolation inequality that unifies:

1. The Fractional Michael-Simon Sobolev inequality (relating the $L^p$ norm of a function to its fractional gradient and fractional mean curvature).
2. The Fractional Hardy inequality (relating weighted norms of a function to its fractional gradient).

The core challenge is to prove these inequalities with a universal constant (independent of the specific geometry of the manifold $M$) and to incorporate the fractional mean curvature $H_\alpha$ as a geometric term on the right-hand side. While such inequalities are well-understood in Euclidean space $\mathbb{R}^n$, extending them to general submanifolds is difficult because standard rearrangement techniques (like Schwarz symmetrization) do not directly apply to curved manifolds without losing the universal constant property. The authors restrict their study to convex hypersurfaces to overcome these geometric obstructions.

2. Methodology

The proof strategy combines techniques from geometric measure theory, fractional calculus, and interpolation theory.

A. Geometric Decomposition (The Convexity Argument)

A central technical hurdle is handling the weighted integral $\int_E |x|^{-\beta} dx$ on a hypersurface $M$. In general manifolds, this integral can diverge or behave unpredictably if the surface oscillates near the origin.

• Lemma 2.2: The authors prove that the boundary of a convex set $\partial\Omega$ can be covered (up to a set of measure zero) by at most $2(n+1)$ graphs of Lipschitz functions $g_i$ with uniformly bounded gradients ($|\nabla g_i| \leq \sqrt{n} + \epsilon$).
• Theorem 1.2: Using this decomposition, they establish a weighted perimeter inequality: for any measurable subset $E \subset \partial\Omega$,
  $$\int_E |x|^{-\beta} dx \leq C |E|^{\frac{n-\beta}{n}}$$
  This result is crucial because it mimics the behavior of the Schwarz rearrangement inequality in $\mathbb{R}^n$ but holds universally for convex hypersurfaces. It relies heavily on the convexity of $\Omega$ to prevent "infinite oscillation" near the origin.

B. Layer Cake Decomposition and Interpolation

The proof of the main inequality (Theorem 1.1) utilizes a layer cake decomposition of the function $u$ (defining level sets $A_i = \{u > 2^i\}$).

• Lemma 3.2: They utilize a key estimate from previous work (Cabr'e, Cozzi, Csató) which bounds the sum of the fractional Gagliardo seminorm and the fractional mean curvature term from below by a series involving the measures of level sets.
• Interpolation: The authors employ a generalized Hölder inequality to interpolate between the fractional Sobolev term and the $L^q$ norm.

C. Handling the Full Parameter Range

The proof is split into two cases based on the relationship between the exponents:

1. Case 1 ($1/\tau \leq a/p + (1-a)/q$): The proof follows directly from the layer cake decomposition, Theorem 1.2 (weighted perimeter), and the fractional Michael-Simon inequality.
2. Case 2 ($1/\tau > a/p + (1-a)/q$): This case requires $\gamma < -as$. The authors use a splitting strategy:
   • They split the domain into the unit ball $B_1(0)$ and its complement.
   • They apply Hölder's inequality twice with auxiliary parameters to reduce the problem to the cases already proven in Case 1.
   • Lemma 4.1: To control the integrals on the split domains, they prove that $\int_{\partial\Omega \cap B_1} |x|^{-\beta} dx$ and $\int_{\partial\Omega \setminus B_1} |x|^{-\beta} dx$ are bounded by universal constants, provided $\beta$ satisfies specific conditions relative to $n$. This again relies on the convexity of $\Omega$ and the monotonicity of the perimeter of convex sets.

3. Key Contributions

1. Fractional Michael-Simon-Hardy Interpolation: The paper establishes a new inequality (Theorem 1.1) that interpolates between the fractional Sobolev inequality and the fractional Hardy inequality on convex hypersurfaces. The inequality takes the form:
   $$\||x|^\gamma u\|_{L^\tau(\partial\Omega)} \leq C \left( [u]_{W^{s,p}(\partial\Omega)}^p + \int_{\partial\Omega} H_\alpha(x) |u(x)|^p dx \right)^{a/p} \|u\|_{L^q(\partial\Omega)}^{1-a}$$
   where $H_\alpha$ is the fractional mean curvature.
2. Universal Constants: The constant $C$ depends only on the dimension $n$ and the parameters ($s, p, q, \tau, \gamma, \alpha$), but not on the specific shape of the convex set $\Omega$.
3. New Geometric Inequality (Theorem 1.2): The authors prove that for convex hypersurfaces, the weighted $L^1$ norm of $|x|^{-\beta}$ is controlled by the measure of the set raised to a power. This is a non-trivial geometric result that fails for non-convex manifolds.
4. Simplified Proof of Fractional Hardy: By restricting their main theorem to the flat case ($M = \mathbb{R}^n$), they provide a new, simple proof of the fractional Hardy inequality in Euclidean space, derived from the combination of the fractional Sobolev inequality and the rearrangement inequality.

4. Main Results

• Theorem 1.1: The main interpolation inequality holds for all $u \in W^{s,p}(\partial\Omega)$ where $\Omega$ is an open convex set. The parameters must satisfy dimensional balance conditions and specific constraints on $\gamma$ (specifically $\gamma \geq -as$ in certain regimes).
• Theorem 1.2: For any measurable $E \subset \partial\Omega$ and $\beta \in [0, n)$, $\int_E |x|^{-\beta} dx \leq C |E|^{(n-\beta)/n}$.
• Lemma 4.1: Establishes the boundedness of weighted integrals over the intersection and complement of the unit ball for convex hypersurfaces, which is essential for the interpolation argument in the full parameter range.

5. Significance

• Unification of Nonlocal Inequalities: The work bridges the gap between fractional Sobolev inequalities (which control regularity) and Hardy inequalities (which control singularity at the origin) in a geometric setting.
• Geometric Analysis: It demonstrates that convexity is a sufficient condition to recover universal constants for nonlocal inequalities on manifolds, a property that generally fails for arbitrary submanifolds. This suggests that convexity effectively "flattens" the geometry enough to allow Euclidean-style rearrangement arguments.
• Applications: These inequalities are fundamental in the study of fractional mean curvature flows, nonlocal minimal surfaces, and the regularity of solutions to nonlocal PDEs on manifolds. The inclusion of the fractional mean curvature term $H_\alpha$ is particularly relevant for the analysis of nonlocal geometric evolution equations.
• Methodological Innovation: The technique of decomposing a convex boundary into a finite number of Lipschitz graphs with controlled gradients to estimate weighted integrals is a novel approach that could be applicable to other geometric analysis problems involving convex sets.

In summary, Csató and Roy successfully extend the classical Michael-Simon and Hardy inequalities to the fractional setting on convex hypersurfaces, providing a robust framework for analyzing nonlocal geometric problems with universal constants.