Bounded Representatives in Critical Sobolev-Hodge Spaces

Technical Summary: Bounded Representatives in Critical Sobolev–Hodge Spaces

Problem Statement
The paper addresses a fundamental question in the analysis of critical Sobolev spaces: whether every element in a critical Sobolev space admits a bounded representative. Specifically, for dimensions $n \ge 2$, form degrees $1 \le \ell \le n-1$, and exponents $1 < p < \infty$, the Sobolev embedding $\dot{W}^{n/p, p}(\mathbb{R}^n) \hookrightarrow L^\infty$ fails. The authors investigate the "selection principle" for differential forms in the critical homogeneous Sobolev–Hodge space $\dot{W}^{n/p, p}(\mathbb{R}^n; \Lambda^\ell)$.

The core question is whether the affine class of forms sharing the same exterior derivative, $\{u : du = dv\}$, always contains a representative $u$ that is bounded ($u \in L^\infty$) with quantitative control, even if the original representative $v$ is unbounded. This extends the seminal work of Bourgain and Brezis, who established this phenomenon for first-order differential constraints, to the higher-order, fractional, and Hodge-decomposition contexts.

Methodology
The proof strategy reduces the selection problem to an endpoint graph estimate for a Riesz potential coupled with the exact Hodge projection. The methodology proceeds through three substantive analytical steps:

1. Operator-Input Maz'ya–$\Phi$ Inequality:
   The authors generalize Stolyarov's scalar-input Maz'ya–$\Phi$ inequality to a finite-dimensional, operator-valued setting. They establish an estimate for homogeneous kernel operators $K$ acting on vector-valued inputs $f$. The key innovation is handling the "multiscale" nature of the problem: vector masses $\int_Q f$ enter the exact atomic cancellation, while error terms are organized by the positive scalar measure $|f|dx$. This involves:

   • Decomposing the kernel into sharp annular pieces.
   • Proving annular operator bounds and exact cancellation for annular atoms.
   • Utilizing spatial localization via dyadic cubes and the "Three Lattice Theorem" to manage the nonlinear integral estimates.
   • Distinguishing between regimes $1 < r \le 2$ and $r > 2$ (where $r = p'$) to handle mixed-scale summations using specific convexity properties of the function $\Gamma_r$.

2. Projected Atomic Profile and Coercivity:
   The general theorem is specialized to the Riesz/Hodge pair. The authors construct a nonlinear spherical profile $A(a)$ derived from the projected kernel.

   • They prove that this profile has "exact atomic cancellation" due to the properties of the Hodge projection $P$.
   • Crucially, they establish the coercivity of this profile using the identity $\int_{S^{n-1}} P(\theta) d\bar{\sigma} = (\ell/n) \text{Id}$. This ensures that the projected term controls the full potential in the $L^r$ norm.

3. Graph Closure and Duality:
   The estimate is initially proven for smooth, compactly supported, mean-zero data. To extend this to the full critical space:

   • The authors avoid the false strong bound $I_\alpha: L^1 \to L^r$ (which fails at the endpoint) by working on the "endpoint graph domain."
   • They utilize frequency-localized graph closure and Hahn–Banach duality. By showing that the graph norm on the sum space $X + L^1_0$ (where $X$ is the dual of the Sobolev space) controls the full potential, they construct a bounded linear functional that yields the desired bounded representative via duality.

Key Results
The main result is Theorem 1.1 (Critical Sobolev–Hodge selection), which states:
Let $n \ge 2$, $1 \le \ell \le n-1$, and $1 < p < \infty$. Then the image of the exterior derivative on the critical Sobolev space coincides with the image on the intersection of the Sobolev space and $L^\infty$:
$$d[\dot{W}^{n/p, p}(\mathbb{R}^n; \Lambda^\ell)] = d[(\dot{W}^{n/p, p} \cap L^\infty)(\mathbb{R}^n; \Lambda^\ell)]$$
Moreover, the induced quotient norms are equivalent. Specifically, there exists a constant $C$ such that for every $v \in \dot{W}^{n/p, p}(\mathbb{R}^n; \Lambda^\ell)$, there exists a representative $u \in (\dot{W}^{n/p, p} \cap L^\infty)(\mathbb{R}^n; \Lambda^\ell)$ satisfying:
$$du = dv, \quad \max\{\|u\|_{\dot{W}^{n/p, p}}, \|u\|_{L^\infty}\} \le C \|v\|_{\dot{W}^{n/p, p}}$$

The paper explicitly notes that the selection operator is inherently nonlinear; no bounded linear selection operator is asserted, consistent with the original Bourgain–Brezis phenomenon.

As a consequence, Corollary 8.1 provides a bounded-potential Hodge decomposition for intermediate form degrees ($2 \le \ell \le n-2$), showing that the space $\dot{W}^{n/p-1, p}(\Lambda^\ell)$ can be decomposed into exact and coexact terms, both admitting bounded representatives.

Significance and Claims
The paper claims to resolve an open problem regarding the selection principle throughout the critical scale, extending previous partial results on higher-order and fractional cases. The significance lies in:

• Restoring Boundedness: Demonstrating that differential constraints (specifically the Hodge structure) allow for the recovery of boundedness in critical Sobolev spaces where standard embeddings fail.
• Analytic Innovation: Introducing a finite-dimensional-input form of the Maz'ya–$\Phi$ inequality for operator-valued kernels, which handles the vector-valued nature of differential forms and the specific geometry of the Hodge projection.
• Quantitative Control: Providing explicit norm estimates that link the size of the bounded representative to the original Sobolev norm, ensuring the selection is not just existential but quantitatively controlled.

The authors emphasize that the proof relies on a delicate interplay between the exact cancellation of the Hodge projection and the coercivity of the projected kernel, avoiding the need for the non-existent strong $L^1 \to L^r$ bound for Riesz potentials at the endpoint.