Bounded Representatives in Critical Sobolev-Hodge Spaces

Abstract

Let $n\ge 2$, $1\le \ell \le n-1$, and $1<p<\infty$. We prove that every $v\in {\dot{W}}^{n/p,p}({\mathbb{R}}^n;{\Lambda }^{\ell })$ has a representative $u\in {\dot{W}}^{n/p,p}\cap L^{\infty }$ with $du=dv$ and $\max |u|{\dot{W}}^{n/p,p},|u|L^{\infty }\lesssim |v|{\dot{W}}^{n/p,p}$. Equivalently, $d[{\dot{W}}^{n/p,p}{\Lambda }^{\ell }]=d[({\dot{W}}^{n/p,p}\cap L^{\infty }){\Lambda }^{\ell }]$ with equivalent quotient norms. The proof reduces the selection problem to an endpoint graph estimate for a Riesz potential and the exact Hodge projection. Its main analytic input is a finite-dimensional-input Maz'ya--$\Phi$ inequality for operator-valued homogeneous kernels. For the Riesz/Hodge pair, a nonlinear spherical profile built from the projected kernel has exact atomic cancellation and is coercive by the identity $\int S^{n-1}P(\theta ),d\bar{\sigma }=(\ell /n)\mathrm{Id}$. Frequency-localized graph closure and Hahn--Banach duality then return a bounded representative.

19 pages