Potential Estimates and Hodge Systems with $L^1$ data on compact manifolds

Abstract

In this paper we establish optimal Lorentz estimates for the Riesz potentials acting on closed or co-closed $k$-forms of finite mass on a smooth, compact Riemannian manifold of dimension $n$: For $\alpha \in (0,n)$ and $k=1,\ldots,n-1$, there exists a constant $C>0$ such that \begin{align*} \| \mathcal{I}_{\alpha,k} F \|_{L^{n/(n-\alpha),1}(\Lambda^k)} \leq C \| F\|_{L^1(\Lambda^k)} \end{align*} for all $k$-forms $F \in L^1(\Lambda^k)$ orthogonal to the space of harmonic $k$-forms and satisfying $\mathrm{d} F=0$ or $\mathrm{d}^* F=0$. We show how this inequality implies analogous Lorentz bounds for solutions of the $k$-form Poisson equation and for the Hodge system with data having finite mass. These results include as a special case the div--curl system on the $3$-dimensional torus, where we answer an open question originally posed by J. Bourgain and H. Brezis.