Endpoint weak-L^p integrability

Determine whether the weak-L^p bounds for the density \theta of the measure \mu extend to the endpoint exponent p=m/(m-1).

Background

Theorem \ref{Lp} establishes a weak-L^p estimate for the density \theta of a measure \mu=\theta\,\mathcal H^m!\llcorner M under a first-order PDE constraint, with some exponent p>1 depending on the dimension and ellipticity parameters. The authors note that a layer-cake argument yields corresponding strong L^q bounds for every q<p.

The unresolved issue is whether the weak estimate can be improved all the way to the natural endpoint exponent p=m/(m-1). This concerns the sharp integrability threshold for densities of measures satisfying the stated PDE constraint and is explicitly left unresolved by the paper.