Recall that a differential $k$-form $\alpha$ on a smooth manifold $M$ is called weakly closed if

$$\int_M \alpha \wedge d\beta = 0,$$

for all smooth forms $\beta$ of degree $n-k-1$, where $n = \dim M$. My question is:

If $\alpha$ is a weakly closed **continuous** 1-form on a closed manifold $M$, can we conclude that $\alpha$ is the sum of a **smooth** closed 1-form and $d\phi$, where $\phi$ is a $C^1$ function on $M$?

This appears to follow from the properties of the de Rham regularization operator(s) and associated homotopy operator(s) on forms; see, for instance, Theorem 12.5 in V. Gol'dshtein and M. Troyanov, *Sobolev inequalities for differential forms and $L_{q,p}$-cohomology*, Journal of Geometric Analysis, vol. 6, no. 4, 2006.

Any thoughts on this would be appreciated.