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This is a cross-post. I am looking for a reference for the regularity of harmonic forms which belong to $L^1(M)$.

Explicitly, let $\M$ be a smooth oriented Riemannian manifold.

Let $\sigma$ be a differential $k$-form on $\M$ with coefficients in $L^1(\M)$. We say $\sigma$ is weakly harmonic if $$ \int_{\M} \langle \sigma , \Delta \alpha \rangle=0 \, \text{ for every smooth compactly supported $\alpha \in \Omega^{k}(\M)$}. $$

**Theorem:** Weakly harmonic forms (in $L^1$) are smooth.

I know a reference for this theorem, when we assume $\sigma \in L^2$ instead of $L^1$ (Warner's book). I am looking for a reference which deals with the $L^1$ case, which naively looks harder.

(Or, if possible, for an easy way to reduce the $L^1$ theorem to the $L^2$ theorem).

If that matters, I don't care about global topology here, i.e. we can assume $M=\mathbb{R}^n$ topologically. (but not "Riemannialy", that is, I don't assume the metric is Euclidean).

I only care about the smoothness, not about any generalized version of Hodge decomposition.

In my case, I actually know that $\sigma \in L^1$ is weakly closed and weakly co-closed (which is stronger than being weakly harmonic, if $M$ is not closed), but I am not sure if it should make the proof easier. For completeness, these are the definitions I am using for these properties:

$$ \text{ weakly closed if } \, \int_{\M} \langle \sigma, \delta \alpha \rangle =0 \, \text{ for every compactly supported $\alpha \in \Omega^{k+1}(\M)$}, $$

$$ \text{ weakly co-closed if } \, \int_{\M} \langle \sigma ,d \alpha \rangle=0 \, \text{ for every compactly supported $\alpha \in \Omega^{k-1}(\M)$}, $$

distributionsare smooth functions. $\endgroup$ – Willie Wong Aug 28 '18 at 13:412more comments