I recently stumbled across the Hodge decomposition theorem, which states that on any compact orientable manifold, for any form the following holds $$ \omega = \text{d}\alpha + \delta \beta + \gamma $$ Where $\alpha,\beta,\gamma$ are uniquely determined forms, $d$ is the exterior derivative, and $\delta = \star d \star$ the Hodge coderivative. What i don't understand is whether there exist a similar decomposition on non compact manifolds, such as $\mathbb{R}^n$, and if that is the case, if we can find $\alpha$ by solving the following equation(in the case $\omega$ is a 1-form), $$ \Delta \alpha = (\delta d + d \delta) \alpha = \delta \omega $$ The equations above works for 1-forms if the Hodge decomposition holds, since $$ \Delta \alpha = (\delta d + d \delta) \alpha =\delta d\alpha = \delta \omega \overset{\text{Hodge}}{=} \delta d\alpha $$
I would like to understand if this way of finding $\alpha$ can work on non-compact spaces.