Let us define the spaces $C_0, C_1$ and $C_2$ of differential $0,1,2$ forms respectively on the sphere $S^2.$
Is it true that $C_1$ is the direct sum of $d(C_0) \oplus \delta^* (C_2)$? I think this is the case, but I'm not sure how to prove it. ($\delta^*$ is the dual operator of $d$, defined via the Hodge star.)
Possibly relevant facts: -since the sphere is simply connected, we know that closed forms are exact (Poincare lemma).
-we know that there are no harmonic 1-forms on the sphere. Hence, given a 1-form $\theta,$ then $d\theta =0$ implies $\delta^* \theta=0.$ If the converse statement were true, then we could use the Poincare lemma to obtain the result.
On one hand, $$C_1=d(C_0)\oplus d(C_0)^{\perp}=d(C_0)\oplus\ker d^*$$ On the other hand, $$C_1=d^*(C_2)\oplus d^*(C_2)^{\perp}=d^*(C_2)\oplus\ker d$$ Since $d(C_0)\subseteq\ker d$, this implies $\ker d=\ker d\cap\left(d(C_0)\oplus\ker d^*\right)=d(C_0)\oplus\left(\ker d\cap\ker d^*\right)$, $\mathrm{H}^1(S^2)=\ker d/ d(C_0)\simeq\ker d\cap\ker d^*$ and $$C_1=d(C_0)\oplus\left(\ker d\cap\ker d^*\right)\oplus d^*(C_2)$$ But, since $S^2$ is simply connected, $\mathrm H^1(S^2)=0$ and hence $$C_1=d(C_0)\oplus d^*(C_2)$$