I have a compactly supported smooth function $g$ on the $(d-1)$-dimensional sphere $\Bbb S^{d-1}$ (thus $g$ is just smooth, since the sphere is compact) and I need to show that $$\int_{\Bbb S^{d-1}} \Delta_{\Bbb S^{d-1}} g(\Omega)\ j(\Omega) d^{d-1}\Omega = 0$$ where $\Delta_{\Bbb S^{d-1}}$ is the spherical Laplacian (angular part of the Laplacian in $\Bbb R^d$) and $j(\Omega)$ is a complete appropriate jacobian determinant.
Since the spherical harmonics are an orthonormal system in $C^\infty(\Bbb S^{d-1})$, one could perhaps decompose $g$ into $\sum_{n\in\Bbb N} c_ng_n$, with $(g_n)_n$ an appropriate sequence of solutions of $\Delta_{\Bbb S^{d-1}} g_n = 0$: if the series symbol is allowed to pass through $\Delta_{\Bbb S^{d-1}}$ that would conclude the proof – but I'm skeptical, because it would mean that every smooth function on the sphere is harmonic. Under what hypotheses is the integral $= 0$? Am I taking the wrong route?
Edit. I somehow confused spherical harmonics (eigenfunctions of the spherical laplacian) with harmonic functions (elements of its kernel). My argument was invalid altogether. Any thoughts?