For a Kahler manifold $M$ we have two well-known Laplacians: the de Rham Laplacian $\Delta_{\text{d}} = $$\Delta_{\text{d}} = ($d$ + $d$^\ast$$^\ast)^2$, and the Dolbeault Laplacian $\Delta_{\overline{\partial}} = \overline{\partial} + \overline{\partial}^\ast$$\Delta_{\overline{\partial}} = (\overline{\partial} + \overline{\partial}^\ast)^2$. Now on smooth functions, these two operators are related by the well-known formula $$ \Delta_{\text{d}}^2 = 2\Delta_{\overline{\partial}}^2 $$$$ \Delta_{\text{d}} = 2\Delta_{\overline{\partial}} $$ Now both these operators act on the exterior algebra. Does there exist a similar formula in this more general setting?
