Laplacian and Hodge Laplacian I am new to the theory of differential forms, but there is one thing that I don't get at all. Imagine that you are on the sphere $\mathbb{S}^2$, then the Laplacian $- \Delta$ is known to be a continuous operator $L(W^k_2,W^{k-2}_2),$ where $W$ is the appropriate Sobolev space.
Now, I found that there is also the concept of a De Rham Laplacian for manifolds, as you can see here for example on wikipedia.
At this point, I read now that this one maps $k-$forms to $k-$forms. So how is this operator different from the Laplacian that I defined in the first place?- I thought that this definition is  a generalization of the Laplacian, but I can't see how these $k-$forms are now related to the Sobolev space.
So is this Hodge Laplacian something completely different or is this one related to my operator defined in the first paragraph?
 A: The first Laplacian you mention (sometimes called the Laplace-Beltrami operator) acts on scalar functions, that is, functions $S^2 \to \mathbb{R}$. The de Rham (a.k.a. Hodge) Laplacian acts on differential forms. In particular, the de Rham Laplacian acts on zero-forms, which are precisely scalar functions $S^2 \to \mathbb{R}$, on which it agrees with the Laplace-Beltrami operator. That is the sense in which the latter Laplacian is a generalization of the former.
One can also study Sobolev spaces of differential forms (or of sections of more general vector bundles), and a statement similar to what you said holds for more general Laplace operators. And of course, all of these things can be studied on Riemannian manifolds other than $S^2$.
But before one learns about Sobolev spaces of differential forms, it is important to understand the basic theory of smooth ($C^\infty$) differential forms (i.e., the usual things that people refer to as differential forms). Then one can start thinking about Laplace operators, etc. The most natural way to view the de Rham Laplacian (or any differential operator on forms, at least at first) is as an operator that maps smooth differential forms to smooth differential forms. After you understand that, you can think about its properties with respect to Sobolev spaces.
One good source for this is Rosenberg's book The Laplacian on a Riemannian Manifold, which is available as a pdf on his website.
