Sobolev space on closed surfaces

Question

I was wondering if anybody here knows how the Sobolev space $H^2(\mathbb{S}^2)$ is defined? I.e. I want to integrate on this space with respect to the surface measure, but since this not the canonical case, I could not find a proper definition of this space.

Answer 1

The most consistent way that I've seen $L^{2}$ Sobolev spaces handled on (relatively)compact manifolds without boundary--and keep in mind that I'm way out of my element when dealing with manifolds--is through domains of powers of the Laplacian or through some similar elliptic PDE. The first Sobolev space, for example, would come from the form domain, which is the closure of $\mathcal{C}^{\infty}$ under the form $$ \|\phi\|^{2}_{H^{1}}=(-\Delta \phi,\phi)+(\phi,\phi). $$ This assumes that $-\Delta$ is essentially selfadjoint on the manifold, meaning that the closure of the graph is a selfadjoint operator $A=-\Delta^{c}$ (c for closure.) Then $H^{n}(M)=\mathcal{D}(A^{n/2})$ is defined in terms of the domain of powers of $A$. For a positive unbounded selfadjoint $A$, $$ A = \int_{0}^{\infty}\lambda dE(\lambda) $$ where $E$ is the spectral measure of $A$. Of course for the sphere $E$ is atomic, which reduces everything to a sum: $$ A = \sum_{n}\lambda_{n}E_{n} $$ In this case, $\lambda_{n}$ are the eigenvalues of the Laplacian on the sphere, and $E_{n}$ is the projection operator on the Spherical Harmonics with the common eigenvalue $\lambda_{n}$. Domains are easily characterized by the spectral theorem. $f \in L^{2}(M)$ is in the domain of $A$ iff $$ \int \lambda^{2}d\|E(\lambda)f\|^{2} < \infty. $$ Similarly, $f \in \mathcal{D}(A^{1/2})=H^{1}(M)$ iff $\int \lambda d\|E(\lambda)f\|^{2} < \infty$. (No absolute value is required on $\lambda$ because $\sigma(A)\subseteq [0,\infty)$.) More generally $f \in \mathcal{D}(A^{n/2})$ iff $\int \lambda^{n} d\|E(\lambda)f\|^{2} < \infty$. Using the spherical harmonics, $$ f \in H^{n}(\mathbb{S}^{2}) \iff \sum_{l,m}\lambda_{l}^{n}|(f,Y_{l,m})|^{2} <\infty $$ This may seem odd or like cheating, but really this is the same as the common cases.

To relate this to the classical case on $\mathbb{R}^{n}$, note that $-\Delta$ is essentially selfadjoint and the closure $A$ from the domain of $\mathcal{C}^{\infty}_{0}$ is a positive operator. The Fourier transform gives the spectral representation for $A$: $$ Af = \mathcal{F}^{-1}(\lambda^{2}\mathcal{F}f) $$ Now assume you start with $f \in L^{2}$. Then $f \in\mathcal{D}(A^{n/2})$ iff $$ \int_{\mathbb{R}^{n}}|\mathcal{F}f(s)|^{2}|s|^{2n}\,ds < \infty. $$ And this is correct in the strictest Functional Analysis sense. Rather than the separate check that $f \in L^{2}$, it is standard to require $(1+|s|)^{n}\mathcal{F}f(s) \in L^{2}$, and this becomes an equivalent norm. (And there are various equivalent expressions such as $(1+|s|^{2})^{n/2}$.) Notice that if such a condition is met, then it is automatically met for all lower $n$ as well, which is one of the odd qualities of Sobolev spaces. Even the mixed terms are finite for $n=2$ because $2|s_j s_k| \le s_j^{2}+s_k^{2} \le |s|^{2}$, which is another peculiarity of the Laplacian. The Laplacian shares this property with other elliptic operators.

Anyway, the abstraction to manifolds seems to work well when dealing with the problem as a spectral problem for a positive elliptic operator of some kind. This makes sense, too, because the typical reason for using such spaces is to study differential operators. So why not study them on the native spectral spaces of the Laplacian in the classical manner of $\mathbb{R}^{n}$.

Reference for Abstract Sobolev Spaces on Manifolds