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I'm working on a smooth $(d-1)$-dimensional surface $M\subset \mathbb{R}^d$. Let $(\phi_k)_{k\in\mathbb{N}}$ be an orthonormal basis of $L^2(M)$ consisting of the eigenfunctions of the Laplace-Beltrami operator $-\triangle_{M}$, with corresponding eigenvalues $(\lambda_k)_{k\in\mathbb{N}}$.

Claim: the $(\phi_k)$ are $H^1$ orthogonal, where $H^s=W^{s,2}$ is an $L^2$ Sobolev space, and $\lVert\phi_k\rVert_{H^1}^2 = 1+\lambda_k$.

Proof: By a Green's formula on $M$ (a generalisation of the divergence theroem; note I'm assuming $\partial M=\emptyset$) $$\int_M \nabla \phi_j \cdot \nabla \phi_k dx= -\int_M \phi_j \triangle \phi_k dx = \lambda_k \int_M \phi_j \phi_k dx= \lambda_k\delta_{jk}.$$

This generalises to $H^n$ for any positive integer $n$. I'm confident the $\phi_k$ are also $H^s$ orthogonal, with norms $\lVert \phi_k \rVert_{H^s}^2 \sim 1+\lambda_k^s$ (with the exact constants depending on how you define the norm) for any $s\in\mathbb{R}$, defining $H^s$ in some appropriate sense. Does someone have a reference for this? I've not done much differential geometry so ideally one which is fairly gentle in that regard!

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    $\begingroup$ Steve Rosenberg's book is the first reference that comes to mind. $\endgroup$ – user10354138 Apr 12 at 15:20
  • $\begingroup$ Having done some more reading I think the key is how to define $H^s(M)$ for $s\not\in \mathbb{N}\cup\{0\}.$ Strichartz, Analysis of the Laplacian on the complete Riemannian manifold, 1 gives the definition $H^s(M)=(\operatorname{Id}-\triangle)^{-s/2} L^2(M)$ from which the desired characterisation of the norms of the eigenfunctions should follow. $\endgroup$ – Kweku A Apr 19 at 13:05
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One definition of $H^s(M)$ is $$H^s(M)= \{u \in \mathcal{S}(M) : \sum_{j=1}^\infty \lambda_j^{2s} \lvert\langle u, \bar{\phi}_j \rangle \rvert^2\equiv \lVert u \rVert_{H^s(M)}^2<\infty\},$$ where $\mathcal{S}$ is the Schwartz space of distributions on $M$ (i.e. the dual of $C^\infty_c(M)$, which is just $C^\infty(M)$ if $M$ is a compact surface). The bar on $\phi_j$ is the complex conjugate, because strictly we should be using complex scalars and inner products.

For this definition of $H^s$, the given scaling of norms of the $\phi_j$ is obvious. So the real question is does the above definition coincide with other commonly used definitions?, to which the answer is yes, as shown in Lions & Magenes, Non-homogeneous boundary value problems and applications, volume I, chapter I, remark 7.6. [1]

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