Reference request: Laplace-Beltrami eigenfunction bases for Sobolev spaces

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!

• Steve Rosenberg's book is the first reference that comes to mind. – user10354138 Apr 12 at 15:20
• 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. – Kweku A Apr 19 at 13:05

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]