Spectral norm

1. The $H^1$ calculation (concrete)

Let $\{\varphi_j\}_{j\ge0}$ be an orthonormal $L^2$ -basis of eigenfunctions of $-\Delta$ with eigenvalues $\lambda_j\ge0$ :

$-\Delta\varphi_j=\lambda_j\varphi_j,\qquad\langle\varphi_j,\varphi_\ell\rangle_{L^2}=\delta_{j\ell}.$

Expand $w=\sum_j w_j\varphi_j$ (convergence in $L^2$ ). Then

$\|(1-\Delta)^{1/2}w\|_{L^2}^2 =\sum_j \big(1+\lambda_j\big)\,|w_j|^2,$

because $(1-\Delta)^{1/2}\varphi_j=(1+\lambda_j)^{1/2}\varphi_j$ and Parseval / Plancherel give the squared $L^2$ -norm as the sum of squared coefficients.

So the spectral sum with weight $(1+\lambda_j)$ is exactly the squared $L^2$ -norm of $(1-\Delta)^{1/2}$ .

Now the usual Sobolev $H^1$ norm on a compact manifold is equivalent to the graph norm of $(1-\Delta)^{1/2}$ ; concretely there exist constants $c,C>0$ depending only on the manifold so that

$c\;\|(1-\Delta)^{1/2}w\|_{L^2}\le \|w\|_{H^1}\le C\;\|(1-\Delta)^{1/2}w\|_{L^2}.$

Combining this with the identity above yields

$\|w\|_{H^1}^2 \simeq \sum_j (1+\lambda_j)\,|w_j|^2,$

which is the $H^1$ instance of the claim.

(Why is the equivalence true? On a compact manifold $\|w\|_{H^1}^2$ is $\|w\|_{L^2}^2+\|\nabla w\|_{L^2}^2$ . Using the eigen-expansion one checks $\|\nabla w\|_{L^2}^2=\sum_j\lambda_j|w_j|^2$ . Hence $\|w\|_{H^1}^2=\sum_j(1+\lambda_j)|w_j|^2$ up to the choice of normalization and conventions — so in fact for $H^1$ one often gets equality (no constants) when you identify norms appropriately.)

2. The general $H^k$ (spectral functional calculus)

Exactly the same spectral computation works for any real $k\ge0$ . By functional calculus

$(1-\Delta)^{k/2}\varphi_j=(1+\lambda_j)^{k/2}\varphi_j,$

so for $w=\sum_j w_j\varphi_j$ ,

$\|(1-\Delta)^{k/2}w\|_{L^2}^2 =\sum_j (1+\lambda_j)^{k}|w_j|^2.$

Thus the right-hand spectral sum is exactly the squared $L^2$ -norm of $(1-\Delta)^{k/2}w$ .

3. Equivalence of $(1-\Delta)^{k/2}$ -graph norm and the usual Sobolev $H^k$ norm

What remains is to explain why the graph norm $\|(1-\Delta)^{k/2}w\|_{L^2}$ is equivalent to the standard Sobolev norm $\|w\|_{H^k}$ . On a compact manifold this is standard:

The operator $1-\Delta$ is a positive, elliptic, self-adjoint pseudodifferential operator of order 2. Its (fractional) powers $(1-\Delta)^{k/2}$ are elliptic pseudo-differential operators of order k.
Elliptic regularity (or basic pseudodifferential calculus) implies that the graph norm of any positive elliptic operator of order k is equivalent to the standard Sobolev $H^k$ -norm. Concretely, there exist constants $c,C>0$ (dependent only on the geometry of $S$ and $k$ ) such that for all $w\in C^\infty(S)$ , $c\;\|(1-\Delta)^{k/2}w\|_{L^2} \le \|w\|_{H^k} \le C\;\big(\|w\|_{L^2}+\|(1-\Delta)^{k/2}w\|_{L^2}\big).$ On a compact manifold the two-sided inequality can be tightened (by adjusting constants) to show full equivalence (no extra low-order term needed if you use $(1-\Delta)^{k/2}$ itself as the defining operator).

If you prefer an elementary (but longer) route: cover the manifold with finitely many coordinate charts, use a partition of unity, and compare the local Euclidean Sobolev norms defined by derivatives up to order k with the global spectral norm. The finiteness of the cover and standard elliptic estimates give the equivalence constants.