Equivalent descriptions of Sobolev spaces on compact manifolds

Question

While reading through a set of lectures on the Laplacian on manifolds, I encountered two descriptions of Sobolev spaces.

The first one, valid only for compact manifolds (because it needs to globalize a result using partitions of unity) claims that the Sobolev space $H^s$ can be defined as the completion in $L^2$ of $\mathcal C ^\infty$ under the norm $\| f \| = \left(\sum \limits _i \| (p_i f) \circ h_i ^{-1} \| ^2 _{H^s (U_i)}\right) ^\frac 1 2$, where $\{ (U_i, h_i) \mid i \in I\}$ is an atlas and $\{ p_i \mid i \in I\}$ is a subordinated partition of unity.

The second one, which works only for natural orders $k$, presents $H^k$ as the completion in $L^2$ of $\mathcal C ^\infty$ under the norm $\| f \| _k = \left( \| f \| _{k-1} ^2 + \| \nabla ^k f \| ^2 \right) ^\frac 1 2$. (This description is, in turn, shown to be equivalent to a third one that uses the eigenvalues of the Laplacian.)

My question is: do the first two constructions above produce the same space?

To make things worse, Grigoryan defines the same spaces slightly differently: only for even orders, $W^k = \{ u \in \mathcal D ' \mid u, \Delta u, \dots, \Delta ^k u \in L^2\}$. Is this yet another space?

Why isn't $H^k$ defined simply as $\{ u \in L^2 \mid X_1 \dots X_i u \in L^2 \forall i \le k \forall X_j \in \Gamma(TM) \}$?

Answer 1

My question is: do the first two constructions above produce the same space?

Yes, these two constructions produce the same space.

Why isn't $H^k$ defined simply as $ \{ u \in L^2 \mid X_1 \dots X_i u \in L^2 \forall i \le k \forall X_j \in \Gamma(TM) \} $?

To be more precise, even this third construction you have cited will also produce the same space.

After pondering about it for a few days, I believe I have succeeded in cooking up a proof after considering the outline of a stronger version of it which presented in the third chapter of the paper Sobolev spaces on Lie manifolds (...).

This equivalence is rather tricky.

We shall prove this result in a slightly more general setting, that of the Sobolev space $W^{k,p}(M)$ where $k$ is a non-negative integer and $p \in [1,\infty[$.

I'll first fix the notation.

Let $(M^n,g)$ be a compact Riemannian manifold with $\nabla$ its associated Levi-Cività connection.

Fix $\tilde{\mathcal{U}}$ a finite smooth atlas for $M$.

Fix $\ \mathcal{U}=\{ (U_r, \psi_r) : 1 \leq r \leq N\}$ a finite smooth atlas for $M$ such that $$\forall r=1,...,N \ \exists (\tilde{U},\tilde{\psi}) \in \tilde{\mathcal{U}} \ \left( \overline{U_r} \subset \tilde{U} \ \& \ \psi_r = \left. \tilde{\psi} \right|_{U_r} \right)$$

Fix $\{ \rho_r : 1 \leq r \leq N\}$ a partition of unity strictly subordinate to $\mathcal{U}$.

Given $r=1,...,N$, we define $V_r=\psi_r(U_r) \subset \mathbb{R}^n$.

Given $u \in C^\infty(M)$, we define

$$ ||u||^p_{W^{k,p}} = \sum_{l=0}^k \int_M |\nabla^l u|^p d\mu_g $$

$$ \lambda(u)^p=\sum_{r=1}^N ||\rho_r u||^p_{W^{k,p}} $$

$$ \nu_{k,p}(u)^p = \sum_{r=1}^N \lVert (\rho_r u) \circ \psi_r^{-1}\rVert^p_{W^{k,p}(V_r)} $$

We shall prove that those norms are equivalent, hence the completion of $C^\infty(M)$ endowed with any of them yields the same space $W^{k,p}(M)$.

We shall procede in two steps which employ (almost) the same tricks: first, we prove that $|| \cdot ||_{W^{k,p}}$ is equivalent to $\lambda$, then that $\lambda$ is equivalent to $\nu_{k,p}$.

First of all, $\{ |\nabla^l \rho_r| : 1 \leq r \leq N; \ 0 \leq l \leq k \}$ is a finite set of continuous functions in the compact space $M$. Therefore, there is $C>0$ such that

$$ \forall r=1,...,N \ \forall l=0,...,k \ \left( ||\nabla^l \rho_i||_{\infty} \leq C \right) $$

Let $L>0$ be such that $$ \forall l=1,...,k \ \left( || \cdot ||_{1, \mathbb{R}^{l}} \leq L || \cdot ||_{p, \mathbb{R}^{l}} \right) $$

Let $$ K = {{k}\choose{\lfloor k/2 \rfloor}} $$

Fix $u \in C^\infty(M)$, $r \in \{1,...,N\}$ and $l \in \{0,...,k\}$.

\begin{align*} \lvert \nabla^l (\rho_r u) \rvert &= \lvert \sum_{m=0}^l {{l}\choose{m}} \nabla^m \rho_r \otimes \nabla^{l-m}u \rvert \\ &\leq \sum_{m=0}^l {{l}\choose{m}} \lvert \nabla^m \rho_r \rvert \lvert \nabla^{l-m}u \rvert \\ &\leq CLK \left( \sum_{m=0}^l \lvert \nabla^m u \rvert^p \right)^{1/p} \end{align*}

We considered a generic $l$ in $\{0,...,k\}$, hence \begin{align*} \lVert \rho_r u \rVert^p_{W^{k,p}} &\leq (CLK)^p \sum_{l=0}^k \int_M \sum_{m=0}^l \lvert \nabla^m u \rvert^p \\ &\leq (CLK)^p \sum_{l=0}^k \lVert u \rVert_{W^{l,p}}^p \\ &\leq (k+1) (CLK)^p \lVert u \rVert_{W^{k,p}}^p \end{align*}

We considered a generic $r$ in $\{1,...,N\}$, hence $$ \lambda(u)^p \leq A \lVert u \rVert_{W^{k,p}}^p $$ where $A=(k+1) N (CLK)^p$.

We obtained our first inequality, so we're halfway there.

Let $T>0$ be such that $|| \cdot ||_{p, \mathbb{R}^N} \geq T || \cdot ||_{1, \mathbb{R}^N}$.

We then obtain \begin{align*} \lambda(u) &= \left( \sum_{r=1}^N \lVert \rho_r u \rVert^p_{W^{k,p}} \right)^{1/p} \\ &\geq T \sum_{i=1}^N \lVert \rho_r u \rVert_{W^{k,p}} \\ &\geq T \lVert \sum_{i=1}^N \rho_r u \rVert_{W^{k,p}} \\ &\geq T \lVert u \rVert_{W^{k,p}} \end{align*}

That is, we have established our first equivalence of norms.

To repeat all that we have done in the first step, we need to analyse the local form of the covariant derivatives of functions $u \in C^\infty(M)$.

Let $k$ be a positive integer, $l \in \{1,...,k\}$ and $(\tilde{U}, \tilde{\varphi}) \in \tilde{\mathcal{U}}$.

Then there exists a set indexed by multi-indices $\alpha$ $$ \{ P_\alpha : 1 \leq |\alpha| \leq k \} \subset C^\infty(\tilde{U}) $$ such that the $k$th covariant derivative of $u \in C^\infty(M)$ can be locally written at $\tilde{U}$ as $$ \nabla^k u = \sum_{1 \leq \lvert \alpha \rvert \leq k} (D^{\alpha} u) P_{\alpha} \ d x_{i_1} \otimes ... \otimes d x_{i_l} $$ where $\alpha=(i_1,...,i_l)$ is a multi-index.

This remark can be easily proved with an induction.

For each chart $(U_r,\psi_r)$, the $P_\alpha$s are continuous functions which have bounded covariant derivatives, so we can repeat the arguments for the first equivalence of norms.