# Extension of Vector Field in the $\mathcal{C}^r$ topology

This question was previously posted on MSE.

Let $$M\subset \mathbb{R}^n$$ be a compact smooth manifold embedded in $$\mathbb{R}^n$$, we define $$\mathfrak{X}(M) := \{X: M \to \mathbb{R}^n;\ X\mbox{ is smooth and }\ X(p) \in T_p M \subset \mathbb{R}^n,\ \forall\ p \in M \}.$$

Choosing an atlas $$\{(\varphi_i,U_i)\}_{i=1}^{n}$$, and compacts $$K_i \subset U_i$$, such that $$\bigcup_{i=1}^n K_i = M,$$ we define the $$\|\cdot \|_r$$ norm as

\begin{align*}\|\cdot\|_r : \mathfrak{X}(M)&\to \mathbb{R}\\ X &\to \max_{\substack{i\in\{1,...,n\} \\ j\in \{0,...,r\}}}\left\{\sup_{x \in \varphi^{-1}_i(K_i)}\left\| \text{d}^{j}\left( X\circ\varphi_i \right) \right\|\right\}, \end{align*}

then we named $$\mathfrak{X}^r(M)$$ as the complete Banach space $$(\mathfrak{X}(M),\|\cdot\|_r)$$ (it is possible to prove that the topology of $$\mathfrak{X}^r(M)$$ does not depend on the selected atlas).

My Question: Let $$X \in \mathfrak{X}(M)$$ and $$Y$$ be a smooth vector field on $$M$$ defined just in a compact $$K \subset M$$ such that $$\max_{\substack{i\in\{1,...,n\} \\ j\in \{0,...,r\}}}\left\{\sup_{x \in \varphi^{-1}_i(K_i\cap K)}\left\| \text{d}^{j}\left( X\circ\varphi_i \right) - \text{d}^{j}\left( Y\circ\varphi_i \right) \right\|\right\}<\varepsilon,$$ is it possible extend $$Y$$ to a vector field $$\tilde{Y}$$ such that

1) $$\left.\tilde{Y}\right|_{K} = Y$$,

2) $$\|X-\tilde Y\|_r < A\cdot\varepsilon$$ , where $$A$$ is a constant that depends only on $$\text {dim }M$$ ?

The compact $$K$$ is a connected submanifold with boundary of $$M$$, such that $$\dim K = \dim M$$.

Edit: I changed $$\|X-\tilde Y\|_r < \varepsilon$$ to $$\|X-\tilde Y\|_r < A\cdot\varepsilon$$ after Moishe Kohan's comment on MSE.

## My ideas

First, I extend $$Y$$ by a smooth vector field $$Z$$ $$\in \mathfrak{X}(M)$$, by the continuity of $$Z$$, so there exists a neighborhood $$U$$ of $$K$$, such that $$\max_{\substack{i\in\{1,...,n\} \\ j\in \{0,...,r\}}}\left\{\sup_{x \in \varphi^{-1}_i(K_i\cap U)}\left\| \text{d}^{j}\left( X\circ\varphi_i \right) - \text{d}^{j}\left( Z\circ\varphi_i \right) \right\|\right\}<\varepsilon,$$

and then choosing a partition of unity $$\{\phi_1, \phi_2\}$$ subordinate to the cover $$\{U,M\setminus K\}$$ we can define $$\tilde{Y} = \phi_1 Z + \phi_2 X,$$ however I could not guarantee that $$\|X - \tilde{Y}\|_r < \varepsilon$$, because I can not control de derivatives of $$\phi_1$$ and $$\phi_2$$. Does anyone know how I should proceed?

• This is a local question, so it reduces to extending the components of the vector field with respect to local coordinates, which are just functions. Stein’s book, Singular Integrals and Differentiability Properties of Functions, has results on this. – Deane Yang Mar 25 at 1:41
• I will check it out, thx for the reference. – Matheus Manzatto Mar 25 at 1:47
• @DeaneYang Do you have any vague recollections or do you know which theorem I am looking for or where should I look for it in the book? – Matheus Manzatto Mar 25 at 2:24
• Look for extension theorems. He does them for both Holder and Sobolev spaces. – Deane Yang Mar 25 at 12:03