Fractional Sobolev metrics on spaces of immersed curves

Abstract

Motivated by applications in the field of shape analysis, we study reparametrization invariant, fractional order Sobolev-type metrics on the space of smooth regular curves $\mathrm {Imm}(\mathrm {S}^{1},\mathbb {R}^d)$ and on its Sobolev completions ${\mathcal {I}}^{q}(\mathrm {S}^{1},{\mathbb {R}}^{d})$. We prove local well-posedness of the geodesic equations both on the Banach manifold ${\mathcal {I}}^{q}(\mathrm {S}^{1},{\mathbb {R}}^{d})$ and on the Fréchet-manifold $\mathrm {Imm}(\mathrm {S}^{1},\mathbb {R}^d)$ provided the order of the metric is greater or equal to one. In addition we show that the $H^s$-metric induces a strong Riemannian metric on the Banach manifold ${\mathcal {I}}^{s}(\mathrm {S}^{1},{\mathbb {R}}^{d})$ of the same order s, provided $s>\frac{3}{2}$. These investigations can be also interpreted as a generalization of the analysis for right invariant metrics on the diffeomorphism group.

Appendices

Appendix A: Smooth curves in ${\mathcal {S}}^{r}(\mathbb {Z})$

The right framework to work with smooth maps on infinite dimensional (other than Banach spaces) are convenient vector spaces (see [19, 25]). Note that every Fréchet space is a convenient vector space. We will need the following lemma about recognising smooth curves in convenient vector spaces.

Lemma A.1

[19, 4.1.19] Let $c{:}\; \mathbb {R}\rightarrow E$ be a curve in a convenient vector space. Let ${\mathcal {V}} \subseteq E'$ be a point-separating subset of bounded linear functionals such that the bornology of E has a basis of $\sigma (E, {\mathcal {V}})$-closed sets. Then the following are equivalent:

• (1) c is smooth;

• (2) There exist locally bounded curves $c^k{:}\; \mathbb {R}\rightarrow E$ such that $\ell \circ c {:}\; \mathbb {R}\rightarrow \mathbb {R}$ is smooth with $(\ell \circ c)^{(k)} = \ell \circ c^k$, for each $\ell \in {\mathcal {V}}$.

To apply this lemma to the space ${\mathcal {S}}^{r}(\mathbb {Z})$ of Fourier multipliers we need to choose a suitable set ${\mathcal {V}}$ of linear functionals. This is accomplished in the following lemma. For $m \in \mathbb {Z}$, let

$$\begin{aligned} {\text {ev}}_m {:}\; {\mathcal {S}}^r(\mathbb {Z}) \rightarrow \mathcal L(\mathbb {C}^d),\quad \mathbf a \mapsto \mathbf a(m), \end{aligned}$$

denote the evaluation mapping.

Lemma A.2

Let $r \in \mathbb {R}$ and ${\mathcal {V}} = \{ \lambda \circ {\text {ev}}_m {:}\; m \in \mathbb {Z},\, \lambda \in {\mathcal {L}}(\mathbb {C}^d)' \} \subset {\mathcal {S}}^{r}(\mathbb {Z})'$. Then the bornology of ${\mathcal {S}}^{r}(\mathbb {Z})$ has a basis consisting of $\sigma ({\mathcal {S}}^{r}(\mathbb {Z}), {\mathcal {V}})$-closed sets.

Proof

We can regard ${\mathcal {V}}$ as a subset of both ${\mathcal {S}}^{r}(\mathbb {Z})'$ as well as $\ell ^\infty (\mathbb {Z}, {\mathcal {L}}(\mathbb {C}^d))'$. It is shown in [19, 4.1.21] that the bornology of $\ell ^\infty (\mathbb {Z}, {\mathcal {L}}(\mathbb {C}^d))$ has a basis of $\sigma (\ell ^\infty (\mathbb {Z}, {\mathcal {L}}(\mathbb {C}^d)), {\mathcal {V}})$-closed sets. We embed ${\mathcal {S}}^{r}(\mathbb {Z})$ as a closed subspace into

$$\begin{aligned} \iota {:}\; {\mathcal {S}}^{r}(\mathbb {Z}) \rightarrow \prod _{\alpha \in \mathbb {N}} \ell ^\infty (\mathbb {Z}, {\mathcal {L}}(\mathbb {C}^d)),\, \left( {\mathbf {a}}(m)\right) _{m\in \mathbb {Z}} \mapsto \left( \langle m \rangle ^{-(r-\alpha )} \Delta ^{\alpha } {\mathbf {a}}(m) \right) _{m \in \mathbb {Z},\, \alpha \in \mathbb {N}}. \end{aligned}$$

Because the bornology of the product is the product bornology it follows that a basis for the bornology on $\prod _{\alpha \in \mathbb {N}} \ell ^\infty (\mathbb {Z}, {\mathcal {L}}(\mathbb {C}^d))$ is given by $\sigma (\prod _{\alpha } \ell ^\infty ,{\mathcal {W}})$-closed sets, where ${\mathcal {W}} = \bigcup _{\alpha \in \mathbb {N}} {\mathcal {V}} \circ {\text {pr}}_{\alpha }$ and ${\text {pr}}_{\alpha }$ denotes the canonical projection onto the $\alpha$-th factor. Here we note that on ${\mathcal {S}}^{r}(\mathbb {Z})'$ the set $\iota ^*({\mathcal {W}})$ is contained in the linear space of ${\mathcal {V}}$ and hence the bornology of ${\mathcal {S}}^{r}(\mathbb {Z})$ has a basis consisting of $\sigma ({\mathcal {S}}^{r}(\mathbb {Z}), {\mathcal {V}})$-closed sets. $\square$

Appendix B: Square-root of an operator in ${\mathcal {E}}^{r}_{\lambda }(\mathbb {Z})$

Since a positive definite Hermitian matrix has a unique positive square root, which depends smoothly on its coefficients, we can define formally the square root $B_{\lambda } = {\mathbf {op}}\left( {\mathbf {a}}(\lambda , m)^{1/2}\right)$ of an element $A_{\lambda } = {\mathbf {a}}_{\lambda }(D)$ in the class ${\mathcal {E}}^{r}_{\lambda }(\mathbb {Z})$. In order to prove this we need the following lemma.

Lemma B.1

[5, Lemma 4.8] Let $a, b, x \in {\mathcal {L}}(\mathbb {C}^d)$ be three matrices satisfying

$$\begin{aligned} bx + xb = a, \end{aligned}$$

with b Hermitian and positive definite. Then

$$\begin{aligned} \left\| x\right\| \le \sqrt{\frac{d}{2}} \left\| b^{-1}\right\| \left\| a\right\| , \end{aligned}$$

where $\left\| \cdot \right\|$ denotes the Frobenius norm, i.e. $\left\| x\right\| = \sqrt{{\text {tr}} xx^*}$.

The following lemma together with its proof is a generalisation of [5, Lemma 4.7] to our situation of one-parameter families of symbols.

Lemma B.2

The positive square root of an operator in the class ${\mathcal {E}}^{r}_{\lambda }(\mathbb {Z})$ belongs to the class ${\mathcal {E}}^{r/2}_{\lambda }(\mathbb {Z})$. Conversely, the square of an operator in the class ${\mathcal {E}}^{r}_{\lambda }(\mathbb {Z})$ belongs to the class ${\mathcal {E}}^{2r}_{\lambda }(\mathbb {Z})$.

Proof

We will prove the estimate

$$\begin{aligned} \left\| \partial ^\beta _{\lambda } \Delta ^{\alpha } {\mathbf {b}}(\lambda , m)\right\| \lesssim \langle m\rangle ^{r/2-\alpha }, \end{aligned}$$

which holds locally uniformly in $\lambda$, by induction over $\alpha + \beta$. If $\alpha + \beta = 0$ the statement is $\Vert {\mathbf {b}}(\lambda , m) \Vert \lesssim \langle m \rangle ^{r/2}$. Assume that it has been proven for $\alpha + \beta \le k$. Then let $\alpha + \beta = k+1$ and, omitting the arguments $(\lambda , m)$, we obtain using the product rule,

$$\begin{aligned} \partial ^\beta _{\lambda } \Delta ^{\alpha } {\mathbf {a}} = {\mathbf {b}}\left( \partial ^\beta _{\lambda } \Delta ^{\alpha } {\mathbf {b}}\right) + \left( \partial ^\beta _{\lambda } \Delta ^{\alpha } {\mathbf {b}} \right) {\mathbf {b}} + \sum _{(i,j) \in X} \left( {\begin{array}{c}\alpha \\ i\end{array}}\right) \left( {\begin{array}{c}\beta \\ j\end{array}}\right) \partial ^{j}_{\lambda } \Delta ^i {\mathbf {b}}\, \partial ^{\beta -j}_{\lambda } \Delta ^{\alpha -i} {\mathbf {b}}, \end{aligned}$$

with

$$\begin{aligned} X = \left\{ (i,j) {:}\; 0 \le i \le \alpha ,\, 0 \le j \le \beta ,\, i + j < \alpha + \beta \right\} . \end{aligned}$$

Then by the induction assumption

$$\begin{aligned} \left\| \partial ^{j}_{\lambda } \Delta ^i {\mathbf {b}}\, \partial ^{\beta -j}_{\lambda } \Delta ^{\alpha -i} {\mathbf {b}}\right\| \lesssim \langle m \rangle ^{r/2-i} \langle m \rangle ^{r/2-\alpha +i} \lesssim \langle m \rangle ^{r-\alpha }, \end{aligned}$$

and hence we obtain via Lemma B.1,

$$\begin{aligned} \left\| \partial ^\beta _{\lambda } \Delta ^{\alpha } {\mathbf {b}}\right\| \lesssim \langle m \rangle ^{-r/2} \langle m \rangle ^{r-\alpha } \lesssim \langle m \rangle ^{r/2-\alpha }. \end{aligned}$$

This completes the induction. $\square$