Definition of Hölder Space on Manifold

Question

Can anyone point me to a reference for the definition of Hölder spaces for manifolds (with boundary)? Every paper I have looked at says these are "defined in the ordinary way" and no one says what that way is.

Thanks in advance!

Answer 1

The local version of the Hölder condition can be defined the same way integral regularity can be defined; see e.g. Ruelle's Elements of Differentiable Dynamics and Bifurcation Theory, pp.138-151, App. B for a convenient summary and Bonic & Frampton's "Smooth Functions on Banach Manifolds" for a more general study. This seems to me to be the "ordinary way".

For the sake of completeness, here is an outline (the above references cover Banach manifolds as well):

Let $\mathbb{E}$ and $\mathbb{F}$ be locally convex topological vector spaces, let $U\subseteq \mathbb{E}$ be open, and fix $(q,\theta)\in\mathbb{Z}_{\geq0}\times ]0,1]$. Call a function $f:U\to \mathbb{F}\,\,\,$ $\theta$-Hölder at $x\in U$ if there is an open subset $x\in U_x\subseteq U$ such that

$$\exists C=C(U_x)\in\mathbb{R}_{>0},\forall y,z\in U_x: |f(y)-f(z)|_{\mathbb{F}}\leq C\, |y-z|_{\mathbb{E}}^\theta.$$

Call $f:U\to \mathbb{F}$ locally $\theta$-Hölder if it is Hölder at any point of $U$. Denote by $C^{(q,\theta)}(U;\mathbb{F})$ the collection of all locally $\theta$-Hölder functions from $U$ to $\mathbb{F}$.

Similarly, call $f:U\to \mathbb{F}\,\,$ a $C^{(q,\theta)}$ function if it is $q$ times continuously differentiable and its $q$-th derivative is locally $\theta$-Hölder, that is,

$$D^qf:U\to \operatorname{Hom}_{\text{TVS}}(\mathbb{E}^{\otimes q};\mathbb{F})$$

is a continuous map, with the target space being the locally convex topological vector space of all continuous $q$-linear maps from $\underbrace{\mathbb{E}\times \mathbb{E}\times\cdots\times \mathbb{E}}_{q \text{ many}}\,$ to $\mathbb{F}$ with norm

$$\Vert B\Vert_{\mathbb{F}\leftarrow\mathbb{E}^{\otimes q}}=\sup\{|B(x_1,x_2,...,x_q)|_{\mathbb{F}}\,\,\,|\,\,\,x_1,x_2,...,x_q\in \mathbb{E}, |x_1|_{\mathbb{E}}\leq 1,|x_2|_{\mathbb{E}}\leq 1,...,|x_q|_{\mathbb{E}}\leq 1\}.$$

Defining a total order on $\mathbb{Z}_{\geq0}\times]0,1]$ by

$$(k,\alpha)< (l,\beta) \iff [k=l \text{ and } \alpha<\beta] \text{ xor } [k<l],$$

thus in particular

\begin{align*} (0,0)\leq (0,\theta) \leq (0,1)<&(1,0) \leq (1,\theta)\leq (1,1) <(2,0)\cdots \\ < &(q,0)\leq (q,\theta)\leq (q,1) <(q+1,0)<\cdots, \end{align*}

we have that if $(q_1,\theta_1)\leq(q_2,\theta_2)$, then $C^{(q_2,\theta_2)}\subseteq C^{(q_1,\theta_1)}$.

For the manifold case, let $M$ and $N$ be $C^\infty$ (wlog; see Motivations of studying $C^r$ manifolds when $r<\infty$) manifolds of dimension $\dim(M)=m\in\mathbb{Z}_{\geq0}$ and $\dim(N)=n\in\mathbb{Z}_{\geq0}$, respectively. Then call a function $f:M\to N\,\,$ a $C^{(q,\theta)}$ function if

$$\forall (U,\phi)\in\operatorname{Chart}(M),\forall (V,\psi)\in \operatorname{Chart}(N): f(U)\cap V\neq\emptyset\implies\psi\circ f\circ \phi^{-1}\in C^{(q,\theta)}(\phi(U); \mathbb{R}^n).$$

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It seems to me global Hölder conditions are varied; in any event one needs some extra conditions; e.g. if $M$ is compact one can optimize the constants of $f$ coming from local Hölder conditions over a finite cover. Another common alternative in the compact case is to fix a $C^\infty$ Riemannian metric on $M$ and an isometric embedding $M\hookrightarrow \mathbb{R}^{d(M)}$ for some $d(M)\in\mathbb{Z}_{\gg0}$ and use the extrinsic Euclidean distance to define global Hölder continuity (see e.g. Brin & Stuck's Introduction to Dynamical Systems (p.143), or Brin's "Hölder Continuity of Invariant Distributions" for this perspective (though in these works local Hölder estimates are all that is needed)).

Alternatively one can choose a $C^0$ fiberwise norm $\mathfrak{q}$ on $M$ (in particular one can take a $C^\infty$ Riemannian metric) and consider the (generalized) intrinsic distance function associated to it. Explicitly, if $\mathfrak{q}:TM\to\mathbb{R}_{\geq0}$ is continuous such that $\forall x\in M: \mathfrak{q}_x=\mathfrak{q}|_{T_xM}:T_xM\to \mathbb{R}_{\geq0}$ is a norm, then

\begin{align*} d_\mathfrak{q}:M\times M&\to [0,\infty],\,\,\\(x,y)&\mapsto\begin{cases} \inf\{L_\mathfrak{q}(\gamma)\,|\, \gamma:([0,1],0,1)\to (M,x,y) \text{ is piecewise }C^\infty\},&\text{ if } x,y \text{ are in the same connected component}\\ \infty&\text{, otherwise} \end{cases}, \end{align*}

where $L_\mathfrak{q}(\gamma)=\int_0^1 \mathfrak{q}_{\gamma(t)}(\gamma'(t))\, dt$ is the length structure associated to $\mathfrak{q}$. For the global Hölder continuity of a function $f:M\to N$ one would need to fix a $C^0$ fiberwise norm on both $M$ and $N$. In this case one could define $f:M\to N$ to be globally $\theta$-Hölder (w/r/t fiberwise norms $\mathfrak{q}^M$ and $\mathfrak{q}^N$) if

$$\exists C\in\mathbb{R}_{>0},\forall y,z\in M: d_{\mathfrak{q}^N}(f(y),f(z))\leq C\, d_{\mathfrak{q}^M}(y,z)^\theta.$$

Defining $C^{(q,\theta)}(M;N)$ for $q\geq 1$ becomes even less canonical; however see Hebey & Robert's "Sobolev Spaces on manifolds" (p.389) (or Lee & Parker's "The Yamabe Problem" (p.44)) for the case when $N=\mathbb{R}$ and $M$ is endowed with a $C^\infty$ Riemannian metric.

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Even though I have never seen this one used, a final probable definition would be via a natural metric (e.g. the Sasaki metric, see e.g. Sasaki Metric for Finsler Manifolds, Riemannian metric of the tangent bundle, Is there a natural connection on $TM$, Why is the Sasaki metric natural?, Completeness of the tangent bundle of riemannian manifold) on the tangent bundle $TM$ induced by a $C^1$ Riemannian metric on $M$; one could then e.g. define $f:M\to N$ globally $C^{(1,\theta)}$ if ($f:M\to N$ is globally $\theta$-Hölder continuous and) $Tf:TM\to TN$ exists and is globally $\theta$-Hölder continuous with respect to the induced natural metrics. Adapting this idea to jet bundles one could define globally $C^{(q,\theta)}$ functions as functions whose $q$-th jets exist and are globally $\theta$-Hölder continuous (see e.g. Higher-order derivatives in manifolds, Higher order differential on a manifold, connections, https://mathoverflow.net/q/222020/66883).

Alternatively, one could fix $C^q$ Riemannian metrics on $M$ and $N$ and consider the global $C^{(q,\theta)}$ regularity of $f$ by saying that for each $i\in\mathbb{0,1,...,q}:$$T^if: T^iM\to T^iN$ exists and is globally $\theta$-Hölder continuous w/r/t the Sasaki distances on $T^iM$ and $T^iN$ induced by the induced Sasaki metrics. Admittedly it's not clear to me, as I'm writing this answer, if these constructions are useful. Here I'm considering $T$ as a monad $T:\text{Man}^\infty\to \text{Man}^\infty$ (see Jubin's "The Tangent Functor Monad and Foliations" at https://arxiv.org/abs/1401.0940): $T^0(M)=M,T^0(f)=f, T^i(M)=T(T^{i-1}(M)), T^i(f)=T(T^{i-1}(f))$.

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Finally, going back to the local version, one could consider $C^{(q,\theta)}$-jets in tandem with $C^{(q,\theta)}$ functions (see Hirsch, Pugh & Shub's Invariant Manifolds (p.26) for a definition of Lipschitz jets; it's straightforward to adapt this to $C^{(q,\theta)}$-jets and develop the theory following Hirsch's Differential Topology (p.60)).