Parabolic and Elliptic Schauder Theory on Manifolds for a Fourth-Order Problem
with a First- and a Third-Order Boundary Condition

Jan-Henrik Metsch$^*$

$^*$ Corresponding author: J.-H. Metsch, Department of Mathematics,
University of Freiburg, Germany<EMAIL_ADDRESS>
These notes provide a self-contained introduction to Schauder theory on manifolds. First, we derive Schauder estimates for a fourth-order parabolic linear problem with a first- and third-order boundary condition on a smooth compact manifold $M$. Then, applying these, we prove the existence of solutions and study the associated elliptic problem.

MSC2020:: Primary: 35B45, 35R01, Secondary: 35-01, 35G16, 35J40

Keywords: Schauder Theory, A Priori Estimates, Fourth Order Equation, PDEs on Manifolds, Initial-Boundary Value Problem, Galerkin Approximation, Introductory Exposition

§ THE PURPOSE OF THESE NOTES

In these notes, I present a self-contained derivation of the Schauder estimates for a fourth-order parabolic problem, which I have encountered during my Ph.D. studies. Even though this problem has essentially been studied in the literature, I could not track down a completely self-contained proof in the literature that does not require potential theory. Among other things, it was, therefore, that my supervisor Prof. Dr. Ernst Kuwert, encouraged me to write up a proof myself and include it as an appendix in my dissertation. In the hope that it will be helpful to others, I then wrote this appendix in a form that resembles more lecture notes rather than an article.

Even though some of the proofs I present here are novel, there is probably nothing in these notes that is new to any expert in the field.

Finally, I wish to mention that the thesis `Biharmonischer Wärmefluss' of Tobias Lamm [12] was immensely helpful throughout the writing of these notes. In the first part of his thesis, Lamm follows a similar ambition and establishes Schauder estimates for higher-order problems with Dirichlet boundary conditions. Still, the problem I study here is not covered by Lamm's analysis and therefore needed `its own proof'.

§ INTRODUCTION

Schauder estimates are a priori estimates for solutions of linear partial differential equations (PDEs) that are immensely useful for studying linear and nonlinear problems. The pioneering work, which includes the first formulation and proof of Schauder estimates, is the 1934 article [17] by Juliusz Schauder himself. Given a sufficiently regular domain $\Omega\subset\R^ n$, sufficiently regular functions $a^ {ij}$, $b^ i$ and $c$ as well as  $f\in C^ {0,\gamma}(\Omega)$ and $\varphi\in C^ {2,\gamma}(\partial\Omega)$, he considers the problem

\begin{equation}\nonumber
\begin{array}{ll}
\left\{
\begin{aligned}
a^{ij}\partial_{ij}u+b^i\partial_i u+cu=f&\hspace{.5cm}\textrm{in }\Omega,\\
u=\varphi&\hspace{.5cm}\textrm{along  }\partial\Omega.\\
\end{aligned}
\right.
\end{array}
\end{equation}

Schauder proves that any solution $u\in C^ {2,\gamma}(\Omega)$ satisfies the estimate

\begin{equation}\nonumber
\|u\|_{C^ {2,\gamma}(\Omega)}\leq C\left(\|f\|_{C^ {0,\gamma}(\Omega)}+\|\varphi\|_{C^ {2,\gamma}(\partial\Omega)}+\|u\|_{L^ \infty(\Omega)}\right).
\end{equation}

Without claiming any completeness or historical accuracy, we collect some well-known and classical sources that have generalized the work of Schauder to include general boundary conditions, higher-order problems, systems of PDEs and parabolic problems. First, we mention the articles [1] and [2], in which Agmon Douglis and Nirenberg generalize to systems of elliptic equations with general boundary conditions. Parabolic equations have, for example, been studied in the book of Ladyženskaja, Solonnikov and Ural'ceva [11], the book of Eidelman[6], the book of Eidelman and Zhitarashu [5] and the book of Friedmann [8].

The classical approach to deriving Schauder estimates is based on potential theoretic estimates. A modern approach that replaced these intricate estimates with a blow-up argument has been introduced by Simon in the pioneering article [18].

Simon first considers constant coefficient operators that he assumes to be hypoelliptic (i.e. every solution $u\in L^2_{\operatorname{loc}}$ is smooth). He then generalizes to more general situations by perturbative arguments.

To execute Simon's method, one generally only has to verify a hypoellipticity assumption, which he uses to give an elegant proof of certain Liouville type theorems. To verify hypoellipticity, the go-to strategy is to develop suitable elliptic or parabolic regularity theory for weak solutions. While these theories are very important, these notes aim to develop Schauder theory and not $L^2$-theory. Therefore the proof we give here follows another approach: First, Liouville-type theorems are proven for smooth solutions and then extended to weaker solutions by mollification. Likewise, when discussing existence theory, we avoid introducing the concept of parabolic Sobolev spaces and weak solutions in favor of a Galerkin approximation. Still, even this approach requires the weak theory of the Neumann Laplacian, which we, therefore, also establish.

I want to thank my supervisor Prof. Dr. Ernst Kuwert, for his encouragement to work out these notes, as doing so massively improved my understanding of the subject. Additionally, I want to thank him for his guidance and the many helpful discussions.

§ STRUCTURE OF THESE NOTES

These notes are split into four chapters. In the following, we summarize their contents. Note that some of the notions will be defined later in the text.

[SchauderCh_Ch01FunctionalSpaces]Chapter 1 recalls the definition of Hölder and Sobolev spaces and extends them to functions defined on manifolds. Additionally, the parabolic Hölder spaces $C^{k,[k/4],\gamma}(\Omega\times[0,T])$ are introduced and generalised to the manifold setting. Lastly, we prove various interpolation estimates for functions in  $C^{k,[k/4],\gamma}(\Omega\times[0,T])$.

[SchauderCh_Ch02NeumannLaplacian]Chapter 2 discusses the Neumann Laplacian on a Riemannian manifold $(M,g)$. We establish the existence of an $L^2_g(M)$-orthonormal basis of $L^2_g(M)$ consisting of eigenfunctions $\phi_k$ – that is solutions to

\displaystyle-\Delta_g \phi_k\hspace{-.2cm}&\displaystyle=\lambda_k \phi_k&\displaystyle\textrm{in $M$},\vspace{.2cm}\\
\displaystyle\frac{\partial \phi_k}{\partial\nu}&\displaystyle=0\hspace{-.2cm}&\displaystyle\textrm{along $\partial M$}.
\end{array}$$

By first working on $\R^n$ and then lifting the results to $M$, we then develop elliptic regularity theory for the more general problem

\left\{
\begin{aligned}
-\Delta_g u&=f\hspace{.5cm}\textrm{in $M$},\\
\frac{\partial u}{\partial\nu}&=h\hspace{.5cm}\textrm{along $\partial M$}.
\end{aligned}
\right.

Thereby we establish smoothness and suitable estimates for the eigenfunctions $\phi_k$. Our main result is Theorem <ref>.

[SchauderCh_Ch03SchauderEstiamtesRn]Chapter 3 establishes parabolic Schauder estimates on domains $V\times[0,T]\subset\R^n\times\R$ by following the strategy from Simon [18]. First, suitable Liouville-type theorems are derived that allow us to execute Simon's blow-up argument. Still following [18], the Schauder estimates are localized to parabolic balls $U_R(p)$ and extended to non-constant operators. Finally, a covering argument is used to deduce Schauder estimates on general domains. The main results here are Corollaries <ref> and <ref>

[SchauderCh_Ch04SchauderTheorMfd]Chapter 4 is split into four parts.  In Section <ref>, the results from Chapter 3 are lifted onto smooth, compact manifolds. The main result is Theorem <ref>. In Section <ref>, we specialize to canonical boundary conditions. These allow for a variational formulation and hence the derivation of an $L^2$ a priori estimate. The main result is Corollary <ref>. In Section <ref>, we study existence of solutions – the main result is Theorem <ref>. In Section <ref>, we refine Corollary <ref> by establishing a decay estimate for the initial boundary value problem

\left\{
\begin{array}{ll}
\displaystyle \dot u+\frac12\Delta(\Delta+2)u=0&\displaystyle \textrm{in $\Sp^2_+\times[0,T]$},\vspace{.15cm}\\
\displaystyle \frac{\partial u}{\partial\nu}=\frac{\partial(\Delta+2)u}{\partial\nu}=0&\displaystyle \textrm{along $\partial\Sp^2_+\times[0,T]$},\vspace{.15cm}\\
\displaystyle u(0)=u_0&\displaystyle \textrm{on $\Sp^2_+$}.
\end{array}
\right.

The author used this result in [15]. Finally, in Section <ref>, we use the parabolic estimates to derive elliptic Schauder estimates. Given a compact Riemannian manifold, we then use the elliptic estimates to analyze the existence of solutions of the problem

\left\{
\begin{aligned}
&\Delta_g^2u=f\hspace{2.73cm}\textrm{in $M$,}\\
&\frac{\partial u}{\partial\nu}=h_1,\ \frac{\partial\Delta_g u}{\partial\nu}=h_2\hspace{.5cm}\textrm{along $\partial M$,}\\
&\int_M ud\mu_g=0.
\end{aligned}
\right.$$

The main results are Lemma <ref> and Corollary <ref>.

§ PREREQUISITES

These notes assume that the reader is familiar with the following concepts:

* Definition and most elementary properties of the Hölder and Sobolev spaces $C^{k,\gamma}(\Omega)$ and $W^{k,p}(\Omega)$. A good reference is the book of Gilbarg and Trudinger [9] (Chapter 4 for Hölder spaces and Chapter 7 for Sobolev spaces). Another excellent reference is Chapter 5 in Evan's book [7].

* Some basic knowledge of functional analysis. We do not require anything not covered in a typical introductory course.

* Some basic notions of differential and Riemannian geometry. These are required only in the fourth Chapter, where we lift Schauder estimates from $\R^n$ to a manifold $M$.

Whenever we use an advanced result from one of these fields, we will provide a literature reference in the text.

§ NOTATION AND CONVENTIONS

* We use the multi-index notation. A multi-index is a vector $\alpha\in\N_0^n$. For such $\alpha$, $x\in\R^n$ and sufficiently regular $f:\R^n\rightarrow\R$, we put

x^\alpha:=x_1^{\alpha_1}\cdot...\cdot x_n^{\alpha_n}
\hspace{.5cm}\textrm{and}\hspace{.5cm}
\nabla_\alpha f:=\frac{\partial}{\partial x_1^{\alpha_1}}...\frac{\partial}{\partial x_n^{\alpha_n}}f.

Additionally for $\alpha=(\alpha_1,...,\alpha_n)$, we put $\alpha!:=\alpha_1!\cdot...\cdot\alpha_n!$ and define the length of $\alpha$ as $|\alpha|:=\alpha_1+...+\alpha_n$.

* As we deal with functions $u(x,t)$ depending on a space variable $x$ and a time variable $t$, we use the symbol $\nabla$ instead of $D$ to denote spatial derivatives, i.e. $\nabla u$, $\nabla^2 u$ etc.

* Given a Riemannian manifold $(M,g)$, the metric in local coordinates is denoted by $g_{ij}$. As usual, $g^{ij}$ denotes the inverse of $g_{ij}$.

* Given a Riemannian manifold $(M,g)$ we denote the Laplace Beltrami operator by $\Delta_g$. If $\partial M\neq\emptyset$, we denote the exterior unit normal along $\partial M$ by $\nu$.

* The Hölder exponents $\gamma$ are always some number in $(0,1)$.

* Given a manifold $M$ with boundary $\partial M$, we say that a chart $\phi:U\rightarrow V$ is an interior chart when $U\cap\partial M=\emptyset$. A boundary chart is a chart $\phi:U\rightarrow V$ where $U\cap\partial M\neq\emptyset$ and $V\subset(\R^{n-1}\times[0,\infty))$. That is, the model for a manifold with boundary is the upper half-space. Therefore, the exterior unit normal along $\partial M$ in local coordinates is $\nu=-\left(g^{nn}\right)^{-\frac12}g^{ni}\partial_i$.

* If $X$ and $Y$ are Banach spaces and $f:X\rightarrow Y$ is a compact embedding, we write $f:X\hookrightarrow\hookrightarrow Y$ or simply $X\hookrightarrow\hookrightarrow Y$, when $f$ is clear from context.

CHAPTER: FUNCTIONAL SPACES

§ HÖLDER SPACES ON RN

§.§ Elliptic Hölder spaces

Let $\Omega\subset\R^n$ be an open set. We denote the set of $m$-times continuously differentiable functions on $\Omega$ by $C^ m(\Omega)$.
Let $\Omega\subset\R^ n$ be open and bounded and $u\in C^ m(\Omega)$ We say that $u\in C^ m(\bar\Omega)$, if for all multiindices $\alpha\in\N_0^ n$ with length $|\alpha|\leq m$ the function $\nabla_ \alpha u$ has a continuous extension onto $\bar\Omega$.
We equip $C^ m(\bar\Omega)$ with the norm

$$\|u\|_{C^m(\bar\Omega)}:=\sum_{k=0}^m\sup_{x\in \Omega}|\nabla^k u(x)|.$$

For $u\in C^0(\Omega)$ and $\gamma\in(0,1)$ we define the Hölder seminorm as

[u]_{\gamma,\Omega}:=\sup_{\substack{x,y\in\Omega\\ x\neq y}}\frac{| u(x)- u(y)|}{|x-y|^\gamma}.

More generally, for $u\in C^m(\Omega)$, we define

$$[\nabla^ m u]_{\gamma,\Omega}:=\sum_{\substack{\alpha\in\N_0^ n\\ |\alpha|=m}}[\nabla_\alpha u]_{\gamma,\Omega}.$$

Let $\Omega\subset\R^ n$ be open and bounded. We say that a function $u\in C^m(\bar\Omega)$ belongs to the space $C^ {m,\gamma}(\Omega)$, if

$$\|u\|_{C^{m,\gamma}(\Omega)}:=\|u\|_{C^m(\Omega)}+[\nabla^m u]_{\gamma,\Omega}<\infty.$$

The spaces $C^{m,\gamma}(\Omega)$ equipped with the $\|\cdot\|_{C^ {m,\gamma}(\Omega)}$-norm are Banach spaces.

Given a function $u\in C^{m,\gamma}(\Omega)$, it is easy to see that $\nabla^ku$ has a unique continuous extension to $\bar\Omega$ for all $0\leq k\leq m$. To make explicit in the notation, that $u$ is in fact defined on $\bar\Omega$, we sometimes write $u\in C^{m,\gamma}(\bar\Omega)$.

The following lemma is an important tool that allows estimating the Hölder seminorm by covering arguments.

Let $\Omega\subset\R^n$ be open and convex and $U_1,...,U_N\subset\Omega$ be open and convex sets such that $\Omega\subset \bigcup_{i=1}^N U_i$. Then for all functions $u\in C^{0,\gamma}(\Omega)$ we have

$$[u]_{\gamma,\Omega}\leq\sum_{i=1}^N[u]_{\gamma, U_i}.$$

For any $x\neq y\in\Omega$ we define $\alpha_{x,y}:[0,1]\rightarrow\Omega,\ \alpha_{x,y}(t):=(1-t)x+ty$ and put

$$S(x,y):=\set{1\leq i\leq N\ |\  U_i\cap\alpha_{x,y}([0,1])\neq\emptyset}
\hspace{.5cm}\textrm{and}\hspace{.5cm}
P(x,y):=|S(x,y)|\leq N.$$

To prove the lemma, we prove the following estimate:

\begin{equation}\label{SchauderCh_SubadditivityToprove}
\frac{|u(x)-u(y)|}{|x-y|^\gamma}\leq\sum_{i\in S(x,y)}[u]_{\gamma,U_i}
\end{equation}

We prove Estimate <ref> by an inductive argument. First, assume that for some points $x\neq y\in\Omega$, we have $|S(x,y)|=1$. Then we can assume without loss of generality that $\alpha_{x,y}(t)\in U_1$ for all $t\in[0,1]$ and, in particular $x,y\in U_1$. In this case, Estimate (<ref>) is trivial.

Now suppose that we already know Estimate (<ref>) to be true for all points $x\neq y\in\Omega$ with $P(x,y)\leq m\leq N$. If $m=N$, the lemma is already proven. If $m\leq N-1$ we argue as follows. Let $x\neq y\in\Omega$ with $P(x,y)=m+1$. Without loss of generality we assume that $S(x,y)=\set{1,...,m+1}$ and that $x\in U_1$. We define

$$t_1:=\sup\set{t\in[0,1]\ |\ \alpha_{x,y}(s)\in U_1\textrm{ for all $s\in[0,t]$}}.$$

If $t_1=1$, then $\alpha_{x,y}(t)\in U_1$ for all $t\in[0,1)$ and Estimate (<ref>) follows as

\frac{|u(y)-u(x)|}{|y-x|^\gamma}
\lim_{\epsilon\rightarrow 0^+}\frac{|u(\alpha_{x,y}(1-\epsilon))-u(x)|}{|\alpha_{x,y}(1-\epsilon)-x|^\gamma}\leq [u]_{\gamma, U_1}.

If $t_1<1$ we define $z:=\alpha_{x,y}(t_1)$. Since $U_1$ is convex and $\alpha_{x,y}(t_1)\not\in U_1$, we deduce $\alpha_{z,y}(t)\not\in U_1$ for all $t\in[0,1]$ and hence $S(z,y)\subset S(x,y)\backslash\set1$. In particular $P(z,y)\leq P(x,y)-1=m$. By the inductive hypothesis, we have

\begin{equation}\label{SchauderCh_Subadditivityproof_InductiveHelp}
\frac{|u(z)-u(y)|}{|z-y|^\gamma}\leq\sum_{i\in S(z,y)}[u]_{\gamma,\gamma}.
\end{equation}

For $\epsilon>0$ small we estimate

$$\frac{|u(x)-u(y)|}{|x-y|^\gamma}\leq \frac{|u(x)-u(\alpha_{x,y}(t_1-\epsilon))|}{|x-y|^\gamma}+\frac{|u(\alpha_{x,y}(t_1-\epsilon))-u(y)|}{|x-y|^\gamma}.$$

Note that $|x-y|\geq |x-\alpha_{x,y}(t_1-\epsilon)|$ and $|x-y|\geq |y-\alpha_{x,y}(t_1-\epsilon)|$. Using that for small $\epsilon>0$ we have $\alpha_{x,y}(t_1-\epsilon)\in U_1$, we get

$$\frac{|u(x)-u(y)|}{|x-y|^\gamma}\leq [u]_{\gamma,U_1,}+\frac{|u(\alpha(t_1-\epsilon))-u(y)|}{|\alpha_{x,y}(t_1-\epsilon)-y|^\gamma}.$$

As $t_1<1$ we can take the limit $\epsilon\rightarrow 0^+$ and, using Estimate (<ref>) as well as $1\in S(x,y)$ and $S(z,y)\subset S(x,y)\backslash\set 1$, we obtain

[u]_{\gamma, U_1}+\sum_{i\in S(z,y)}[u]_{\gamma,U_i}
\leq\sum_{i\in S(x,y)}[u]_{\gamma,U_i\gamma}.$$

§.§ Parabolic Hölder Spaces

Let $\Omega\subset\R^n$ be open and $T>0$. We consider $\Omega\times(0,T)\subset\R^n\times\R$. Given a function $u:\Omega\times(0,T)\rightarrow\R$, we denote derivatives with respect to the time variable $t\in(0,T)$ by a dot $\cdot$ or by $\partial_t$ and derivatives with respect to the space variable $x$ by  $\partial_i$, $\partial_{x_i}$, etc. or simply by $\nabla$.

Let $\Omega\subset\R^n$ be open, $T>0$ and $m\in\N_0$. We define

$$C^{m,[m/4]}(\Omega\times(0,T)):=\set{u:\Omega\times(0,T)\rightarrow\R\ |\ \textrm{$\partial_t^j\nabla^k u$ exists for all $4j+k\leq m$ and is continuous}}.$$

It can quickly be checked that for $u\in C^{m,[m/4]}(\Omega\times(0,T))$ we have $\nabla u\in C^{m-1,[(m-1)/4]}(\Omega\times(0,T))$ and $\partial_t u\in C^{m-4,[m/4]-1}(\Omega\times(0,T))$.

Next, we formulate the analog of Definition <ref>.

Let $\Omega\subset\R^n$ be open, $T>0$ and $u\in C^{m,[m/4]}(\Omega\times(0,T))$. We say that $u\in C^{m,[m/4]}(\bar\Omega\times[0,T])$ if for all multiindices $\alpha\in\N_0^n$ and $j\in\N_0$ such that $|\alpha|+4j\leq m$ the function $\partial_t^j\nabla_\alpha u$ has a continuous extension onto $\bar\Omega\times[0,T]$. We equip $C^{m,[m/4]}(\bar\Omega\times[0,T])$ with the norm

$$\|u\|_{C^{m,[m/4]}(\Omega\times(0,T))}:=\sum_{4j+|\alpha|\leq m}\sup_{(x,t)\in\Omega\times(0,T)}|\partial_t^j\nabla_\alpha u(x,t)|.$$

Let $\Omega\subset\R^n$ be open, $T>0$, $\alpha,\beta\in(0,1)$ and $u\in C^{0,0}(\Omega\times(0,T))$. We define the spatial and temporal Hölder seminorm by

\begin{align*}
[u]^{\operatorname{space}}_{\alpha,\Omega\times(0,T)}&:=\sup_{\substack{x,y\in\Omega,\ t\in(0,T)\\ x\neq y}}\frac{|u(x,t)-u(y,t)|}{|x-y|^{\alpha}},\\
[u]^{\operatorname{time}}_{\beta,\Omega\times(0,T)}&:=\sup_{\substack{x\in\Omega,\ t,s\in(0,T)\\ t\neq s}}\frac{|u(x,t)-u(x,s)|}{|t-s|^{\beta}}.
\end{align*}

Let $\Omega\subset\R^n$ be open and bounded, $T>0$ and $m\in\N_0$. For a function $u\in C^{m,[m/4]}(\Omega\times(0,T))$, we define

\begin{equation}\label{SchauderCh_parabolicHölderseminormdefinition}
[u]_{\gamma,\Omega\times(0,T)}^{(m)}:=\sum_{4j+k= m}[\partial_t^j\nabla^k u]_{\gamma,\Omega\times(0,T)}^{\operatorname{space}}
+\sum_{0<\frac{m+\gamma-k}4-j<1}[\partial_t^j \nabla^k u]^{\operatorname{time}}_{\frac{m+\gamma-k}4-j,\Omega\times(0,T)}.
\end{equation}

We say that $u\in C^{m,[m/4],\gamma}(\Omega\times(0,T))$, if

$$\|u\|_{C^{m,[m/4],\gamma}(\Omega\times(0,T))}:=\|u\|_{C^{m,[m/4]}(\Omega\times(0,T))}+ [u]_{\gamma,\Omega\times(0,T)}^{(m)}<\infty.$$

It can quickly be checked, that $C^{m,[m/4],\gamma}(\Omega\times(0,T))$ equipped with $\|\cdot\|_{C^{m,[m/4],\gamma}(\Omega\times(0,T))}$ is Banach.

Given a function $u\in C^{m,[m/4],\gamma}(\Omega\times[0,T])$, it is easy to see that $\partial_t^l\nabla^ku$ has a unique continuous extension to $\bar\Omega\times[0,T]$ whenever $k+4l\leq m$. To make explicit in the notation, that $u$ is in fact defined on $\bar\Omega\times[0,T]$, we sometimes write $u\in C^{m,\gamma}(\bar\Omega\times[0,T])$.

Motivation of Definition <ref>

We give a brief justification for the spatial and temporal Hölder norms appearing in Equation (<ref>). Given a function $u\in C^{m,[m/4]}(\Omega\times(0,T))$ and $\lambda>0$, we consider $u_\lambda(x,t):=u(\lambda x, \lambda^4 t)$. A quick computation shows

\begin{align*}
[\partial_t^j\nabla^k u_\lambda]_{\alpha,\Omega\times(0,T)}^{\operatorname{space}}&=\lambda^{4j+k}\lambda^\alpha[\partial_t^j\nabla^k u]_{\alpha,\Omega\times(0,T)}^{\operatorname{space}},\\
[\partial_t^j\nabla^k u_\lambda]_{\beta,\Omega\times(0,T)}^{\operatorname{time}}&=\lambda^{4j+k}\lambda^{4\beta}[\partial_t^j\nabla^k u]_{\beta,\Omega\times(0,T)}^{\operatorname{time}}.
\end{align*}

The expected scaling behaviour of `a $(m+\gamma)$-th derivative' is $\lambda^{m+\gamma}$. As $\alpha,\gamma\in(0,1)$ and $j,k,m\in\N_0$, this already implies that for the spatial seminorm we require $4j+k=m$ and $\alpha=\gamma$. In the temporal case, we require $\beta=\frac{m+\gamma-k}4-j$.

The following proposition can be proven by following the same arguments that lead to Lemma <ref>:

Let $\Omega\subset\R^n$ be open and convex, $T>0$, $U_1,...,U_N\subset\Omega\times(0,T)$ be open and convex sets and $I_1,...,I_N\subset(0,T)$ be intervals such that $\Omega\times(0,T)\subset \bigcup_{i=1}^N U_i\times I_i$. Then for all functions $u\in C^{0,0}(\Omega\times(0,T))$ and $\alpha,\beta\in(0,1)$

$$[u]_{\alpha,\Omega\times(0,T)}^{\operatorname{space}}\leq\sum_{i=1}^N[u]^{\operatorname{space}}_{\alpha, U_i\times I_i}
\hspace{.5cm}\textrm{and}\hspace{.5cm}
[u]_{\beta,\Omega\times(0,T)}^{\operatorname{time}}\leq\sum_{i=1}^N[u]^{\operatorname{time}}_{\beta, U_i\times I_i}.$$

Finally, we give the following proposition.

Let $\Omega$ be a bounded domain. For $u\in C^{4,1,\gamma}(\Omega\times(0,T))$, we define

[\nabla^4 u]_{\gamma,\Omega\times(0,T)}^{(0)}
[\dot u]_{\gamma,\Omega\times(0,T)}^{(0)}.$$

Then, the following norm is equivalent to $\|\cdot\|_{C^{4,1,\gamma}(\Omega\times(0,T))}$:

In view of Proposition <ref>, this is a consequence of Theorem <ref> and Lemma <ref>

[colback=white!20!white,colframe=black!100!white,sharp corners, breakable]
In the following we will always use the norm $\|u\|'_{C^{4,1,\gamma}(\Omega\times(0,T)}$ from Proposition <ref> without including the `$'$' in the notation.

§ TECHNICAL LEMMAS

Let $p_0=(x_0, t_0)\in\R^n\times\R$ and $\rho>0$. We define the parabolic ball of radius $\rho$ with center $p_0$ as

$$U_\rho(p_0):=\set{p=(x,t)\in\R^n\times\R\ |\ \max\{|x-x_0|, |t-t_0|^{\frac14}\}<\rho}=B_\rho(x_0)\times(t_0-\rho^4,t_0+\rho^4).$$

Additionally, we define the following truncated parabolic balls:

$$\begin{array}{l l }
\end{array}$$

For $U\subset\R^n\times\R$ we define the floor boundary and side boundary respectively as

$$\partial_FU:=U\cap(\R^n\times 0)
\hspace{.5cm}\textrm{and}\hspace{.5cm}
\partial_SU:=U\cap(\R^{n-1}\times 0\times\R).$$

Let $p_0\in\R^n\times\R$, $\rho>0$, $U_\rho:=U_\rho(p_0)$ and $u\in C^{4,1,\gamma}(U_\rho)$. There exists $\epsilon_0(n)>0$ such that for all $\epsilon\in (0,\epsilon_0)$ and $0\leq k\leq 4$

\begin{align*}
\rho^k \|\nabla^k u\|_{L^\infty(U_\rho)}&\leq \epsilon\rho^{4+\gamma}[D^{4,1} u]^{(0)}_{\gamma,U_\rho}+C(n)\epsilon^{-\frac{k}{4-k+\gamma}}\|u\|_{L^\infty(U_\rho)},\\
\rho^4 \|\dot u\|_{L^\infty(U_\rho)}&\leq \epsilon\rho^{4+\gamma}[D^{4,1} u]^{(0)}_{\gamma,U_\rho}+C(n)\epsilon^{-\frac{k}{\gamma}}\|u\|_{L^\infty(U_\rho)}.
\end{align*}

Additionally, for $0\leq l\leq 3$ and $\epsilon\in (0,\epsilon_0)$

\begin{align*}
\rho^{l+\gamma} [\nabla^l u]_{\gamma,U_\rho}^{(0)}&\leq \epsilon\rho^{4+\gamma}[D^{4,1} u]^{(0)}_{\gamma,U_\rho}+C(n)\epsilon^{-\frac{l+\gamma}{4-l}}\|u\|_{L^\infty(U_\rho)}.
\end{align*}

By translation and scaling, we can assume that $p_0=0$ and $\rho=1$.
Let $ f\in C^{1,0,\gamma}(U_{1})$, $q=(x_1,t_1)\in U_1:=U_1(0)$ and $\sigma\in(0,1)$ such that $U_\sigma(q)\subset U_1$. Denoting the exterior normal along $\partial B_\sigma(x_1)$ by $\nu$ we use the divergence theorem to estimate for $(x,t)\in U_\sigma(q)$

\begin{align}
|\partial_ if(x,t)|\leq &\frac1{|B_\sigma(x_1)|}
\left|\int_{B_\sigma(x_1)} \partial_i f(y,t)dy\right|+\frac1{|B_\sigma(x_1)|}\int_{B_\sigma(x_1)}|\partial_i f(x,t)-\partial_i f(y,t)|dy\label{SchauderCh_InterpolationProof02}\\
\leq & \frac1{|B_\sigma(x_1)|}
\left|\int_{\partial B_\sigma(x_1)} f(y,t)\nu_i dy\right|+\frac1{|B_\sigma(x_1)|}\int_{B_\sigma(x_1)}[\nabla f]_{\gamma,B_\sigma(x_1)}^{\operatorname{space}}|x-y|^\gamma dy\nonumber\\
\leq & \frac{|\partial B_\sigma(x_1)|}{|B_\sigma(x_1)|}
\|f\|_{L^\infty(B_\sigma(x_1))}+[\nabla f]_{\gamma,B_\sigma(x_1)}^{\operatorname{space}}\sigma^\gamma.\nonumber
\end{align}

We have $|B_\sigma(x_1)|=c(n)\sigma^n$ and $|\partial B_\sigma(x_1)|=c'(n)\sigma^{n-1}$. So, taking the supremum over all $(x,t)$ in $U_\sigma(p)$ and $1\leq i\leq n$, we obtain

\begin{equation}\label{SchauderCh_Interpolationproof01}
\|\nabla f\|_{L^\infty(U_\sigma(q))}\leq \frac{C(n)}\sigma \|f\|_{L^\infty(U_\sigma(q))}+C(n)\sigma^\gamma[\nabla f]_{\gamma,U_\sigma(q)}^{(0)}.
\end{equation}

Since for all $q'\in U_1$ there exists $q\in U_1$ such that $q'\in U_\sigma(q)\subset U_1$, we deduce that for all $\sigma\in(0,1)$

\begin{equation}\label{SchauderCh_InterpolationProof03}\|\nabla f\|_{L^\infty(U_1)}\leq C(n)\left(\sigma^\gamma[\nabla f]_{\gamma,U_1}^{(0)}+\sigma^{-1}\|f\|_{L^\infty(U_1)}\right).
\end{equation}

Instead of estimating $|\partial_i f(x,t)-\nabla_i f(y,t)|$ with the Hölder seminorm, we can write $|\partial_i f(x,t)-\partial_i f(y,t)|\leq C(n)\|\nabla^2 f\||x-y|$ if $f$ is $C^{2,0,\gamma}(U_1)$. Following the same reasoning that led to (<ref>), we then obtain

\begin{equation}\label{SchauderCh_InterpolationProof04}
\|\nabla f\|_{L^\infty(U_1)}\leq C(n)\left(\sigma\|\nabla^2 f\|_{L^\infty(U_1)}+\sigma^{-1}\|f\|_{L^\infty(U_1)}\right).
\end{equation}

Taking $f=u$ in Estimate (<ref>), we get

\begin{equation}\label{SchauderCh_InterpolationProof04_02}
\|\nabla u\|_{L^\infty(U_1)}
\leq C(n)\left(\sigma \|\nabla^2 u\|_{L^\infty(U_1)}
\sigma^{-1}\| u\|_{L^\infty(U_1)}\right).
\end{equation}

Next, taking $f=\nabla u$ in Estimate (<ref>) and subsequently inserting Estimate (<ref>) with $\sigma\rightarrow\mu$, implies

\begin{align}
\|\nabla^2 u\|_{L^\infty(U_1)}
&\leq C(n)\left(\sigma \|\nabla^3 u\|_{L^\infty(U_1)}
\sigma^{-1}\| \nabla u\|_{L^\infty(U_1)}\right)\nonumber\\
\ &\leq C(n)\left(\sigma \|\nabla^3 u\|_{L^\infty(U_1)}
\sigma^{-1}\left(
\mu \|\nabla^2 u\|_{L^\infty(U_1)}
\mu^{-1}\| u\|_{L^\infty(U_1)}\label{SchauderCh_InterpolationProof05}
\right)\right).
\end{align}

Choosing $\mu=\epsilon(n)\sigma$ with a small enough number $\epsilon(n)\in(0,1)$, we obtain

\begin{equation}\label{SchauderCh_InterpolationProof06}
\|\nabla^2 u\|_{L^\infty(U_1)}\leq C(n)\left(
\sigma\|\nabla^{3} u\|_{L^\infty(U_1)}
\sigma^{-2}\| u\|_{L^\infty(U_1)}
\right).
\end{equation}

Inserting Estimate (<ref>) into Estimate (<ref>), we get

\begin{equation}\label{SchauderCh_InterpolationProof07}
\|\nabla u\|_{L^\infty(U_1)}\leq C(n)\left(
\sigma^2\|\nabla^{3} u\|_{L^\infty(U_1)}
\sigma^{-1}\| u\|_{L^\infty(U_1)}
\right).
\end{equation}

Next, we take $f=\nabla^2 u$ in Estimate (<ref>) and use Estimate (<ref>) with $\sigma\rightarrow\mu$ to estimate

\begin{align*}
\|\nabla^3 u\|_{L^\infty(U_1)}
&\leq C(n)\left(\sigma \|\nabla^4 u\|_{L^\infty(U_1)}
\sigma^{-1}\|  \nabla^2u\|_{L^\infty(U_1)}\right)\\
&\leq C(n)\left(\sigma \|\nabla^4 u\|_{L^\infty(U_1)}
\sigma^{-1}\left(
\mu \|\nabla^3 u\|_{L^\infty(U_1)}
\mu^{-2}\| u\|_{L^\infty(U_1)}
\right)\right).
\end{align*}

Choosing again $\mu=\epsilon(n)\sigma$ with $\epsilon(n)\in(0,1)$ small enough yields

\begin{equation}\label{SchauderCh_InterpolationProof08}
\|\nabla^3 u\|_{L^\infty(U_1)}
\leq C(n)\left(\sigma \|\nabla^4 u\|_{L^\infty(U_1)}
\sigma^{-3}\| u\|_{L^\infty(U_1)}\right).
\end{equation}

Inserting Estimate (<ref>) back into Estimates (<ref>) and (<ref>), we have shown that for $k=1,2,3$

\begin{equation}\label{SchauderCh_InterpolationProof09}
\|\nabla^k u\|_{L^\infty(U_1)}
\leq C(n)\left(\sigma^{4-k} \|\nabla^4 u\|_{L^\infty(U_1)}
\sigma^{-k}\|   u\|_{L^\infty(U_1)}\right).
\end{equation}

Finally, we use Estimate (<ref>) with $f\rightarrow\nabla^3 u$ and subsequently Estimate (<ref>) with $k=3$ to deduce that

\begin{align*}
\|\nabla^4 u\|_{L^\infty(U_1)} \leq &C(n)\left(\sigma^{\gamma} [\nabla^4 u]_{\gamma,U_1}^{(0)}
\sigma^{-1}\|   \nabla^3u\|_{L^\infty(U_1)}\right)\\
\leq &C(n)\left(\sigma^{\gamma} [\nabla^4 u]_{\gamma,U_1}^{(0)}
\sigma^{-1}  \left(
\mu  \|\nabla^4 u\|_{L^\infty(U_1)}+\mu^{-3} \|u\|_{L^\infty(U_1)}
\right)\right).
\end{align*}

Taking $\mu=\epsilon(n)\sigma$ with small enough $\epsilon(n)\in(0,1)$ gives

\begin{equation}\label{SchauderCh_InterpolationProof10}
\|\nabla^4 u\|_{L^\infty(U_1)}\leq C(n)\left(\sigma^{\gamma} [\nabla^4 u]_{\gamma,U_1}^{(0)}
\sigma^{-4}\| u\|_{L^\infty(U_1)}\right).
\end{equation}

Estimates (<ref>) and (<ref>) together imply that for $0\leq k\leq 4$

\begin{equation}\label{SchauderCh_InterpolationProof10NEW}
\|\nabla^k u\|_{L^\infty(U_1)}\leq C(n)\left(\sigma^{4-k+\gamma} [\nabla^4 u]_{\gamma,U_1}^{(0)}
\sigma^{-k}\| u\|_{L^\infty(U_1)}\right),
\end{equation}

which is the first estimate claimed in the theorem.
To prove the second estimate, let again $q=(x_1,t_1)\in U_1$ and $\sigma>0$ such that $U_\sigma(q)\subset U_1$. Then, for any $(x,t)\in U_\sigma(q)$ we have $t\in(t_1-\sigma^4, t_1+\sigma^4)$. Consequently

\begin{align}
|\dot u(x,t)| &\leq \frac1{2\sigma^4}\left|\int_{t_1-\sigma^4}^{t_1+\sigma^4}\dot u(x,s)ds\right|+\frac1{2\sigma^4}\int_{t_1-\sigma^4}^{t_1+\sigma^4}|\dot u(x,t)-\dot u(x,s)|ds\nonumber\\
&\leq C(n)\sigma^{-4}\| u\|_{L^\infty(U_1)}
+\frac1{2\sigma^4}\int_{t_1-\sigma^4}^{t_1+\sigma^4}[\dot u]^{\operatorname{time}}_{\frac\gamma4, U_1}|t-s|^{\frac\gamma4}ds\nonumber\\
&\leq C(n)\left(\sigma^{-4}\|\dot u\|_{L^\infty(U_1)}
+\sigma^{\gamma}[u]_{\frac\gamma4, U_1}^{\operatorname{time}}\right).\label{SchauderCh_InterpolationProof11}
\end{align}

For any $q'\in U_1$ there exists $q\in U_1$ such that $q'\in U_\sigma(q)\subset U_1$. Thus, Estimate (<ref>) implies the second estimate claimed in the theorem.

We are left with estimating the spatial and temporal Hölder seminorms of $\nabla^l u$ for $l=0,1,2,3$. To do so, we consider two points $q=(x,t)\neq (y,s)=q'\in U_1$. We first consider the case where $|t-s|^{\frac14}\geq\sigma$ and $|x-y|\geq\sigma$. Using Estimate (<ref>), we estimate

\begin{align}
\frac{|\nabla^lu(x,t)-\nabla^lu(y,t)|}{|x-y|^\gamma}\nonumber\\
\leq& 4\sigma^{-\gamma}\|\nabla^l u\|_{L^\infty(U_1)}\nonumber\\
\leq &C(n)\left(
\sigma^{4-l}[D^{4,1} u]_{\gamma, U_1}^{(0)}+\sigma^{-l-\gamma}\|u\|_{L^\infty(U_1)}
\right)\label{SchauderCh_InterpolationProof12}.
\end{align}

Now we consider the case where $|t-s|^{\frac14}\leq\sigma$ or $|x-y|\leq\sigma$ respectively. If $|x-y|\leq\sigma$, we use Estimates (<ref>) to estimate

\begin{align}
\frac{|\nabla^lu(x,t)-\nabla^lu(y,t)|}{|x-y|^\gamma}&\leq\|\nabla^{l+1} u\|_{L^\infty(U_1)} |x-y|^{1-\gamma}\nonumber\\
&\leq C(n)\left(\sigma^{4-l}[D^{4,1} u]_{\gamma, U_1}^{(0)}+\sigma^{-l-\gamma}\|u\|_{L^\infty(U_1)}
\right).\label{SchauderCh_InterpolationProof122}
\end{align}

Estimates (<ref>) and (<ref>) already imply the claimed estimates for the spatial seminorms. It remains to consider the case $|t-s|\leq \sigma^4$. We use Estiamte (<ref>) to estiamte

\begin{align}
\frac{ |u(x,t)-u(x,s)|}{|t-s|^{\frac\gamma4}}\leq& \|\dot u\|_{L^\infty(U_1)}|t-s|^{1-\frac{\gamma}4}\nonumber\\
\leq &C(n)(\sigma^\gamma[D^{4,1} u]_{\gamma, U_1}^{(0)}+\sigma^{-4}\|u\|_{L^\infty(U_1)})\sigma^{4-\gamma}\nonumber\\
\leq & C(n)\left(\sigma^4 [D^{4,1} u]_{\gamma, U_1}^{(0)}+\sigma^{-\gamma}\|u\|_{L^\infty(U_1)}\right).\label{SchauderCh_InterpolationProof1222}
\end{align}

Taking $l=0$ in Estimates  (<ref>) and (<ref>) and combining with Estimate (<ref>), we obtain

\begin{equation}\label{SchauderCh_InterpolationProofuTimeSemi}
[u]_{\gamma,U_1}^{(0)}\leq C(n)\left(\sigma^4 [D^{4,1} u]_{\gamma, U_1}^{(0)}+\sigma^{-\gamma}\|u\|_{L^\infty(U_1)}\right).
\end{equation}

Next, we consider $l=1,2,3$. Let $x\in B_1(0)$, $s,t\in (-1,1)$ such that $|t-s|\leq\sigma^4$ and let $x_1\in B_1(0)$ such that $x\in B_\sigma:= B_\sigma(x_1)\subset B_1(0)$. Denoting the exterior normal along $\partial B_\sigma$ by $\nu$, we estimate

\begin{align}
|\nabla^lu (x,t)-\nabla^l u(x,s)|\leq & \frac1{|B_\sigma|}\left|\int_{B_\sigma} \nabla^l u(x,t)-\nabla^l u(y,t) dy
-\int_{B_\sigma} \nabla^l u(x,s)-\nabla^l u(y,s) dy\right|\nonumber\\
+\frac1{|B_\sigma|}\left|\int_{B_\sigma}\nabla^l u(y,s)-\nabla^l (y,t) dy\right|.\label{SchauderCh_LongINTERPOLATIONEstimate}
\end{align}

To estimate the integral in the first line of Estimate (<ref>), we write

\begin{align}
&\left|\int_{B_\sigma} \nabla^l u(x,t)-\nabla^l u(y,t) dy
-\int_{B_\sigma} \nabla^l u(x,s)-\nabla^l u(y,s) dy\right|\nonumber\\
\leq &\left|\int_0^1 \int_{B_\sigma}\nabla ^{l+1} u(y+\lambda(x-y),t)(x-y)- \nabla ^{l+1} u(y+\lambda(x-y),s)(x-y)dy d\lambda\right|\nonumber\\
\leq &\left|\int_0^1 \int_{B_\sigma}[\nabla^{l+1} u]_{\frac\gamma4, U_1}^{\operatorname{time}}||t-s|^{\frac\gamma4}|x-y|dy d\lambda\right|\nonumber\\
\leq & |B_\sigma|[\nabla^{l+1} u]_{\frac\gamma4, U_1}^{\operatorname{time}}||t-s|^{\frac\gamma4}\sigma.\label{SchauderCh_LongINTERPOLATIONEstimate1}
\end{align}

To estimate the integral in the second line of Estimate (<ref>), we use the divergence theorem to write

\begin{align}
& \left|\int_{B_\sigma}\nabla^l u(y,s)-\nabla^l (y,t) dy\right|\nonumber\\
\leq & \left|\int_{\partial B_\sigma}\langle \nabla^{l-1} u(y,s)-\nabla^{l-1} u(y,t),\nu\rangle dS(y)\right|\nonumber\\
\leq &\int_{\partial B_\sigma}| \nabla^{l-1} u(y,s)-\nabla^{l-1} u(y,t)| dS(y)\nonumber\\
\leq &|\partial B_\sigma|[\nabla^{l-1} u]_{\frac\gamma4, U_1}^{\operatorname{time}}|t-s|^{\frac\gamma4}\label{SchauderCh_LongINTERPOLATIONEstimate2}
\end{align}

Inserting Estimates (<ref>) and (<ref>) into Estimate (<ref>), we get

\begin{equation}\label{SchauderCh_LongINTERPOLATIONEstimate3}
|\nabla^lu (x,t)-\nabla^l u(x,s)| \leq C(n)\left(\sigma[\nabla^{l+1} u]_{\frac\gamma4, U_1}^{\operatorname{time}}+\sigma^{-1}[\nabla^{l-1} u]_{\frac\gamma4, U_1}^{\operatorname{time}}\right)|t-s|^{\frac\gamma4}.
\end{equation}

Combining Estimates (<ref>) and (<ref>) shows that for $l=1,2,3$ and $\sigma>0$

\begin{equation}\label{SchauderCh_integralestimateSemiNorm}
\hspace{-.2cm}[\nabla ^l u]_{\frac\gamma4, U_1}^{\operatorname{time}}\hspace{-.1cm}\leq\hspace{-.05cm} C(n)
\left(
\sigma[\nabla^{l+1} u]_{\frac\gamma4, U_1}^{\operatorname{time}}
\hspace{-.1cm}+\sigma^{-1}[\nabla^{l-1} u]_{\frac\gamma4, U_1}^{\operatorname{time}}
\hspace{-.1cm}+\sigma^{4-l}[D^{4,1} u]_{\gamma, U_1}^{(0)}
\hspace{-.1cm}+\sigma^{-l-\gamma}\|u\|_{L^\infty(U_1)}
\right).
\end{equation}

Using Estimate (<ref>), we obtain

\begin{align}
[\nabla u]_{\frac\gamma4, U_1}^{\operatorname{time}}&\leq C(n)\left(\sigma[\nabla^{2} u]_{\frac\gamma4, U_1}^{\operatorname{time}}+\sigma^{3}[D^{4,1} u]_{\gamma, U_1}^{(0)}+\sigma^{-1-\gamma}\|u\|_{L^\infty(U_1)}\right),\label{SchauderCh_nabla1uEstimate}\\
[\nabla ^2 u]_{\frac\gamma4, U_1}^{\operatorname{time}}&\leq C(n)\left(\sigma[\nabla^{3} u]_{\frac\gamma4, U_1}^{\operatorname{time}}+\sigma^{-1}[\nabla u]_{\frac\gamma4, U_1}^{\operatorname{time}}+\sigma^{2}[D^{4,1} u]_{\gamma, U_1}^{(0)}+\sigma^{-2-\gamma}\|u\|_{L^\infty(U_1)}\right).\label{SchauderCh_nabla2uEstimate}
\end{align}

Let $\epsilon(n)\in(0,1)$ be a small number that is allowed to depend on $n$. We insert Estimate (<ref>) $\sigma\rightarrow \epsilon(n)\sigma$ into Estimate (<ref>). Choosing $\epsilon(n)$ small enough then shows

\begin{equation}\label{SchauderCh_Improvednabla2uEstimate}
[\nabla ^2 u]_{\frac\gamma4, U_1}^{\operatorname{time}}\leq C(n)\left(\sigma[\nabla^{3} u]_{\frac\gamma4, U_1}^{\operatorname{time}}+\sigma^{2}[D^{4,1} u]_{\gamma, U_1}^{(0)}+\sigma^{-2-\gamma}\|u\|_{L^\infty(U_1)}\right).
\end{equation}

Taking $l=3$ in Estimate (<ref>) gives

\begin{equation}\label{SchauderCh_Nabla3uEstimate}
[\nabla ^3 u]_{\frac\gamma4, U_1}^{\operatorname{time}}\leq C(n)\left(\sigma^{-1}[\nabla^{2} u]_{\frac\gamma4, U_1}^{\operatorname{time}}+\sigma[D^{4,1} u]_{\gamma, U_1}^{(0)}+\sigma^{-3-\gamma}\|u\|_{L^\infty(U_1)}\right).
\end{equation}

Again, let $\epsilon(n)\in(0,1)$ be a small number that is allowed to depend on $n$. We insert Estimate (<ref>) with  $\sigma\rightarrow \epsilon(n)\sigma$ into Estimate (<ref>). Choosing $\epsilon(n)$ small enough gives

\begin{equation}\label{SchauderCh_nabla3uFinalEst}
[\nabla ^3 u]_{\frac\gamma4, U_1}^{\operatorname{time}}\leq C(n)\left(\sigma[D^{4,1} u]_{\gamma, U_1}^{(0)}+\sigma^{-3-\gamma}\|u\|_{L^\infty(U_1)}\right).
\end{equation}

Inserting  Estimate (<ref>) back into Estimate (<ref>), we deduce

\begin{equation}\label{SchauderCh_Finalnabla2uEst}
[\nabla ^2 u]_{\frac\gamma4, U_1}^{\operatorname{time}}\leq C(n)\left(\sigma^{2}[D^{4,1} u]_{\gamma, U_1}^{(0)}+\sigma^{-2-\gamma}\|u\|_{L^\infty(U_1)}\right).
\end{equation}

Finally, inserting Estimate (<ref>) into Estimate (<ref>) gives

\begin{equation}\label{SchauderCh_Finalnabla1uEst}
[\nabla  u]_{\frac\gamma4, U_1}^{\operatorname{time}}\leq C(n)\left(\sigma^{3}[D^{4,1} u]_{\gamma, U_1}^{(0)}+\sigma^{-1-\gamma}\|u\|_{L^\infty(U_1)}\right).
\end{equation}

Estimates (<ref>), (<ref>), (<ref>) and (<ref>) are the claimed estimates for the temporal seminorms.

Let $\rho>0$, $p_0\in\R^n\times\R$, $U_\rho:=U_\rho(p_0)$ and $u\in C^ {4,1,\gamma}(U_\rho)$. Then, for $l\in\set{0,1,2,3}$

$$[\nabla^ l u]^ {\operatorname{time}}_{\frac{4-l+\gamma}4, U_\rho}\leq C(n)\left([D^ {4,1} u]_{\gamma,U_\rho}^{(0)} +\rho^{-4-\gamma}\|u\|_{L^\infty(U_\rho)}\right).$$

By scaling and translation, we may assume $\rho=1$ and $p_0=0$. Let $(x,t)\neq(x,s)\in U_1$ and $\epsilon:=|t-s|^{\frac14}$. We assume, without loss of generality, that $t<s$. If $\epsilon\geq \frac16$ we use Theorem <ref> to estimate

\begin{equation}\label{SchauderCh_HoldereSemiNorm4thorderTrivialEstimate}
\frac{|\nabla^l u(x,t)-\nabla u(x,s)|}{|t-s|^{\frac{4-l+\gamma}4}}
\leq 2\|\nabla ^l u\|_{L^\infty(U_1)}6^{\frac{4-l+\gamma}4}\leq C(n)\left([D^{4,1} u]_{\gamma, U_1}^{(0)}+\|u\|_{L^\infty(U_1)}\right).
\end{equation}

So we may also assume that $\epsilon\leq \frac16$. Given $i\in\set{1,...,n}$ and a function $f: B_{1-\epsilon}(0)\rightarrow\R$, we put

$$D_i^\epsilon f(x,t):= \frac{f(x+\epsilon e_i, t)- f(x,t)}\epsilon=\frac1\epsilon\int_0^\epsilon \partial_i f(x+\rho e_i,t)d\rho.$$

For higher difference quotients, we write e.g. $D_{ij}^\epsilon f:=D_i^\epsilon D_j^\epsilon f$. For $x\in B_{1-3\epsilon}(0)$, the difference quotients $D_i^\epsilon u(x,t)$, $D_{ij}^\epsilon u(x,t)$ and $D_{ijk}^\epsilon u(x,t)$ are all well-defined and we have the following formulas:

\begin{align}
D_i^\epsilon u(x,t)&=\frac1\epsilon\int_0^\epsilon \partial_i u(x+\rho_1 e_i,t)d\rho_1\label{SchauderCh_BetterHoelderRegularityFormula1order}\\
D_{ij}^\epsilon u(x,t)&=\frac1{\epsilon^2}\int_0^\epsilon \int_0^\epsilon \partial_{ij} u(x+\rho_1 e_i+\rho_2 e_j,t)d\rho_1d\rho_2\label{SchauderCh_BetterHoelderRegularityFormula1orde2}\\
D_{ijk}^\epsilon u(x,t)&=\frac1{\epsilon^3}\int_0^\epsilon \int_0^\epsilon\int_0^\epsilon \partial_{ijk} u(x+\rho_1 e_i+\rho_2e_j+\rho_3e_k,t)d\rho_1d\rho_2d\rho_3\label{SchauderCh_BetterHoelderRegularityFormula1orde3}
\end{align}

Improved Hölder Continuity of $\nabla^3 u$

Using Equation (<ref>), we write

\begin{align*}
&D_{ijk}^\epsilon u(x,t)-\partial_{ijk} u(x,t)\nonumber\\
=&\frac1{\epsilon^3}\int_0^\epsilon \int_0^\epsilon\int_0^\epsilon u(x+\rho_1 e_i+\rho_2e_j+\rho_3e_k,t)d\rho_1d\rho_2d\rho_3- \partial_{ijk} u(x,t)\nonumber\\
=&\frac1{\epsilon^3}\int_0^\epsilon \int_0^\epsilon\int_0^\epsilon\int_0^1  \langle\nabla \partial_{ijk} u(x+\mu\left(\rho_1 e_i+\rho_2e_j+\rho_3e_k\right),t),\rho_1 e_i+\rho_2e_j+\rho_3e_k\rangle d\mu d\rho_1d\rho_2d\rho_3.
\end{align*}

So, using the temporal Hölder continuity of the fourth derivative $\nabla^4 u$ and recalling that $\epsilon^4=|t-s|$, we may estimate

\begin{align}
&\left|\partial_{ijk} u(x,t)-D_{ijk}^\epsilon u(x,t)-\partial_{ijk} u(x,s)+D_{ijk}^\epsilon u(x,s)\right|\nonumber\\
\leq & \frac1{\epsilon^3}\int_0^\epsilon\int_0^\epsilon\int_0^\epsilon\int_0^1 [\nabla^4 u]_{\frac\gamma4,U_1}^{\operatorname{time}}|t-s|^{\frac\gamma4}\mu(\rho_1+\rho_2+\rho_3)d\mu d\rho_1d\rho_2d\rho_3\nonumber\\
\leq & \frac1{\epsilon^3} [\nabla^4 u]_{\frac\gamma4,U_1}^{\operatorname{time}}|t-s|^{\frac\gamma4} \int_0^\epsilon\int_0^\epsilon\int_0^\epsilon\int_0^13\epsilon d\mu d\rho_1d\rho_2d\rho_3\nonumber\\
= &3 [\nabla^4 u]_{\frac\gamma4, U_1}^{\operatorname{time}}|t-s|^{\frac{1+\gamma}4}.\label{SchauderCh_BetterHoelderRegularityFormula1orde3_2}
\end{align}

Recalling that $t<s$, we estimate

\begin{align}
|D_{ijk}^\epsilon u(x,t)- D_{ijk}^\epsilon u(x,s)|\leq &|t-s| \sup_{t\leq \tau\leq s}\sup_{x\in B_{3-\epsilon}(0)} |D_{ijk} \dot u(x,\tau)|\nonumber\\
\leq & |t-s| \sup_{t\leq \tau\leq s} \sup_{x\in B_{1-3\epsilon}(0)} \frac1\epsilon \left(|D_{jk} \dot u(x+\epsilon e_i,\tau)|+|D_{jk} \dot u(x,\tau)|\right)\nonumber\\
\leq & |t-s| \frac2\epsilon \sup_{t\leq \tau\leq s} \sup_{x\in B_{1-2\epsilon}(0)}|D_{jk} \dot u(x,\tau)|\nonumber\\
\leq & |t-s| \frac4{\epsilon^2} \sup_{t\leq \tau\leq s} \sup_{x\in B_{1-\epsilon}(0)}|D_{k} \dot u(x,\tau)|\nonumber\\
\leq & |t-s| \frac4{\epsilon^2} \sup_{t\leq \tau\leq s} \sup_{x\in B_{1-\epsilon}(0)}| \frac{|\dot u(x+\epsilon e_k,\tau)-\dot u(x,\tau)|}\epsilon\nonumber\\
\leq & |t-s| \frac4{\epsilon^3}  [\dot u]^{\operatorname{space}}_{\frac\gamma4, U_1}\epsilon^\gamma\nonumber\\
\leq &  4 [\dot u]^{\operatorname{space}}_{\frac\gamma4, U_1}|t-s|^{\frac{1+\gamma}4}.\label{SchauderCh_BetterHoelderRegularityFormula1orde3_3}
\end{align}

Combining Estimates (<ref>) and (<ref>) implies that for $x\in B_{1-3\epsilon}(0)$ and $|t-s|\leq\frac16$ we have

\begin{equation}\label{SchauderCh_ClaimedTimeHölderForNabla3}
\frac{|\nabla^3 u(x,t)-\nabla^3 u(x,s)|}{|t-s|^{\frac{1+\gamma}4}}\leq C(n)[D^{4,1} u]^{(0)}_{\gamma, U_1}.
\end{equation}

If now $x\not\in B_{1-3\epsilon}(0)$ but still $|t-s|\leq\frac16$, we can find $z\in B_{1-3\epsilon}(0)$ such that $|x-z|<3\epsilon$. We write

$$\partial_{ijk} u(x,t)=\partial_{ijk} u(z,t)+\int_0^1\langle\nabla \partial_{ijk} u(z+\rho(x-z),t), x-z\rangle d\rho.$$

Making use of Estimate (<ref>) and $|z-x|\leq 3\epsilon=3|t-s|^{\frac14}$, we get

\begin{align}
\frac{|\nabla^3 u(x,t)-\nabla^3 u(x,s)|}{|t-s|^{\frac{1+\gamma}4}}
\leq &C(n)[D^{4,1} u]^{(0)}_{\gamma, U_1}+\frac1{|t-s|^{\frac{1+\gamma}4}}\int_0^1[\nabla^4 u]_{\frac\gamma4, U_1}^{\operatorname{time}}|t-s|^{\frac\gamma4}|x-z|d\rho\nonumber\\
\leq & C(n)[D^{4,1} u]^{(0)}_{\gamma, U_1}.\label{SchauderCh_ClaimedTimeHölderForNabla3FullDisk}
\end{align}

Combining Estimates (<ref>) and (<ref>) we deduce the lemma for $l=3$.

Improved Hölder Continuity of $\nabla^2 u$ and $\nabla u$

Estimating the temporal seminorms for $\nabla^2 u$ and $\nabla u$ is very much analogous to the argument we have shown for $\nabla^3 u$. We begin with $\nabla^2 u$. Using Equation (<ref>) instead of (<ref>) as well as the already established Hölder regularity of $\nabla^3 u$, we can derive that for $x\in B_{1-3\epsilon}(0)$ and $|t-s|\leq\frac16$

\begin{equation}\label{SchauderCh_BetterHoelderRegularityFormula1orde2_2}
\left|\partial_{ij} u(x,t)-D_{ij}^\epsilon u(x,t)-\partial_{ij} u(x,s)+D_{ij}^\epsilon u(x,s)\right|
\leq 2[\nabla^3 u]_{\frac{1+\gamma}4, U_1}^{\operatorname{time}}|t-s|^{\frac{1+\gamma}4}
\end{equation}

instead of Estimate (<ref>). Next, following the analysis that led to Estimate (<ref>), we derive

\begin{equation}\label{SchauderCh_BetterHoelderRegularityFormula1orde2_3}
|D_{ij}^\epsilon u(x,t)- D_{ij}^\epsilon u(x,s)|\leq   4 [\dot u]_{\frac\gamma4, U_1}^{\operatorname{space}}|t-s|^{\frac{2+\gamma}4}.
\end{equation}

Finally, we generalise to $x\in B_1(0)$; this time by using that

$$\partial_{ij} u(x,t)=\partial_{ij} u(z,t)+\int_0^1 \langle\nabla\partial_{ij}u(z+\rho(x-z),t), x-z\rangle d\rho.$$

Combining with Estimate (<ref>), the lemma follows for $l=2$. Finally, for $l=1$, we first use Equation (<ref>) and the now established Hölder continuity of $\nabla^2 u$ to establish the following analog of Estimate (<ref>). For $x\in B_{1-3\epsilon}(0)$

\begin{equation}\label{SchauderCh_BetterHoelderRegularityFormula1orde1_2}
\left|\partial_{i} u(x,t)-D_{i}^\epsilon u(x,t)-\partial_{i} u(x,s)+D_{i}^\epsilon u(x,s)\right|\nonumber\\
\leq[\nabla^2 u]_{\frac{2+\gamma}4, U_1}^{\operatorname{time}}|t-s|^{\frac{3+\gamma}4}.
\end{equation}

Next, following the analysis that lead to Estimate (<ref>), we derive

\begin{equation}\label{SchauderCh_BetterHoelderRegularityFormula1orde1_3}
|D_{i}^\epsilon u(x,t)- D_{i}^\epsilon u(x,s)|\leq   [\dot u]_{\frac\gamma4, U_1}^{\operatorname{space}}|t-s|^{\frac{3+\gamma}4}.
\end{equation}

We deduce the lemma for $l=1$ by following the same arguments as for $l=2,3$.

There exists $\epsilon_0(n)>0$ such that for $\epsilon\in(0,\epsilon_0)$, $p_0\in\R^n$, $\rho>0$, $U_\rho:=U_\rho(p_0)$, $u\in C^{4,1,\gamma}(U_\rho)$, $l\in\set{0,1,2,3}$ and $k\in\set{0,...,3-l}$

$$\rho^{l+k+\gamma}[\nabla^l u]^{\operatorname{time}}_{\frac{k+\gamma}4,U_\rho} \leq\epsilon\rho^{4+\gamma}[D^{4,1}u]^{(0)}_{\gamma, U_\rho}+C(n,\gamma)\epsilon^{-\frac{l+k+\gamma}{4-l-k}}\|u\|_{L^\infty(U_\rho)}.$$

Without loss of generality $\rho=1$ and $p_0=0$. Let $\sigma\in(0,1)$ and $(x,t), (x,s)\in U_1:=U_1(0)$. We first assume that $|t-s|\geq \sigma^4$. Using Theorem <ref> we deduce that for $\sigma<\sigma_0(n)$

\begin{align}
\frac{|\nabla^lu(x,t)-\nabla^lu(x,s)|}{|t-s|^ {\frac{k+\gamma}4}}&\leq 2\sigma^{-k-\gamma}\|\nabla^l u\|_{L^\infty(U_1)}\nonumber\\
&\leq 2\sigma^{-k-\gamma}\left(
\sigma^{4+\gamma-l}[D^{4,1} u]_{\gamma, U_1}^{(0)}+\sigma^{-l}\|u\|_{L^\infty(U_1)}
\right)\nonumber\\
&\leq C(n)\left(
\sigma^{4-l-k}[D^{4,1} u]_{\gamma, U_1}^{(0)}+\sigma^{-l-k-\gamma}\|u\|_{L^\infty(U_1)}
\right)\label{SchauderCh_Interpolation2Proof12}.
\end{align}

So, it suffices to prove the lemma for $|t-s|\leq \sigma^4$. Let $t_0\in(0,1)$ such that $t,s\in(t_0-\sigma^4, t_0+\sigma^4)\subset(0,1)$. If $B_\sigma(x)\subset B_1(0)$, we can use Lemma <ref> and estimate

\begin{align}
\frac{|\nabla^lu(x,t)-\nabla^lu(x,s)|}{|t-s|^ {\frac{k+\gamma}4}}\leq& \frac{|\nabla^lu(x,t)-\nabla^lu(x,s)|}{|t-s|^ {\frac{4-l+\gamma}4}}\frac{|t-s|^ {\frac{4-l+\gamma}4}}{|t-s|^ {\frac{k+\gamma}4}}\nonumber\\
&\leq [\nabla^ l u]_{\frac{4-l+\gamma}4,U_\sigma((x,t_0))}^ {\operatorname{time}}|t-s|^ {\frac{4-l-k}4}\nonumber\\
&\leq C(n)\left(\sigma^ {4-l-k}[D^ {4,1} u]_{\gamma, U_1}^{(0)}+\sigma^ {-l-k-\gamma}\|u\|_{L^ \infty(U_1)}\right).\label{SchauderCh_Interpolation2_CaseReduction}
\end{align}

If $B_\sigma(x)\not\subset B_1(0)$ there is a point $y\in B_1(0)$ such that $x\in B_\sigma(y)\subset B_1(0)$. Using Theorem <ref>, we estimate for $\tau\in\set{t,s}$

$$\frac{|\nabla ^l u(x,\tau)-\nabla ^l u(y,\tau)|}{|t-s|^{\frac{k+\gamma}4}}
\leq
\sigma^{-k-\gamma}[\nabla^l u]_{\gamma, U_1}^{(0)}|x-y|^\gamma
\leq
\sigma^{-k}\left(\sigma^{4-l}[D^{4,1} u]_{\gamma, U_1}^{(0)}+\sigma^{-l-k-\gamma}\|u\|_{L^\infty(U_1)}\right).

Hence Estimate (<ref>) is valid for all $x\in B_1(0)$. Recalling Estimate (<ref>), the lemma follows.

The following proposition can be established by following the same arguments that lead to Theorem <ref> and Lemmas <ref> and <ref>.

Theorem <ref> and Lemmas <ref> and <ref> are also true, when the parabolic balls $U_\rho$ are replaced by the truncated parabolic balls $U_\rho^\pm(p)$, $U_{\rho+}(p)$ and $U_{\rho+}^{\pm}(p)$.

§ HÖLDER SPACES ON MANIFOLDS

§.§ Elliptic Hölder Spaces on Manifolds

Throughout this subsection, we fix a smooth, compact, $n$-dimensional manifold $M$, possibly with boundary $\partial M\neq\emptyset$.

A finite atlas $\mathcal A=\set{(\varphi_i, U_i)\ |\ 1\leq i\leq m}$ of $M$ consisting of charts $\varphi_i:U_i\subset M\rightarrow V_i\subset\R^n$ or $\R^{n-1}\times[0,\infty)$ is called a good atlas of $M$, if:

* For all $1\leq i\leq m$ the sets $V_i$ are convex.

* For all $1\leq i\leq m$ there exist an open set $\tilde U_i\supset\supset U_i$, an open and convex set $\tilde V_i\supset\supset V_i$ and a smooth extension $\tilde\varphi_i:\tilde U_i\rightarrow\tilde V_i$ of $\varphi_i$.

If $\partial M\neq\emptyset$, a good atlas on $M$ provides us with a good atlas on $\partial M$ by considering only those charts $U_i$ that contain boundary points i.e. $U_i\cap\partial M\neq\emptyset$ and restricting to the boundary. That is

$$\varphi_i\big|_{U_i\cap \partial M}:U_i\cap\partial M\rightarrow V_i\cap (\R^{n-1}\times0).$$

In the following definition, we define Hölder spaces on $M$. Afterwards, we will have to justify this definition.

Let $\mathcal A=\set{(\varphi_i, U_i)\ |\ 1\leq i\leq m}$ be a good atlas of $M$, $V_i:=\varphi_i(U_i)$ and $\gamma\in(0,1)$. We say that $u\in C^k(M)$ is in $C^{k,\gamma}_{\mathcal A}(M)$, if

$$\|u\|_{C^{k,\gamma}_\mathcal A(M)}:=\max_{1\leq i\leq m}\|u\circ\varphi_i^{-1}\|_{C^{k,\gamma}(V_i)}<\infty.$$

It is easy to see that all the usual properties of Hölder spaces remain true. For example $(C^{k,\gamma}_\mathcal A(M),\|\cdot\|_{C^{k,\gamma}_\mathcal A(M)})$ is a Banach space. In the next lemma, we show that the space $C^{k,\gamma}_\mathcal A(M)$ is independent of the choice of good atlas.

Let $\mathcal A$ and $\mathcal B$ be two good atlases on a smooth manifold $M$. Then $C^{k,\gamma}_{\mathcal A}(M)=C^{k,\gamma}_{\mathcal B}(M)$ for all $k\in \N_0$ and $\gamma\in (0,1)$. Additionally there exists a constant $C=C(\mathcal A, \mathcal B, k,\gamma)$ such that

\|u\|_{C^ {k,\gamma}_{\mathcal A}(M)} \leq C(\mathcal A,\mathcal B,k,\gamma)\|u\|_{C^ {k,\gamma}_{\mathcal B}(M)}.

It suffices to prove the estimate. Let $\varphi:U\rightarrow V$ be a good chart from $\mathcal A$ and $\hat\varphi:\hat U\rightarrow\hat V$  from $\mathcal B$ such that $U\cap\hat U\neq\emptyset$. We put $V':=\varphi(U\cap\hat U)\subset V$ and put $\psi:=\hat\varphi\circ\varphi^{-1}:V'\rightarrow \hat V$. Then $\psi\in C^\infty(V')$ with bounded derivatives. For $x\in V'$ and a multi-index $\alpha$ of length $l$ we compute

\begin{equation}\label{SchauderCh_HolderSpaceEstimate_GoodAtlas0}
\nabla_\alpha  (u\circ \varphi^{-1})(x)
=\nabla_\alpha (u\circ\hat\varphi^{-1}\circ\psi)(x)
=\sum_{|\beta|\leq l} c_\beta (\psi)(x)\nabla_\beta (u\circ \hat \varphi^{-1})(\psi(x)).
\end{equation}

Here $c_\beta(\psi)(x)$ are smooth functions that depend on $\psi$ and derivatives of $\psi$. Using this formula it is easily seen that

\begin{equation}\label{SchauderCh_HolderSpaceEstimate_GoodAtlas1}
\|u\circ\varphi^{-1}\|_{C^{k}(V')}
\leq\hspace{-.1cm} C(\psi,k)\|u\circ\tilde \varphi^{-1}\|_{C^{k}(\psi(V'))}
\leq\hspace{-.1cm}  C(\psi,k)\|u\circ\hat \varphi^{-1}\|_{C^{k}(\hat V)}
\leq\hspace{-.1cm}  C(\psi,k)\|u\|_{C^{k}_{\mathcal B}(M)}.
\end{equation}

As $\mathcal B$ is a good atlas, we can cover $U$ by good chart domains $\hat U_1,...,\hat U_m$ in $\mathcal B$. Using Estimate (<ref>) we then get

\begin{equation}\label{SchauderCh_NormEquivalence_NormPartResult}
\|u\circ\varphi^{-1}\|_{C^{k}(V)}\leq \sum_{i=1}^m\|u\circ\varphi^{-1}\|_{C^{k}(\varphi(U\cap\tilde U_i))}
\leq C(\mathcal A,\mathcal B,k)\|u\|_{C^{k}_{\mathcal B}(M)}.
\end{equation}

Since this is true for all good charts in $\mathcal A$, we have proven $\|u\|_{C^{k}_{\mathcal A}(M)}\leq C(\mathcal A,\mathcal B,k)\|u\|_{C^{k}_{\mathcal B}(M)}$. It remains to estimate the seminorms.

Let again $\varphi:U\rightarrow V$ be a good chart from $\mathcal A$, $f\in C^{0,\gamma}(V)$,  $\hat\varphi:\hat U\rightarrow\hat V$ be a good chart from $\mathcal B$ such that $\hat U\cap U\neq\emptyset$ and $S\subset\varphi( U\cap \hat U)$. We estimate

\begin{align}
[f]_{\gamma, S}=&\sup_{x\neq y\in S}
\frac{|f(x)-f(y)|}{|x-y|^\gamma}\nonumber\\
\leq &\sup_{x\neq y\in S}
\frac{|(f\circ\varphi\circ\hat \varphi^{-1})(\hat\varphi(\varphi^{-1}(x))-(f\circ\varphi\circ\hat \varphi^{-1})(\hat\varphi(\varphi^{-1}(y))|}{|x-y|^\gamma}\nonumber\\
\leq &[f\circ\varphi\circ\hat\varphi^{-1}]_{\gamma, \hat\varphi(\varphi^{-1}(S))}\sup_{x\neq y\in S}\left|\frac{\hat\varphi(\varphi^{-1}(x))-\hat\varphi(\varphi^{-1}(y))}{|x-y|}\right|^\gamma\nonumber\\
\leq & [f\circ\varphi\circ\hat\varphi^{-1}]_{\gamma, \hat\varphi(\varphi^{-1}(S))}\left(\|\nabla(\hat\varphi\circ\varphi^{-1})\|_{C^0(\varphi(U\cap\hat U))}\right)^\gamma\nonumber\\
\leq& C(\mathcal A,\mathcal B,\gamma) [f\circ\varphi\circ\hat\varphi^{-1}]_{\gamma, \hat\varphi(\varphi^{-1}(S))}.\label{SchauderCh_SchauderNromEquivBasicEstimate}
\end{align}

For general $x,y\in V$ there doesn't exist a good chart in $\mathcal B$ such that $\varphi^{-1}(x),\varphi^{-1}(y)\in \hat U$. Since $\mathcal A$ is a good atlas, there is a smooth extension $\varphi:\tilde U\rightarrow \tilde V$ with $U\subset\subset \tilde U$ and $V\subset\subset\tilde V$. Let $\mathcal B=\set{(\hat\varphi_i, \hat U_i)\ |\ 1\leq i\leq m}$ and put $\Omega_i:=\varphi( \tilde U\cap \hat U_i)$. Then $\Omega_i\subset \tilde V$ are open and cover $\bar V$. Additionally, we can write

$$\Omega_i=\bigcup_{x\in\Omega_i} B_{r_{x,i}}(x)\hspace{.5cm}\textrm{with $r_{x,i}>0$ such that $B_{r_{x,i}}(x)\subset \Omega_i$}.$$

This implies that $\bar V\subset\bigcup_{i=1}^m\Omega_i\subset \bigcup_{i=1}^m\bigcup_{x\in\Omega_i} B_{r_{x,i}}(x)$. By compactness, we can select points $x_1,...., x_J$ and radii $r_j$ such that $V\subset\bar V\subset \bigcup_{j=1}^J B_{r_j}(x_j)$. Note that for each $1\leq j\leq J$ there exists $1\leq i(j)\leq m$ such that $B_{r_j}(x_j)\subset\Omega_{i(j)}$. Using Estimate (<ref>), we obtain

$$[f]_{\gamma, B_{r_j}(x_j)}
\leq \max_{1\leq i\leq m}[f]_{\Omega_i,\gamma}
\leq C(\mathcal A,\mathcal B,\gamma)\max_{1\leq i\leq m}[f\circ\varphi\circ\hat\varphi_i^{-1}]_{\gamma, \hat V_i}.$$

Since $V$ is convex, the sets $B_{r_j}(x_j)\cap V$ are convex and cover $V$. So, using Lemma <ref> and noting that $J$ is a number only depending on $\mathcal A$ and $\mathcal B$, we obtain

\begin{equation}\label{SchauderCh_EquivalenceHolderNormsBASICestimate}
[f]_{\gamma, V}\leq \sum_{j=1}^J[f]_{\gamma, B_{r_j}(x_j)}
\leq C(\mathcal A,\mathcal B,\gamma)\max_{1\leq i\leq m}[f\circ\varphi\circ\hat\varphi_i^{-1}]_{\gamma, \hat V_i}.
\end{equation}

Let $\alpha\in \N_0^n$ be a multi-index of length $k$. We apply Estimate (<ref>) with $f=\nabla_\alpha(u\circ\varphi^{-1})$. In view of Equation (<ref>), we deduce that for all good charts $\varphi:U\rightarrow V$ from $\mathcal A$

\begin{equation}\label{SchauderCh_SemiNormsestimaedResult_EquivalenceNorms}
[\nabla_\alpha (u\circ\varphi)^{-1}]_{\gamma, U}\leq C(\mathcal A,\mathcal B,k,\gamma)\max_{1\leq i\leq m}[\nabla_\alpha(u\circ\hat\varphi_i^{-1})]_{\gamma, \hat V_i}

The lemma follows by combining Estimates (<ref>) and (<ref>).

[colback=white!20!white,colframe=black!100!white,sharp corners, breakable]
We sometimes write $\|u\|_{C^{k,\gamma}(M)}$ without making the choice of good atlas explicit in the notation. This is now justified by
Lemma <ref>.

§.§ Parabolic Hölder Spaces on Manifolds

Throughout this subsection we fix $T\in(0,\infty)$ and  a smooth, compact, $n$-dimensional manifold $M$, possibly with boundary $\partial M\neq\emptyset$. For $m\in\N_0$, we say that a function $u$ is $C^{m,[m/4]}(M\times(0,T))$, if for $k+4l$ the derivatives $\partial_t^l\nabla^k u$ exist and are continuous.

Let $k\in\N_0$ and $\mathcal A=\set{(\varphi_i, U_i)\ |\ 1\leq i\leq m}$  be a good atlas on $M$. We say that a function $u\in C^{k,[k/4]}(M\times(0,T))$ is in $C^{k,[k/4],\gamma}(M\times[0,T])$, if

$$\|u\|_{C^{k,[k/4],\gamma}_{\mathcal A}(M\times[0,T])}:=\max_{1\leq i\leq m}\|u\circ\varphi_i^{-1}\|_{C^{k,[k/4],\gamma}(V_i\times[0,T])}<\infty.$$

Here, in a slight abuse of notation, $(u\circ\varphi_i^{-1})(x,t):=u(\varphi_i^{-1}(x),t).$

As in the elliptic case, we need to justify that this definition does not depend on the choice of good atlas.

Let $M$ be a smooth, $n$-dimensional manifold and let $\mathcal A$ and $\mathcal B$ be good atlases on $M$. There exists a constant $C(\mathcal A,\mathcal B,k,\gamma)$ such that for any $T>0$

\|u\|_{C^ {k,[k/4],\gamma}_\mathcal A(M\times[0,T])}
\leq
C(\mathcal A,\mathcal B,k,\gamma)
\|u\|_{C^ {k,[k/4],\gamma}_\mathcal B(M\times[0,T])}

[colback=white!20!white,colframe=black!100!white,sharp corners, breakable]
The important observation in Lemma <ref> lies in the fact that the constant $C(\mathcal A,\mathcal B,k,\gamma)$ does not depend on $T$.

Let $\varphi:U\rightarrow V$ be a good chart from $\mathcal A$ and $\hat \varphi:\hat U\rightarrow\hat V$ be a good chart from $\mathcal B$ with $U\cap\hat U\neq\emptyset$. We define $\psi:=\hat\varphi\circ\varphi^{-1}:\varphi(U\cap\hat U)\rightarrow \hat\varphi(U\cap\hat U)$. For $4j+l\leq k$ and a multi index $\alpha$ of length $l$, we note the analogue of Equation (<ref>)

\begin{equation}\label{SchauderCh_HolderSpaceEstimate_Parabolic_GoodAtlas0}
\nabla_\alpha\partial_t^j  (u\circ \varphi^{-1})(x,t)
=\nabla_\alpha\partial_t^j (u\circ\hat\varphi^{-1}\circ\psi)(x,t)
=\sum_{|\beta|\leq l} c_\beta (\psi)(x)\nabla_\beta\partial_t^j (u\circ \hat \varphi^{-1})(\psi(x),t).
\end{equation}

For a multi-index $\alpha$, we can use Equation (<ref>) to conclude

∇_α∂_t^j  (u∘φ^-1)(x,t)_C^0(V×[0,T])
≤C(𝒜,ℬ,k,γ)u_C^k,[k/4]_ℬ(M×[0,T])    if |α|+4j≤k,

[∇_α∂_t^j  (u∘φ^-1)]_γ,V×[0,T]^space
≤C(𝒜,ℬ,k,γ)u_C^k,[k/4],γ_ℬ(M×[0,T])    if |α|+4j=k ,

by following the same arguments that we presented in the proof of Lemma <ref>.
It remains to estimate the temporal Hölder seminorms. Let $t\neq s\in [0,T]$ and $j,l\in\N_0$ such that $\eta:=\frac{k+\gamma-l}4-j\in(0,1)$. Let $\varphi:U\rightarrow V$ be a good chart in $\mathcal A$ and $x\in V$. We aim to prove

\begin{equation}\label{SchauderCh_HolderSpaceEstimate_Parabolic_GoodAtlas1GOAL}
\left|\frac{\partial_t^j\nabla^l (u\circ\varphi^{-1})(x,t)-\partial_t^j\nabla^l (u\circ\varphi^{-1})(x,s)}{|t-s|^\eta}\right|
\leq
C(\mathcal A,\mathcal B, k,\gamma)\|u\|_{C^{k,[k/4],\gamma}_{\mathcal B}(M\times[0,T])}.
\end{equation}

Once this is established, the lemma follows by combining Estimates (<ref>), (<ref>) and (<ref>). If $|t-s|\geq 1$, we use Estimate (<ref>) to estimate

\begin{align}
&\left|\frac{\partial_t^j\nabla^l (u\circ\varphi^{-1})(x,t)-\partial_t^j\nabla^l (u\circ\varphi^{-1})(x,s)}{|t-s|^\eta}\right|\nonumber\\
\leq &2\sup_{\tau\in[0,T]}\left|\partial_t^j\nabla^l (u\circ\varphi^{-1})(x,\tau)\right|\nonumber\\
\leq &2\|u\circ\varphi^{-1}\|_{C^{k,[k/4]}(V\times[0,T])}\nonumber\\
\leq &C(\mathcal A,\mathcal B,k)\|u\|_{C^{k,[k/4],\gamma}_{\mathcal B}(M\times[0,T])}.\label{SchauderCh_HolderSpaceEstimate_Parabolic_GoodAtlas4}
\end{align}

Next, we consider the case where $|t-s|\leq 1$. Let $\hat\varphi:\hat U\rightarrow\hat V$ be a good chart from $\mathcal B$ such that $\varphi^{-1}(x)\in\hat U$ and put $\psi:=\hat\varphi\circ\varphi$. Using Equation (<ref>), we estimate

\begin{align}
&\left|\frac{\partial_t^j\nabla^l (u\circ\varphi^{-1})(x,t)-\partial_t^j\nabla^l (u\circ\varphi^{-1})(x,s)}{|t-s|^\eta}\right|\nonumber\\
\leq &\sum_{|\beta|\leq l} |c_\beta (\psi)(x)| \left|\frac{\partial_t^j\nabla_\beta (u\circ\hat \varphi^{-1})(\psi(x),t)-\partial_t^j\nabla_\beta (u\circ\hat \varphi^{-1})(\psi(x),s)}{|t-s|^\eta}\right|.\label{SchauderCh_HolderSpaceEstimate_Parabolic_GoodAtlas5}
\end{align}

We estimate the terms individually. If $|\beta|+4j\leq k-4$, then $\partial_t^j \nabla_\beta u$ is $C^1$ in time and hence, using $|t-s|\leq 1$ and Estimate (<ref>), we can estimate

\begin{align}
&\left|\frac{\partial_t^j\nabla_\beta (u\circ\hat\varphi^{-1})(\psi(x),t)-\partial_t^j\nabla_\beta (u\circ\hat\varphi^{-1})(\psi(x),s)}{|t-s|^\eta}\right|\nonumber\\
\leq
&\sup_{\tau\in[0,T]}\left|\partial_t^{j+1}\nabla_\beta (u\circ\hat\varphi^{-1})(\psi(x),\tau)\right|\nonumber\\
\leq
&\|\partial_t^{j+1}\nabla_\beta (u\circ\hat\varphi^{-1})\|_{C^0(\hat V\times[0,T])}\nonumber\\
\leq &
\|u\|_{C^{k,[k/4],\gamma}_{\mathcal B}(M\times[0,T])}.\label{SchauderCh_HolderSpaceEstimate_Parabolic_GoodAtlas6}
\end{align}

Finally, we need to consider the case where $k-4<|\beta|+4j\leq k$ or, equivalently
$1>\frac{k-|\beta|}4-j\geq 0$. Then $1>\frac{k+\gamma-|\beta|}4-j> 0$ and since $|\beta|\leq l$ we get
$0<\eta\leq \frac{k+\gamma-|\beta|}4-j<1$.
Using that $|t-s|\leq 1$ and the definition of the $C^{k,[k/4],\gamma}$-norm we then get

\begin{align}
&\left|\frac{\partial_t^j\nabla_\beta (u\circ\hat\varphi^{-1})(\psi(x),t)-\partial_t^j\nabla_\beta (u\circ\hat\varphi^{-1})(\psi(x),s)}{|t-s|^\eta}\right|\nonumber\\
\leq &
\left|\frac{\partial_t^j\nabla_\beta (u\circ\hat\varphi^{-1})(\psi(x),t)-\partial_t^j\nabla_\beta (u\circ\hat\varphi^{-1})(\psi(x),s)}{|t-s|^{\frac{k+\gamma-|\beta|}4-j}}\right|\nonumber\\
\leq&
\|u\|_{C^{k,[k/4],\gamma}_{\mathcal B}(M\times[0,T])}.\label{SchauderCh_HolderSpaceEstimate_Parabolic_GoodAtlas7}
\end{align}

Estimate (<ref>) and hence the lemma follows by combining Estimates

[colback=white!20!white,colframe=black!100!white,sharp corners, breakable]
We sometimes write $\|u\|_{C^{k,[k/4],\gamma}(M\times[0,T])}$ without making the choice of good atlas explicit in the notation. This is now justified by
Lemma <ref>.

§ SOBOLEV SPACES ON MANIFOLDS

Throughout this section, let $M$ be a smooth, $n$-dimensional, compact manifold possibly with boundary $\partial M\neq\emptyset$. In this section, we define the notion of Sobolev spaces on $M$.

We say that a set $A\subset M$ is measurable if $\phi(A)$ is Lebesgue measurable for all charts $\phi:U\subset M\rightarrow V\subset\R^n$. We say that $A\subset M$ is a null set if $\phi(A)$ is a Lebesgue null set for all charts $\phi:U\subset M\rightarrow V\subset\R^n$.

We denote by $\sim$ the equivalence relation that identifies functions $u$ and $v$ if they agree apart from a null set. As usual, we will often not distinguish between a function $u$ and its equivalence class $[u]_\sim$.

A function $u:M\rightarrow\R$ is in $W^{k,p}_{\operatorname{loc}}(M)$ if $u\circ\phi^{-1}\in W^{k,p}_{\operatorname{loc}}(V)$ for all charts $\phi:U\subset M\rightarrow V\subset\R^n$.

* In case of a boundary chart where $V\subset\R^ {n-1}\times[0,\infty)$ and $V\cap(\R^ {n-1}\times 0)\neq\emptyset$, we say that $v\in W^ {k,p}(V)$ if $v\in W^ {k,p}(V^+)$, where $V^+:=\set{x\in V\ |\ x_n>0}$.

* We stress that for $V\subset\R^ {n-1}\times[0,\infty)$ and $V\cap(\R^ {n-1}\times 0)\neq\emptyset$ the `$\operatorname{loc}$' refers to sets $W\subset V$ open in the relative topology on $\R^ {n-1}\times[0,\infty)$ with closure $\bar W\subset V$. Such $W$ are allowed to contain points of the form $(x,0)$.

To define a sensible $W^{k,p}(M)$-norm, we can not use an arbitrary chart. For example we consider $M=(0,1)$ and $u:M\rightarrow\R,\ u(x)=x$. A sensible definition of $W^{k,p}(M)$ must surely include $u$. However, taking the chart $\varphi:(0,1)\rightarrow (0,\infty), \varphi(x)=\frac1x$ produces the chart representation $u(\varphi^{-1}(t))=\frac1t$, which is not even in $L^1((0,\infty))$. The issue is, of course, that the chosen chart is not good. To eliminate this problem, we make the following definition:

Let $\mathcal A=\set{(\varphi_i, U_i)\ |\ 1\leq i\leq m}$ be a good atlas on $M$. It is easy to see that for all $u\in W^{k,p}_{\operatorname{loc}}(M)$ the expression

\begin{equation}\label{SchauderCh_SobolevSpaceDefinition}
\|u\|_{W^{k,p}_\mathcal A(M)}:=\sum_{i=1}^m\|u\circ\varphi_i^{-1}\|_{W^{k,p}(V_i)}
\end{equation}

is finite. We define $W^{k,p}(M):=W^{k,p}_{\operatorname{loc}}(M)$,  equipped with $\|\cdot \|_{W^{k,p}_\mathcal A(M)}$.

Indeed, any good chart $\varphi_i:U_i\rightarrow V_i$ can be extended to $\varphi_i:\tilde U_i\rightarrow\tilde V_i$ with $V_i\subset\subset \tilde V_i$ and by definition of $W^{k,p}_{\operatorname{loc}}(M)$ we get $u\circ\varphi_i^{-1}\in W^{k,p}(V_i)$.

Similar to the proof of Lemma <ref>, we can establish the following proposition, which shows, that Definition <ref> does not depend on the choice of the good atlas.

Let $M$ be a compact $n$-dimensional manifold and $\mathcal A$ and $\mathcal B$ be two good atlases on $M$. Then, for all $u\in W^{k,p}_{\operatorname{loc}}(M)$ we have

$$\|u\|_{W^{k,p}_\mathcal A(M)}\leq C(\mathcal A,\mathcal B,k,p)\|u\|_{W^{k,p}_{\mathcal B}(M)}.$$

It is easy to see that the usual properties of Sobolev spaces, e.g. their completeness, carry over to the manifold setting. Also, we often use the following embedding theorems. For a proof in $\R^n$, we refer to Evan's book [7] (see Chapter 5, Section 7, Theorem 1 with the subsequent remark for Rellich's theorem and Chapter 5, Section 6 Theorem 4 for Morrey's inequality).

Let $M$ be a  compact, $n$-dimensional manifold, possibly with boundary. The following hold:

* (Rellich's Theorem)  The inclusion $W^{1,p}(M)\hookrightarrow L^p(M)$ is compact for all $1\leq p <\infty$.

* (Morrey's Inequality) If $r\in\N_0$, $\alpha\in(0,1)$ and $k-\frac np=r+\alpha$, then the inclusion $W^{k,p}(M)\hookrightarrow C^{r,\alpha}(M)$ is continuous.

Another result that carries over from $\R^n$ is the trace theorem. For $\R^n$, this result can be found in Evan's book [7] (see Chapter 5, Section 5, Theorem 1).

Let $M$ be a compact manifold with boundary $\partial M\neq\emptyset$ and $p\in[1,\infty)$. There exists a linear and continuous map $T:W^{1,p}(M)\rightarrow L^p(\partial M)$ such that $Tu=u|_{\partial M}$ for $u\in C^0(M)$.

As usual, we write $u$ instead of $Tu$.

§.§ The Weak Gradient

Let $(M,g)$ be a compact, Riemannian manifold, $\mathcal A=\set{(\varphi_i, U_i)\ |\ 1\leq i\leq m}$ be a good atlas and $u\in W^ {1,p}(M)$. For each $1\leq i\leq m$, we can define the vector field

$$X_i:U_i\rightarrow TU_i,\ X_i(p):=g^ {ab}(\varphi_i(p))\partial_a (u\circ\varphi_i^ {-1})(\varphi_i(p))\frac{\partial}{\partial x^b}\bigg|_p.$$

Given two overlapping chart domains $U_i\cap U_j\neq\emptyset$, it is easy to see that $X_i$ and $X_j$ coincide on $U_i\cap U_j$ apart from perhaps a null set. So, we may define the weak gradient $\nabla u$ globally on $M$ by the local coordinates formula

$$\nabla_g u(p)=g^ {ab}(\varphi_i(p))\partial_a (u\circ\varphi_i^ {-1})(\varphi_i(p))\frac{\partial}{\partial x^b}\bigg|_p.$$

Let $(M,g)$ be a compact Riemannian manifold, possibly with boundary. The norms

\|u\|_{L^p_g(M)}:=\left(\int_M |u|^p d\mu_g\right)^{\frac1p}
\hspace{.5cm}\textrm{and}\hspace{.5cm}
\|u\|_{W^{1,p}_g(M)}:=\left(\int_M |u|^p d\mu_g+\int_M \|\nabla_g u\|_g^p d\mu_g\right)^{\frac1p}

provide norms on $L^p(M)$ and $W^{1,p}(M)$ that are equivalent to the $L^p(M)$- and $W^{1,p}(M)$-norm from Definition <ref>. Here $\|\nabla_g u\|_g:=(g(\nabla_g u,\nabla_g u))^ {\frac12}$.

For $p=2$, Proposition <ref> provides us with scalar products on $L^2(M)$ and $W^{1,2}(M)$ that makes these spaces Hilbert spaces. Concretely

\langle u,v\rangle_{L^2_g(M)}:=\int_M uv d\mu_g
\hspace{.5cm}\textrm{and}\hspace{.5cm}
\langle u,v\rangle_{W^{1,2}_g(M)}:=\int_M uv+g(\nabla_g u,\nabla_g v)d\mu_g.

The weak gradient satisfies the following expected property.

Let $(M,g)$ be a compact Riemannian manifold, possibly with boundary $\partial M$. We denote the induced measure on $M$ by $\mu_g$ and the induced measure on the boundary by $S_g$. For any function $u\in W^{1,p}(M)$ and vector field $Y\in C^1(TM)$

\begin{equation}\label{NeumannLaplacianCh_DivergenceTheorem}
\int_M g(\nabla_g u,Y)d\mu_g=-\int_M u\operatorname{div}_g Y d\mu_g+\int_{\partial M}u g(Y,\nu_g)dS_g.
\end{equation}

$\nu_g$ denotes the exterior unit normal along $\partial M$ with respect to $g$ if $\partial M\neq\emptyset$. If $\partial M=\emptyset$, the boundary integral in Equation (<ref>) vanishes.

For $u\in C^1(M)$, this result is standard and can, for example, be found in [14] (see Chapter 16, Theorem 16.32). For general $u\in W^{1,p}(M)$, we choose a sequence $(u_k)\subset C^1(M)$ such that $u_k\rightarrow u$ in $W^{1,p}(M)$. Equation (<ref>) holds with $u$ replaced by $u_k$. By the continuity of the trace operator from Proposition <ref>, we can send $k\rightarrow\infty$ and obtain (<ref>) for $u\in W^{1,p}(M)$.

Let $(M,g)$ be a compact Riemannian manifold with boundary $\partial M\neq\emptyset$. Then, for any $\epsilon>0$ there exists a constant $C(\epsilon)$ such that for all $u\in W^ {1,2}(M)$

$$\|u\|_{L^2_g(\partial M)}^2\leq \epsilon\int_M \|\nabla_g u\|_g^2 d\mu_g+C(\epsilon)\|u\|_{L^2_g(M)}^2.$$

Let $\nu$ denote the exterior unity normal along $\partial M$ and let $\bar\nu$ denote a smooth vector field on $M$ such that $\bar\nu|_{\partial M}=\nu$. Given any smooth function $u\in C^\infty(M)$ we have

$$\int_{\partial M} u^2dS_g=\int_{\partial M}g(u^2\bar\nu,\nu)dS_g=\int_{M}\operatorname{div}_g(u^2\bar\nu)d\mu_g=\int_M u^2\operatorname{div}_g\bar\nu+2ug(\bar\nu,\nabla_g u)d\mu_g.$$

Consequently, there exists a constant $C(M)$ such that

\begin{equation}\label{SchauderCh_TraceEstimateEq1}
\int_{\partial M} u^2 dS_g\leq C\left(\int_M u^2 d\mu_g+\int_M |u|\ \|\nabla_g u\|_gd\mu_g\right).
\end{equation}

By density, Estimate (<ref>) is true for all $u\in W^{1,2}(M)$. The lemma follows by employing Young's inequality.

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