In this appendix, we prove a series of properties of the norms introduced in Section 2 (see (2.4), (2.5), (2.16) and (2.17)). But, first, we need to recall the notion of Hölder classes of functions \(C^{\sigma}\) and some properties.
We consider \(G\) as an open subset of \(\mathbb{R}^{n}\). Let \(k\geqslant 0\) be a positive integer, we define \(C^{k}(G)\) as the spaces of functions \(f:G\to\mathbb{R}\) with continuous partial derivatives \(\partial^{\alpha}f\in C^{0}(G)\) for all \(\alpha\in\mathbb{N}^{n}\) with \(|\alpha|=\alpha_{1}+\ldots+\alpha_{n}\leqslant k\) and verifying
$$|f|_{C^{k}}=\sup_{|\alpha|\leqslant k}|\partial^{\alpha}f|_{C^{0}}=\sup_{|\alpha|\leqslant k}\sup_{x\in G}|\partial^{\alpha}f(x)|<\infty.$$
Given
\(\sigma=k+\mu\), with
\(k\in\mathbb{Z}\),
\(k\geqslant 0\) and
\(0<\mu<1\), we define the Hölder spaces
\(C^{\sigma}(G)\) as the spaces of functions
\(f\in C^{k}(G)\) verifying
$$|f|_{C^{\sigma}}=\sup_{|\alpha|\leqslant k}|\partial^{\alpha}f|_{C^{0}}+\sup_{|\alpha|=k}\sup_{x,y\in G,x\neq y}{|\partial^{\alpha}f(x)-\partial^{\alpha}f(y)|\over|x-y|^{\mu}}<\infty.$$
(A.1)
In the case of functions
\(f=(f_{1},\ldots,f_{n})\) with values in
\(\mathbb{R}^{n}\), we define
\(|f|_{C^{\sigma}}=\max_{1\leqslant i\leqslant n}|f_{i}|_{C^{\sigma}}\) and, in agreement with the convention made above, if
\(M\) is an
\(n\times n\) matrix we set
\(|M|_{C^{\sigma}}=\max_{1\leqslant i,j\leqslant n}|M_{ij}|_{C^{\sigma}}\).
The following proposition contains some properties about Hölder classes of functions.
We recall that \(C(\cdot)\) stands for constants depending on the parameters in brackets.
Proposition 6
We consider \(f\), \(g\in C^{\sigma}(G)\) and \(\sigma\geqslant 0\).
-
1)
For all \(\beta\in\mathbb{N}^{n}\) and \(s\geqslant 0\), if \(|\beta|+s\leqslant\sigma\) then \(\left|{\partial^{|\beta|}\over\partial{x_{1}}^{\beta_{1}}\ldots\partial{x_{n}}^{\beta_{n}}}f\right|_{C^{s}}\leqslant C(n)|f|_{C^{\sigma}}\).
-
2)
\(|fg|_{C^{\sigma}}\leqslant C(n,\sigma)\left(|f|_{C^{0}}|g|_{C^{\sigma}}+|f|_{C^{\sigma}}|g|_{C^{0}}\right)\).
Concerning composite functions, let \(z\) be defined on \(G_{1}\subset\mathbb{R}^{n}\) and let it take its
values on \(G_{2}\subset\mathbb{R}^{n}\) where \(f\) is defined.
If \(\sigma\geqslant 1\) and \(f\in C^{\sigma}(G_{2})\), \(z\in C^{\sigma}(G_{1})\), then \(f\circ z\in C^{\sigma}(G_{1})\)
-
3)
\(|f\circ z|_{C^{\sigma}}\leqslant C(n,\sigma)\left(|f|_{C^{\sigma}}|z|^{\sigma}_{C^{1}}+|f|_{C^{1}}|z|_{C^{\sigma}}+|f|_{C^{0}}\right)\).
Proof
We refer to [13] for the proof.
\(\square\)
Here, we want to prove Proposition 2. For this purpose, we recall the definition of the space of functions \(\mathcal{B}_{\sigma,k}\) (see Definition 4), the norms (2.4) and (2.5), and the statement of Proposition 2.
Definition
We consider \(\sigma\geqslant 0\), a positive integer \(k\geqslant 0\), \(B\subset\mathbb{R}^{n}\) an open ball centered at the origin, and an open interval \(I\subset\mathbb{R}\). Let \(\mathcal{B}_{\sigma,k}\) be the space of functions \(f:\mathbb{T}^{n}\times B\times I\to\mathbb{R}\) such that
$$\displaystyle\begin{aligned} &\displaystyle f_{p}^{t}\in C^{\sigma+k}(\mathbb{T}^{n})\quad\mbox{for all fixed $(p,t)\in B\times I$},\\ &\displaystyle\partial^{i}_{q}f,\partial^{j}_{(q,p)}\left(\partial_{q}^{i}f\right)\in C(\mathbb{T}^{n}\times B\times I)\\ &\displaystyle\mbox{for all $(j,i)\in\mathbb{N}^{2n}\times\mathbb{N}^{n}$ with $|j|=1$ and $0\leqslant|i|\leqslant k$}.\end{aligned}$$
Concerning the norms \(|\cdot|_{\sigma,k,l}\) and \(|\cdot|_{\sigma,k,0}\) (see (2.4) and (2.5), respectively), we recall that, for all \(f\in\mathcal{B}_{\sigma,k}\) and \(l>1\),
$$\displaystyle\begin{aligned} &\displaystyle|f|_{\sigma,k,l}=\sup_{(p,t)\in B\times I}|f^{t}_{p}|_{C^{\sigma+k}}(1+|t|^{l})+\max_{0\leqslant|i|\leqslant k}\sup_{(p,t)\in B\times I}|\left(\partial_{p}\partial_{q}^{i}f\right)^{t}_{p}|_{C^{0}}(1+|t|^{l-1})\\ &\displaystyle|f|_{\sigma,k,0}=\sup_{(p,t)\in B\times I}|f^{t}_{p}|_{C^{\sigma+k}}+\max_{0\leqslant|i|\leqslant k}\sup_{(p,t)\in B\times I}|\left(\partial_{p}\partial_{q}^{i}f\right)^{t}_{p}|_{C^{0}}.\end{aligned}$$
Now, we have everything we need to rewrite the statement of Proposition 2 and prove it.
Proposition 1
Given \(\sigma\geqslant 0\) and a positive integer \(k\geqslant 0\), for all \(f\), \(g\in\mathcal{B}_{\sigma,k}\) and positive \(l\), \(m\geqslant 1\) and \(d\geqslant 0\)
-
1)
\(|f|_{\sigma^{\prime},k^{\prime},l}\leqslant C(n)|f|_{\sigma,k,l}\) for all \(0\leqslant\sigma^{\prime}\leqslant\sigma\) and \(k^{\prime}\in\mathbb{Z}^{n}\) with \(0\leqslant k^{\prime}\leqslant k\).
-
2)
\(|f|_{\sigma,k,l}\leqslant C(n,l,d)|f|_{\sigma,k,l+d}\)
-
3)
\(|fg|_{\sigma,k,l+m}\leqslant C(n,\sigma,k)\left(|f|_{0,0,l}|g|_{\sigma,k,m}+|f|_{\sigma,k,l}|g|_{0,0,m}\right)+C(n,k)|f|_{0,k,l}|g|_{0,k,m}\).
Moreover, we consider \(\tilde{g}\in\mathcal{B}_{\sigma,k}\) such that \(\tilde{g}:\mathbb{T}^{n}\times B\times I\to\mathbb{T}^{n}\times B\times I\) with \(\tilde{g}(q,p,t)=(g(q,p,t),p,t)\). Then \(f\circ\tilde{g}\in\mathcal{B}_{\sigma,k}\) and
-
4)
\(|f\circ\tilde{g}|_{\sigma,k,l+m}\leqslant C(n,\sigma,k)\left(|f|_{\sigma,k,l}|g|^{\sigma+k}_{1,0,m}+|f|_{1,0,l}|g|_{\sigma,k,m}+|f|_{0,0,l+m}\right)\) \(\hskip 65.441339pt+C(n,k)\left(|f|_{1,k,l}|g|_{0,0,m}+|f|_{0,k,l}|g|_{0,k,m}+|f|_{0,k,l+m}\right)\left(1+|g|_{0,k,0}^{k}\right)\).
Before the proof, we observe that the previous properties are still verified when \(l=m=0\) or only one of the two parameters, \(l\) or \(m\), is zero.
Proof
The proof rests on Proposition 6. The proof of 1 is a consequence of property 1 of Proposition 6. In fact, for all \(0\leqslant\sigma^{\prime}\leqslant\sigma\) and \(k^{\prime}\in\mathbb{Z}^{n}\) with \(0\leqslant k^{\prime}\leqslant k\), we note that
$$\displaystyle\begin{aligned} &\displaystyle|f|_{\sigma^{\prime},k^{\prime},l}=\sup_{(p,t)\in B\times I}|f^{t}_{p}|_{C^{\sigma^{\prime}+k^{\prime}}}(1+|t|^{l})+\max_{0\leqslant|i|\leqslant k^{\prime}}\sup_{(p,t)\in B\times I}|\left(\partial_{p}\partial_{q}^{i}f\right)^{t}_{p}|_{C^{0}}(1+|t|^{l-1})\\ &\displaystyle\leqslant C(n)\left(\sup_{(p,t)\in B\times I}|f^{t}_{p}|_{C^{\sigma+k}}(1+|t|^{l})+\max_{0\leqslant|i|\leqslant k}\sup_{(p,t)\in B\times I}|\left(\partial_{p}\partial_{q}^{i}f\right)^{t}_{p}|_{C^{0}}(1+|t|^{l-1})\right)\\ &\displaystyle=C(n)|f|_{\sigma,k,l}.\end{aligned}$$
Concerning the proof of 2 we observe that
$$\displaystyle\begin{aligned} &\displaystyle|f|_{\sigma,k,l}=\sup_{(p,t)\in B\times I}|f^{t}_{p}|_{C^{\sigma+k}}(1+|t|^{l})+\max_{0\leqslant|i|\leqslant k}\sup_{(p,t)\in B\times I}|\left(\partial_{p}\partial_{q}^{i}f\right)^{t}_{p}|_{C^{0}}(1+|t|^{l-1})\\ &\displaystyle\leqslant C(l,d)\left(\sup_{(p,t)\in B\times I}|f^{t}_{p}|_{C^{\sigma+k}}(1+|t|^{l+d})+\max_{0\leqslant|i|\leqslant k}\sup_{(p,t)\in B\times I}|\left(\partial_{p}\partial_{q}^{i}f\right)^{t}_{p}|_{C^{0}}(1+|t|^{l+d-1})\right)\\ &\displaystyle=C(l,d)|f|_{\sigma,k,l+d},\end{aligned}$$
where in the first inequality of the latter we used
\((1+|t|^{l})\leqslant C(l,d)(1+|t|^{l+d})\) and
\((1+|t|^{l-1})\leqslant C(l,d)(1+|t|^{l+d-1})\).
The proof of 3 relies on property 2 of Proposition 6. In fact, for all fixed \((p,t)\in B\times\mathbb{R}\)
$$\displaystyle\begin{aligned} \displaystyle\left|f^{t}_{p}g^{t}_{p}\right|_{C^{\sigma+k}}\left(1+|t|^{l+m}\right)&\displaystyle\leqslant C(\sigma,k)\left(|f^{t}_{p}|_{C^{0}}|g^{t}_{p}|_{C^{\sigma+k}}+|f^{t}_{p}|_{C^{\sigma+k}}|g^{t}_{p}|_{C^{0}}\right)\left(1+|t|^{l}\right)\left(1+|t|^{m}\right)\\ &\displaystyle\leqslant C(\sigma,k)\Big{(}|f_{p}^{t}|_{C^{0}}\left(1+|t|^{l}\right)|g_{p}^{t}|_{C^{\sigma+k}}\left(1+|t|^{m}\right)\\ &\displaystyle{}\quad+|f_{p}^{t}|_{C^{\sigma+k}}\left(1+|t|^{l}\right)|g_{p}^{t}|_{C^{0}}\left(1+|t|^{m}\right)\Big{)}\\ &\displaystyle\leqslant C(\sigma,k)\left(|f|_{0,0,l}|g|_{\sigma,k,m}+|f|_{\sigma,k,l}|g|_{0,0,m}\right),\end{aligned}$$
where in the first line we used
\(\left(1+|t|^{l+m}\right)\leqslant\left(1+|t|^{l}\right)\left(1+|t|^{m}\right)\).
Taking the sup for all
\((p,t)\in B\times I\) on the left-hand side of the latter, we obtain
$$\sup_{(p,t)\in B\times\mathbb{R}}\left|f^{t}_{p}g^{t}_{p}\right|_{C^{\sigma+k}}\left(1+|t|^{l+m}\right)\leqslant C(\sigma,k)\left(|f|_{0,0,l}|g|_{\sigma,k,m}+|f|_{\sigma,k,l}|g|_{0,0,m}\right).$$
It remains to estimate the second term of the norm (see the right-hand side of
(2.4)). To accomplish this, we utilize the multi-index notation. For all
\(\alpha\),
\(i\in\mathbb{N}^{n}\) with
\(|i|\leqslant k\), we observe that
$$\partial_{q}^{i}(fg)=\sum_{\alpha\leqslant i}{i\choose\alpha}\partial_{q}^{\alpha}f\partial_{q}^{i-\alpha}g,$$
(A.2)
where
\(\partial_{q}^{i}=\partial_{q_{1}}^{i_{1}}\ldots\partial_{q_{n}}^{i_{n}}\),
\({i\choose\alpha}={i_{1}\choose\alpha_{1}}{i_{2}\choose\alpha_{2}}\ldots{i_{n}\choose\alpha_{n}}\) and
\(\alpha\leqslant i\) is equivalent to
\(\alpha_{j}\leqslant i_{j}\) for all
\(j=1,\ldots,n\). Taking the derivative with respect to
\(p\) of
(A.2), we obtain
$$\partial_{p}\partial_{q}^{i}(fg)=\sum_{\alpha\leqslant i}{i\choose\alpha}\left(\partial_{p}\partial_{q}^{\alpha}f\partial_{q}^{i-\alpha}g-\partial_{q}^{\alpha}f\partial_{p}\partial_{q}^{i-\alpha}g\right).$$
(A.3)
Thanks to the latter, we have the following estimate for all fixed
\((p,t)\in B\times I\):
$$\displaystyle\begin{aligned} &\displaystyle\left|\left(\partial_{p}\partial_{q}^{i}\left(fg\right)\right)_{p}^{t}\right|_{C^{0}}\left(1+|t|^{l+m-1}\right)\\ &\displaystyle\leqslant\sum_{\alpha\leqslant i}{i\choose\alpha}\left(\left|\left(\partial_{p}\partial_{q}^{\alpha}f\right)^{t}_{p}\left(\partial_{q}^{i-\alpha}g\right)^{t}_{p}\right|_{C^{0}}+\left|\left(\partial_{q}^{\alpha}f\right)^{t}_{p}\left(\partial_{p}\partial_{q}^{i-\alpha}g\right)^{t}_{p}\right|_{C^{0}}\right)\left(1+|t|^{l+m-1}\right)\\ &\displaystyle\leqslant C(n,k)\max_{0\leqslant|i|\leqslant k}\sup_{(p,t)\in B\times I}\left|\left(\partial_{p}\partial_{q}^{i}f\right)^{t}_{p}\right|_{C^{0}}\left(1+|t|^{l-1}\right)\left|g^{t}_{p}\right|_{C^{k}}\left(1+|t|^{m}\right)\\ &\displaystyle+C(n,k)\left|f^{t}_{p}\right|_{C^{k}}\left(1+|t|^{l}\right)\max_{0\leqslant|i|\leqslant k}\sup_{(p,t)\in B\times I}\left|\left(\partial_{p}\partial_{q}^{i}g\right)^{t}_{p}\right|_{C^{0}}\left(1+|t|^{m-1}\right).\end{aligned}$$
The first inequality in the latter is due to
(A.3). For the second inequality, one has to follow these steps.
Firstly, we need to estimate the terms inside the sum such that they no longer depend on
\(\alpha\). Secondly, one can estimate the sum by a constant
\(C(n,k)\). It is obtained by taking
the maximum for all
\(0\leqslant\alpha\leqslant|i|\leqslant k\), i. e.,
\(\sum_{\alpha\leqslant i}{i\choose\alpha}\leqslant C(n,k)\).
Now, thanks to the latter,
$$\displaystyle\begin{aligned} &\displaystyle\max_{0\leqslant|i|\leqslant k}\sup_{(p,t)\in B\times I}\left|\left(\partial_{p}\partial_{q}^{i}\left(fg\right)\right)_{p}^{t}\right|_{C^{0}}\left(1+|t|^{l+m-1}\right)\\ &\displaystyle\leqslant C(n,k)\left(\max_{0\leqslant|i|\leqslant k}\sup_{(p,t)\in B\times I}\left|\left(\partial_{p}\partial_{q}^{i}f\right)^{t}_{p}\right|_{C^{0}}\left(1+|t|^{l-1}\right)\right)|g|_{0,k,m}\\ &\displaystyle+C(n,k)|f|_{0,k,l}\left(\max_{0\leqslant|i|\leqslant k}\sup_{(p,t)\in B\times I}\left|\left(\partial_{p}\partial_{q}^{i}g\right)^{t}_{p}\right|_{C^{0}}\left(1+|t|^{m-1}\right)\right)\leqslant C(n,k)|f|_{0,k,l}|g|_{0,k,m}.\end{aligned}$$
This concludes the proof of
3. We now proceed to verify
4. First, we prove
4 for the case
\(k=0\). Subsequently, we consider the case
\(k\geqslant 1\).
Let
\(k=0\), for all fixed
\((p,t)\in B\times\mathbb{R}\) and, thanks to property
3 of
Proposition
6,
$$\displaystyle\begin{aligned} \displaystyle|f^{t}_{p}\circ g_{p}^{t}|_{C^{\sigma}}\left(1+|t|^{l+m}\right)&\displaystyle\leqslant C(n,\sigma)\left(\left|f_{p}^{t}\right|_{C^{\sigma}}\left|g_{p}^{t}\right|^{\sigma}_{C^{1}}+\left|f_{p}^{t}\right|_{C^{1}}\left|g_{p}^{t}\right|_{C^{\sigma}}+\left|f_{p}^{t}\right|_{C^{0}}\right)\left(1+|t|^{l+m}\right)\\ &\displaystyle\leqslant C(n,\sigma)\Big{(}\left|f_{p}^{t}\right|_{C^{\sigma}}\left(1+|t|^{l}\right)\left|g_{p}^{t}\right|^{\sigma}_{C^{1}}\left(1+|t|^{m}\right)^{\sigma}\\ &\displaystyle{}\quad+\left|f_{p}^{t}\right|_{C^{1}}\left(1+|t|^{l}\right)\left|g_{p}^{t}\right|_{C^{\sigma}}\left(1+|t|^{m}\right)\\ &\displaystyle{}\quad+\left|f_{p}^{t}\right|_{C^{0}}\left(1+|t|^{l+m}\right)\Big{)}\\ &\displaystyle\leqslant C(n,\sigma)\left(|f|_{\sigma,0,l}|g|^{\sigma}_{1,0,m}+|f|_{1,0,l}|g|_{\sigma,0,m}+|f|_{0,0,l+m}\right),\end{aligned}$$
where in the second line we used
\(\left(1+|t|^{m}\right)\leqslant\left(1+|t|^{m}\right)^{\sigma}\). Taking the sup for all
\((p,t)\in B\times I\) on the left-hand side of the latter,
$$\sup_{(p,t)\in B\times I}|f^{t}_{p}\circ g_{p}^{t}|_{C^{\sigma}}\left(1+|t|^{l+m}\right)\leqslant C(n,\sigma)\left(|f|_{\sigma,0,l}|g|^{\sigma}_{1,0,m}+|f|_{1,0,l}|g|_{\sigma,0,m}+|f|_{0,0,l+m}\right).$$
Concerning the second term of the norm (see
(2.4)), for all fixed
\((p,t)\in B\times I\)
$$\displaystyle\begin{aligned} \displaystyle\left|\left(\partial_{p}\left(f\circ\tilde{g}\right)\right)_{p}^{t}\right|_{C^{0}}\left(1+|t|^{l+m-1}\right)&\displaystyle=\left|\left(\partial_{q}f\right)^{t}_{p}\circ g^{t}_{p}\left(\partial_{p}g\right)^{t}_{p}\right|_{C^{0}}\left(1+|t|^{l+m-1}\right)\\ &\displaystyle{}\quad+\left|\left(\partial_{p}f\right)^{t}_{p}\circ g^{t}_{p}\right|_{C^{0}}\left(1+|t|^{l+m-1}\right)\\ &\displaystyle\leqslant C(n)\left|f^{t}_{p}\right|_{C^{1}}\left(1+|t|^{l}\right)\left|\left(\partial_{p}g\right)^{t}_{p}\right|_{C^{0}}\left(1+|t|^{m-1}\right)\\ &\displaystyle{}\quad+\left|\left(\partial_{p}f\right)^{t}_{p}\right|_{C^{0}}\left(1+|t|^{l+m-1}\right)\\ &\displaystyle\leqslant C(n)\left(|f|_{1,l}|g|_{\sigma,m}+|f|_{0,l+m}\right).\end{aligned}$$
Taking the sup for all
\((p,t)\in B\times I\) on the left-hand side of the above inequality,
$$\sup_{(p,t)\in B\times I}\left|\left(\partial_{p}\left(f\circ\tilde{g}\right)\right)_{p}^{t}\right|_{C^{0}}\left(1+|t|^{l+m-1}\right)\leqslant C(n)\left(|f|_{1,0,l}|g|_{0,0,m}+|f|_{0,0,l+m}\right).$$
This proves
4 when
\(k=0\). Now, we consider
\(k\geqslant 1\). Similarly to the previous case,
$$\sup_{(p,t)\in B\times I}|f^{t}_{p}\circ g_{p}^{t}|_{C^{\sigma+k}}\left(1+|t|^{l+m}\right)\leqslant C(n,\sigma,k)\left(|f|_{\sigma,k,l}|g|^{\sigma+k}_{1,0,m}+|f|_{1,0,l}|g|_{\sigma,k,m}+|f|_{0,0,l+m}\right).$$
For the second term of the norm
(2.4), we employ Faà di Bruno’s formula.
Let
\(d\),
\(r\in\mathbb{N}\) and denote by
\(\Omega_{r}^{d}\) the set of partitions of
\(\{1,\ldots,r\}\subset\mathbb{N}\) into
\(d\) nonempty subsets. For all
\(i\in\mathbb{N}^{n}\) with
\(1\leqslant|i|\leqslant k\), let
\(\{\gamma_{j}\}_{j=1}^{|i|}\subset\mathbb{N}^{n}\) be any sequence of bases such that
\(i=\sum_{j=1}^{|i|}\gamma_{j}\) with
\(|\gamma_{j}|=1\). Here, we recall that, for all
\(i\in\mathbb{N}^{n}\),
\(|i|=i_{1}+\ldots+i_{n}\).
Now, for all
\((p,t)\in B\times I\), Faà di Bruno’s formula yields the following expression:
$$\left(\partial_{q}^{i}\left(f\circ\tilde{g}\right)\right)^{t}_{p}=\sum_{d=1}^{|i|}\sum_{j_{1}=1}^{n}\ldots\sum_{j_{d}=1}^{n}\underbrace{{\partial^{d}\over\partial q_{j_{1}}\ldots\partial q_{j_{d}}}f^{t}_{p}\circ g_{p}^{t}}_{A_{d}}\underbrace{\sum_{\beta\in\Omega_{|i|}^{d}}\prod_{u=1}^{d}\partial_{q}^{\sum_{v\in\beta_{u}}\gamma_{v}}\left(g_{j_{u}}\right)^{t}_{p}}_{B_{d}},$$
where we point out that, for all
\(\beta\in\Omega_{|i|}^{d}\),
\(\beta=\{\beta_{1},\ldots,\beta_{d}\}\) is a partition of
\(\Omega_{|i|}^{d}\) into
\(d\) nonempty subsets. We refer to [
20] for the above formula. For all
\(1\leqslant h\leqslant n\), taking the derivative with respect to
\(p_{h}\) of the latter,
$$\left(\partial_{p_{h}}\partial_{q}^{i}\left(f\circ\tilde{g}\right)\right)^{t}_{p}=\sum_{d=1}^{|i|}\sum_{j_{1}=1}^{n}\ldots\sum_{j_{d}=1}^{n}\tilde{A}_{d}B_{d}+\sum_{d=1}^{|i|}\sum_{j_{1}=1}^{n}\ldots\sum_{j_{d}=1}^{n}A_{d}\tilde{B}_{d},$$
(A.4)
where
$$\displaystyle\begin{aligned} \displaystyle\tilde{A}^{d}&\displaystyle=\left(\sum_{r=1}^{n}\left({\partial^{d+1}\over\partial q_{r}\partial q_{j_{1}}\ldots\partial q_{j_{d}}}f^{t}_{p}\circ g_{p}^{t}\left(\partial_{p_{h}}g_{r}\right)_{p}^{t}\right)+{\partial^{d+1}\over\partial p_{h}\partial q_{j_{1}}\ldots\partial q_{j_{d}}}f^{t}_{p}\circ g_{p}^{t}\right)\\ \displaystyle\tilde{B}^{d}&\displaystyle=\sum_{\beta\in\Omega_{|i|}^{d}}\sum_{r=1}^{d}\partial_{p_{h}}\partial_{q}^{\sum_{v\in\beta_{u}}\gamma_{v}}\left(g_{j_{r}}\right)^{t}_{p}\prod_{u=1,u\neq r}^{d}\partial_{q}^{\sum_{v\in\beta_{u}}\gamma_{v}}\left(g_{j_{u}}\right)^{t}_{p}.\end{aligned}$$
Now, for all fixed
\((p,t)\in B\times I\), we want to estimate the norm
\(|\left(\partial_{p_{h}}\partial_{q}^{i}\left(f\circ\tilde{g}\right)\right)^{t}_{p}|_{C^{0}}\). To achieve this, thanks to
(A.4), it is sufficient to estimate the following quantities for all fixed
\(d=1,\ldots,|i|\). In the upcoming analysis, we will make use of the properties in Proposition
6:
$$\displaystyle\begin{aligned} \displaystyle|\tilde{A}^{d}|&\displaystyle\leqslant C(n)\left(|f^{t}_{p}|_{C^{1+k}}|\partial_{p}g^{t}_{p}|_{C^{0}}+\max_{0\leqslant|i|\leqslant k}|\left(\partial_{p}\partial_{q}^{i}f\right)^{t}_{p}|_{C^{0}}\right)\\ \displaystyle|B^{d}|&\displaystyle\leqslant C(n)\sum_{\beta\in\Omega_{|i|}^{d}}\prod_{u=1}^{d}|g^{t}_{p}|_{C^{k}}\leqslant C(n)\sum_{\beta\in\Omega_{|i|}^{d}}|g^{t}_{p}|_{C^{k}}^{d}\leqslant C(n,k)\left(1+|g^{t}_{p}|_{C^{k}}^{k}\right)\\ \displaystyle|A^{d}|&\displaystyle\leqslant C(n)|f^{t}_{p}|_{C^{k}}\\ \displaystyle|\tilde{B}^{d}|&\displaystyle\leqslant\sum_{\beta\in\Omega_{|i|}^{d}}\sum_{r=1}^{d}\max_{0\leqslant|i|\leqslant k}|\left(\partial_{p}\partial_{q}^{|i|}g\right)^{t}_{p}|_{C^{0}}\prod_{u=1,u\neq r}^{d}|g^{t}_{p}|_{C^{k}}\\ &\displaystyle\leqslant C(n,k)\max_{0\leqslant|i|\leqslant k}|\left(\partial_{p}\partial_{q}^{|i|}g\right)^{t}_{p}|_{C^{0}}\left(1+|g^{t}_{p}|_{C^{k}}^{k}\right).\end{aligned}$$
To avoid a flow of useless parameters, we have taken the maximum on the right-hand side of the above estimates for all
\(1\leqslant d\leqslant|i|\leqslant k\). Consequently, the constants involved depend only on
\(n\) and
\(k\). Thanks to
(A.4) and the latter estimates, for all
\((p,t)\in B\times I\),
$$\displaystyle\begin{aligned} &\displaystyle\left|\left(\partial_{p}\partial_{q}^{i}\left(f\circ\tilde{g}\right)\right)^{t}_{p}\right|_{C^{0}}\left(1+|t|^{l+m-1}\right)\\ &\displaystyle\leqslant C(n,k)\left(|f^{t}_{p}|_{C^{1+k}}|\partial_{p}g^{t}_{p}|+\max_{0\leqslant|i|\leqslant k}|\left(\partial_{p}\partial_{q}^{i}f\right)^{t}_{p}|_{C^{0}}\right)\left(1+|g^{t}_{p}|_{C^{k}}^{k}\right)\left(1+|t|^{l+m-1}\right)\\ &\displaystyle+C(n,k)|f^{t}_{p}|_{C^{k}}\max_{0\leqslant|i|\leqslant k}|\left(\partial_{p}\partial_{q}^{|i|}g\right)^{t}_{p}|_{C^{0}}\left(1+|g^{t}_{p}|_{C^{k}}^{k}\right)\left(1+|t|^{l+m-1}\right)\\ &\displaystyle\leqslant C(n,k)\left(|f|_{1,k,l}|g|_{0,0,m}+|f|_{0,k,l}|g|_{0,k,m}+|f|_{0,k,l+m}\right)\left(1+|g|_{0,k,0}^{k}\right).\end{aligned}$$
This concludes the proof of
4.
\(\square\)
The second part of this appendix is dedicated to the proof of Proposition 4. First, let us recall the definition of the space of functions \(\mathcal{D}_{\sigma,k}\) (see Definition 5), the norms (2.16) and (2.17) and the statement of Proposition 4.
Definition
We consider \(\sigma\geqslant 0\), a positive integer \(k\geqslant 0\), \(A\subset\mathbb{R}^{n}\) and an open interval \(I\subset\mathbb{R}\). Let \(\mathcal{D}_{\sigma,k}\) be the space of functions \(f:\mathbb{T}^{n}\times A\times I\to\mathbb{R}\) such that
$$\displaystyle\begin{aligned} &\displaystyle f_{p}^{t}\in C^{\sigma+k}(\mathbb{T}^{n})\quad\mbox{for all fixed $(p,t)\in A\times I$},\\ &\displaystyle\partial^{i}_{q}f\in C(\mathbb{T}^{n}\times A\times I)\quad\mbox{for all $i\in\mathbb{N}^{n}$ with $0\leqslant|i|\leqslant k$}.\end{aligned}$$
The norms \(|\cdot|_{\sigma,k,l,L(A)}\) and \(|\cdot|_{\sigma,k,0,L(A)}\) (see (2.16) and (2.17), respectively) are defined as follows for all \(f\in\mathcal{D}_{\sigma,k}\) and \(l\geqslant 1\):
$$\displaystyle\begin{aligned} &\displaystyle|f|_{\sigma,k,l,L(A)}=\sup_{(p,t)\in A\times I}|f^{t}_{p}|_{C^{\sigma+k}}(1+|t|^{l})+\max_{0\leqslant|i|\leqslant k}\sup_{t\in I}|\partial_{q}^{i}f^{t}|_{L(A)}(1+|t|^{l-1}),\\ &\displaystyle|f|_{\sigma,k,0,L(A)}=\sup_{(p,t)\in A\times I}|f^{t}_{p}|_{C^{\sigma+k}}+\max_{0\leqslant|i|\leqslant k}\sup_{t\in I}|\partial_{q}^{i}f^{t}|_{L(A)}.\end{aligned}$$
In the last part of this appendix, we recall the statement of Proposition 4 and provide its proof.
Proposition 2
Given \(\sigma\geqslant 0\) and a positive integer \(k\geqslant 0\), for all \(f\), \(g\in\mathcal{D}_{\sigma,k}\) and positive \(l\), \(m\geqslant 1\) and \(d\geqslant 0\)
-
1)
\(|f|_{\sigma^{\prime},k^{\prime},l,L(A)}\leqslant C(n)|f|_{\sigma,k,l,L(A)}\) for all \(0\leqslant\sigma^{\prime}\leqslant\sigma\) and \(k^{\prime}\in\mathbb{Z}^{n}\) with \(0\leqslant k^{\prime}\leqslant k\).
-
2)
\(|f|_{\sigma,k,l,L(A)}\leqslant C(n,l,d)|f|_{\sigma,k,l+d,L(A)}.\)
-
3)
\(|fg|_{\sigma,k,l+m,L(A)}\leqslant C(n,\sigma,k)\left(|f|_{0,0,l,L(A)}|g|_{\sigma,k,m,L(A)}+|f|_{\sigma,k,l,L(A)}|g|_{0,0,m,L(A)}\right)\) \(\hskip 76.822441pt+C(n,k)|f|_{0,k,l,L(A)}|g|_{0,k,m,L(A)}.\)
Moreover, we consider \(\tilde{g}\in\mathcal{D}_{\sigma,k}\) such that \(\tilde{g}:\mathbb{T}^{n}\times A\times I\to\mathbb{T}^{n}\times A\times I\) with \(\tilde{g}(q,p,t)=(g(q,p,t),p,t)\). Then \(f\circ\tilde{g}\in\mathcal{D}_{\sigma,k}\) and
-
4)
\(|f\circ\tilde{g}|_{\sigma,l+m,L(A)}\leqslant C(n,\sigma,k)\left(|f|_{\sigma,k,l,L(A)}|g|^{\sigma+k}_{1,0,m,L(A)}+|f|_{1,0,l,L(A)}|g|_{\sigma,k,m,L(A)}\right)\) \(\hskip 28.452756pt+C(n,\sigma,k)|f|_{0,0,l+m,L(A)}+C(n,k)|f|_{1,k,l,L(A)}|g|_{0,0,m,L(A)}\left(1+|g|_{0,k,0,L(A)}^{k}\right)\) \(\hskip 28.452756pt+C(n,k)\left(|f|_{0,k,l,L(A)}|g|_{0,k,m,L(A)}+|f|_{0,k,l+m,L(A)}\right)\left(1+|g|_{0,k,0,L(A)}^{k}\right)\).
The previous properties are still verified when \(l=m=0\) or only one of the two parameters, \(l\) or \(m\), is zero.
Proof
The proofs of properties 1 and 2 closely resemble those of properties 1 and 2 in Proposition 2. Due to this similarity, we omit the proofs.
We prove 3. Concerning the first term of the norm (2.16), using property 2 of Proposition 6,
$$\sup_{(p,t)\in A\times I}\left|f^{t}_{p}g^{t}_{p}\right|_{C^{\sigma+k}}\left(1+|t|^{l+m}\right)\leqslant C(\sigma,k)|f|_{0,0,l,L(A)}|g|_{\sigma,k,m,L(A)}$$
$${}\quad+C(\sigma,k)|f|_{\sigma,k,l,L(A)}|g|_{0,0,m,L(A)}.$$
(A.5)
Now, we consider the second term of the norm (see the right-hand side of
(2.16)). For all
\(q\in\mathbb{T}^{n}\),
\(\alpha\),
\(i\in\mathbb{N}^{n}\) with
\(|i|\leqslant k\) and for all fixed
\(t\in I\) and
\(x\),
\(y\in A\) such that
\(x\neq y\)
$$\displaystyle\begin{aligned} &\displaystyle{|\partial_{q}^{i}(fg)^{t}(q,x)-\partial_{q}^{i}(fg)^{t}(q,y)|\over|x-y|}(1+|t|^{l+m-1})\\ &\displaystyle\leqslant\sum_{\alpha\leqslant i}{i\choose\alpha}{|\partial_{q}^{\alpha}f^{t}(q,x)-\partial_{q}^{\alpha}f^{t}(q,y)|\over|x-y|}(1+|t|^{l-1})|\partial_{q}^{i-\alpha}g(q,x)|(1+|t|^{m})\\ &\displaystyle+\sum_{\alpha\leqslant i}{i\choose\alpha}{|\partial_{q}^{i-\alpha}g^{t}(q,x)-\partial_{q}^{i-\alpha}g^{t}(q,y)|\over|x-y|}(1+|t|^{m-1})|\partial_{q}^{\alpha}f(q,y)|(1+|t|^{l})\\ &\displaystyle\leqslant C(n,k)|f|_{0,k,l,L(A)}|g|_{0,k,m,L(A)}.\end{aligned}$$
Thanks to the latter and
(A.5), one can conclude the proof of
3. Now, we prove
4. Similarly to the proof of Proposition
2, first we prove
4 when
\(k=0\). By property
3 of Proposition
6
$$\displaystyle\begin{aligned} \displaystyle\sup_{(p,t)\in A\times I}|f^{t}_{p}\circ g_{p}^{t}|_{C^{\sigma}}\left(1+|t|^{l+m}\right)&\displaystyle\leqslant C(n,\sigma)|f|_{\sigma,0,l,L(A)}|g|^{\sigma}_{1,0,m,L(A)}\\ &\displaystyle{}\quad+C(\sigma)\left(|f|_{1,0,l,L(A)}|g|_{\sigma,0,m,L(A)}+|f|_{0,0,l+m,L(A)}\right).\end{aligned}$$
Now, we estimate the second term of the norm (see
(2.16)). For all
\(q\in\mathbb{T}^{n}\) and for all fixed
\(t\in I\) and
\(x\),
\(y\in A\) such that
\(x\neq y\)
$$\displaystyle\begin{aligned} &\displaystyle{|f^{t}(g^{t}(q,x),x)-f^{t}(g^{t}(q,y),y)|\over|x-y|}(1+|t|^{l+m-1})\\ &\displaystyle={|f^{t}(g^{t}(q,x),x)-f^{t}(g^{t}(q,x),y)+f^{t}(g^{t}(q,x),y)-f^{t}(g^{t}(q,y),y)|\over|x-y|}(1+|t|^{l+m-1})\\ &\displaystyle\leqslant{|f^{t}(g^{t}(q,x),x)-f^{t}(g^{t}(q,x),y)|\over|x-y|}(1+|t|^{l+m-1})\\ &\displaystyle+{|f^{t}(g^{t}(q,x),y)-f^{t}(g^{t}(q,y),y)|\over|x-y|}(1+|t|^{l+m-1})\\ &\displaystyle\leqslant C(n)\left(|f^{t}|_{L(A)}\left(1+|t|^{l+m-1}\right)+|f^{t}_{y}|_{C^{1}}\left(1+|t|^{l}\right)|g^{t}|_{L(A)}\left(1+|t|^{m-1}\right)\right)\\ &\displaystyle\leqslant C(n)\left(|f|_{0,0,l+m,L(A)}+|f|_{1,0,l,L(A)}|g|_{0,0,m,L(A)}\right),\end{aligned}$$
where we used
\(|f^{t}(g^{t}(q,x),y)-f^{t}(g^{t}(q,y),y)|\leqslant|\left(\partial_{q}f\right)^{t}_{y}|_{C^{0}}|g^{t}|_{L(A)}|x-y|\).
Taking the sup for all
\(q\in\mathbb{T}^{n}\) and
\(x\),
\(y\in A\) with
\(x\neq y\) on the left-hand side of the latter and then for all
\(t\in I\), we conclude the proof of
4 when
\(k=0\). Now, let
\(k\geqslant 1\). Similarly to the previous case,
$$\displaystyle\begin{aligned} \displaystyle\sup_{(p,t)\in A\times I}|f^{t}_{p}\circ g_{p}^{t}|_{C^{\sigma+k}}\left(1+|t|^{l+m}\right)&\displaystyle\leqslant C(n,\sigma,k)|f|_{\sigma,k,l,L(A)}|g|^{\sigma+k}_{1,0,m,L(A)}\\ &\displaystyle{}\quad+C(n,\sigma,k)\left(|f|_{1,0,l,L(A)}|g|_{\sigma,k,m,L(A)}+|f|_{0,0,l+m,L(A)}\right).\end{aligned}$$
Now, we need to estimate the second term of the norm
(2.16). For this reason, for all
\(i\in\mathbb{N}^{n}\) with
\(|i|\leqslant k\), and for all fixed
\(t\in\mathbb{R}\) and
\(x\),
\(y\in A\) such that
\(x\neq y\), we consider the following decomposition:
$$\displaystyle\hspace{-6mm}{|\partial_{q}^{i}\left(f_{x}^{t}\circ g_{x}^{t}\right)-\partial_{q}^{i}\left(f_{y}^{t}\circ g_{y}^{t}\right)|\over|x-y|}(1+|t|^{l+m-1})$$
$$\leqslant{|\partial_{q}^{i}\left(f_{x}^{t}\circ g_{x}^{t}\right)-\partial_{q}^{i}\left(f_{y}^{t}\circ g_{x}^{t}\right)|\over|x-y|}(1+|t|^{l+m-1})$$
(A.6)
$$+{|\partial_{q}^{i}\left(f_{y}^{t}\circ g_{x}^{t}\right)-\partial_{q}^{i}\left(f_{y}^{t}\circ g_{y}^{t}\right)|\over|x-y|}(1+|t|^{l+m-1}).$$
(A.7)
We will estimate
(A.6) and
(A.7) separately, employing Fàa di Bruno’s formula. The notation used will be consistent with that in the proof of property
4 of Proposition
2.
We begin with the estimation of
(A.6). For all
\(i\in\mathbb{N}^{n}\) with
\(1\leqslant|i|\leqslant k\), let
\(\{\gamma_{j}\}_{j=1}^{|i|}\subset\mathbb{N}^{n}\) be any sequence of bases such that
\(i=\sum_{j=1}^{|i|}\gamma_{j}\) with
\(|\gamma_{j}|=1\).
For all
\((p,t)\in B\times I\), thanks to Fàa di Bruno’s formula
$$\displaystyle\hspace{-86mm}{|\partial_{q}^{i}\left(f_{x}^{t}\circ g_{x}^{t}\right)-\partial_{q}^{i}\left(f_{y}^{t}\circ g_{x}^{t}\right)|\over|x-y|}(1+|t|^{l+m-1})$$
$$\displaystyle\leqslant\left|\sum_{d=1}^{|i|}\sum_{j_{1}=1}^{n}\ldots\sum_{j_{d}=1}^{n}{\partial^{d}\over\partial q_{j_{1}}\ldots\partial q_{j_{d}}}\left(f^{t}_{x}\circ g_{x}^{t}-f^{t}_{y}\circ g_{x}^{t}\right)\sum_{\beta\in\Omega_{|i|}^{d}}\prod_{u=1}^{d}\partial_{q}^{\sum_{v\in\beta_{u}}\gamma_{v}}\left(g_{j_{u}}\right)^{t}_{x}\right|{(1+|t|^{l+m-1})\over|x-y|}$$
$$\displaystyle\hspace{-58mm}\leqslant C(n,k)\max_{0\leqslant|i|\leqslant k}\sup_{t\in\mathbb{R}}|\partial_{q}^{i}f^{t}|_{L(A)}(1+|t|^{l+m-1})\left(1+|g_{x}^{t}|^{k}_{C^{k}}\right)$$
$$\displaystyle\hspace{-84mm}\leqslant C(n,k)|f|_{0,k,l+m,L(A)}\left(1+|g|^{k}_{0,k,0,L(A)}\right),$$
(A.8)
where we recall that
\(\Omega_{|i|}^{d}\) is the set of partitions of
\(\{1,\ldots,|i|\}\subset\mathbb{N}\) into
\(d\) nonempty subsets. Now, we estimate
(A.7). First, we observe that by Fàa di Bruno’s formula for all
\(i\in\mathbb{N}^{n}\) with
\(|i|\leqslant k\), and for all fixed
\(t\in I\) and
\(x\),
\(y\in A\) such that
\(x\neq y\)
$$\displaystyle\begin{aligned} &\displaystyle\partial_{q}^{i}\left(f_{y}^{t}\circ g_{x}^{t}\right)-\partial_{q}^{i}\left(f_{y}^{t}\circ g_{y}^{t}\right)=\sum_{d=1}^{|i|}\sum_{j_{1}=1}^{n}\ldots\sum_{j_{d}=1}^{n}{\partial^{d}\over\partial q_{j_{1}}\ldots\partial q_{j_{d}}}f^{t}_{y}\circ g_{x}^{t}\sum_{\beta\in\Omega_{|i|}^{d}}\prod_{u=1}^{d}\partial_{q}^{\sum_{v\in\beta_{u}}\gamma_{v}}\left(g_{j_{u}}\right)^{t}_{x}\\ &\displaystyle-\sum_{d=1}^{|i|}\sum_{j_{1}=1}^{n}\ldots\sum_{j_{d}=1}^{n}{\partial^{d}\over\partial q_{j_{1}}\ldots\partial q_{j_{d}}}f^{t}_{y}\circ g_{y}^{t}\sum_{\beta\in\Omega_{|i|}^{d}}\prod_{u=1}^{d}\partial_{q}^{\sum_{v\in\beta_{u}}\gamma_{v}}\left(g_{j_{u}}\right)^{t}_{y}\\ &\displaystyle=\sum_{d=1}^{|i|}\sum_{j_{1}=1}^{n}\ldots\sum_{j_{d}=1}^{n}\left(A^{d}_{yx}-A^{d}_{yy}\right)B^{d}_{x}+\sum_{d=1}^{|i|}\sum_{j_{1}=1}^{n}\ldots\sum_{j_{d}=1}^{n}A^{d}_{yy}\left(B^{d}_{x}-B^{d}_{y}\right)\end{aligned}$$
with
$$\displaystyle\begin{aligned} \displaystyle A^{d}_{yx}&\displaystyle=&\displaystyle{\partial^{d}\over\partial q_{j_{1}}\ldots\partial q_{j_{d}}}f^{t}_{y}\circ g_{x}^{t},\quad B^{d}_{x}=\sum_{\beta\in\Omega_{|i|}^{d}}\prod_{u=1}^{d}\partial_{q}^{\sum_{v\in\beta_{u}}\gamma_{v}}\left(g_{j_{u}}\right)^{t}_{x}\\ \displaystyle A^{d}_{yy}&\displaystyle=&\displaystyle{\partial^{d}\over\partial q_{j_{1}}\ldots\partial q_{j_{d}}}f^{t}_{y}\circ g_{y}^{t},\quad B^{d}_{y}=\sum_{\beta\in\Omega_{|i|}^{d}}\prod_{u=1}^{d}\partial_{q}^{\sum_{v\in\beta_{u}}\gamma_{v}}\left(g_{j_{u}}\right)^{t}_{y}.\end{aligned}$$
This means that we can rewrite
(A.7) as
$$\displaystyle\hspace{-23mm}{|\partial_{q}^{i}\left(f_{y}^{t}\circ g_{x}^{t}\right)-\partial_{q}^{i}\left(f_{y}^{t}\circ g_{y}^{t}\right)|\over|x-y|}(1+|t|^{l+m-1})$$
$$\displaystyle\hspace{-11mm}\leqslant\sum_{d=1}^{|i|}\sum_{j_{1}=1}^{n}\ldots\sum_{j_{d}=1}^{n}{|A^{d}_{yx}-A^{d}_{yy}|\over|x-y|}|B^{d}_{x}|(1+|t|^{l+m-1})$$
(A.9)
$$+\sum_{d=1}^{|i|}\sum_{j_{1}=1}^{n}\ldots\sum_{j_{d}=1}^{n}|A^{d}_{yy}|(1+|t|^{l}){|B^{d}_{x}-B^{d}_{y}|\over|x-y|}(1+|t|^{m-1}).$$
(A.10)
By the latter, we can see that, in order to estimate
(A.7), we need to estimate
(A.9) and
(A.10). Following this purpose, we observe that
$${|A^{d}_{yx}-A^{d}_{yy}|\over|x-y|}(1+|t|^{l+m-1})\leqslant\left|{\partial^{d}\over\partial q_{j_{1}}\ldots\partial q_{j_{d}}}f^{t}_{y}\circ g_{x}^{t}-{\partial^{d}\over\partial q_{j_{1}}\ldots\partial q_{j_{d}}}f^{t}_{y}\circ g_{y}^{t}\right|{(1+|t|^{l+m-1})\over|x-y|}$$
$${}\quad\leqslant C(n)|f^{t}_{y}|_{C^{1+k}}(1+|t|^{l})|g^{t}|_{L(A)}(1+|t|^{m-1})\leqslant C(n)|f|_{1,k,l,L(A)}|g|_{0,0,m,L(A)}$$
(A.11)
$$\displaystyle\hspace{-53mm}|B^{d}_{x}|\leqslant C(n)\sum_{\beta\in\Omega_{|i|}^{d}}\prod_{u=1}^{d}|g^{t}_{x}|_{C^{k}}\leqslant C(n,k)\left(1+|g|^{k}_{0,k,0,L(A)}\right)$$
(A.12)
$$\displaystyle\hspace{-53mm}|A^{d}_{yy}|(1+|t|^{l})\leqslant C(n)|f^{t}_{y}|_{C^{k}}(1+|t|^{l})\leqslant C(n)|f|_{0,k,l,L(A)}$$
(A.13)
$$\displaystyle\hspace{-3mm}{|B^{d}_{x}-B^{d}_{y}|\over|x-y|}(1+|t|^{m-1})\leqslant\sum_{\beta\in\Omega_{|i|}^{d}}{\prod_{u=1}^{d}|\partial_{q}^{\sum_{v\in\beta_{u}}\gamma_{v}}\left(g_{j_{u}}\right)^{t}_{x}-\partial_{q}^{\sum_{v\in\beta_{u}}\gamma_{v}}\left(g_{j_{u}}\right)^{t}_{y}|\over|x-y|}(1+|t|^{m-1})$$
$$\displaystyle\hspace{-50mm}{}\quad\leqslant\sum_{\beta\in\Omega_{|i|}^{d}}{|\partial_{q}^{\sum_{v\in\beta_{1}}\gamma_{v}}\left(g_{j_{1}}\right)^{t}_{x}-\partial_{q}^{\sum_{v\in\beta_{1}}\gamma_{v}}\left(g_{j_{1}}\right)^{t}_{y}|\over|x-y|}(1+|t|^{m-1})$$
$$\displaystyle\hspace{-74mm}{}\quad\times\prod_{u=2}^{d}|\partial_{q}^{\sum_{v\in\beta_{u}}\gamma_{v}}\left(g_{j_{u}}\right)^{t}_{x}-\partial_{q}^{\sum_{v\in\beta_{u}}\gamma_{v}}\left(g_{j_{u}}\right)^{t}_{y}|$$
$$\displaystyle\hspace{-50mm}{}\quad\leqslant\sum_{\beta\in\Omega_{|i|}^{d}}\max_{0\leqslant|i|\leqslant k}\sup_{t\in I}|\partial_{q}^{i}g^{t}|_{L(A)}(1+|t|^{m-1})\prod_{u=2}^{d}2|g|_{0,k,0,L(A)}$$
$$\displaystyle\hspace{-77mm}{}\quad\leqslant C(n,k)|g|_{0,k,m,L(A)}\left(1+|g|^{k}_{0,k,0,L(A)}\right).$$
(A.14)
More specifically, by
(A.9)–
(A.14) we can estimate
(A.7) in the following form:
$$\displaystyle\hspace{-63mm}{|\partial_{q}^{i}\left(f_{y}^{t}\circ g_{x}^{t}\right)-\partial_{q}^{i}\left(f_{y}^{t}\circ g_{y}^{t}\right)|\over|x-y|}(1+|t|^{l+m-1})$$
$$\leqslant C(n,k)\left(|f|_{1,k,l,L(A)}|g|_{0,0,m,L(A)}+|f|_{0,k,l,L(A)}|g|_{0,k,m,L(A)}\right)\left(1+|g|^{k}_{0,k,0,L(A)}\right).$$
(A.15)
This concludes the proof of
4 because, thanks to
(A.6)–
(A.8) and
(A.15),
$$\displaystyle\begin{aligned} &\displaystyle\max_{0\leqslant|i|\leqslant k}\sup_{t\in I}|\partial_{q}^{i}\left(f\circ\tilde{g}\right)^{t}|_{L(A)}(1+|t|^{l-1})\\ &\displaystyle\leqslant C(n,k)|f|_{0,k,l+m,L(A)}\left(1+|g|^{k}_{0,k,0,L}\right)\\ &\displaystyle+C(n,k)\left(|f|_{1,k,l,L(A)}|g|_{0,0,m,L(A)}+|f|_{0,k,l,L(A)}|g|_{0,k,m,L(A)}\right)\left(1+|g|^{k}_{0,k,0,L(A)}\right).\end{aligned}$$
\(\square\)
