Regularity of Characteristic Exponents and Linear Response for Transfer Operator Cocycles

Abstract

We consider cocycles obtained by composing sequences of transfer operators with positive weights, associated with uniformly expanding maps (possibly having countably many branches) and depending upon parameters. Assuming $C^k$ regularity with respect to coordinates and parameters, we show that when the sequence is picked within a certain uniform family the top characteristic exponent and generator of top Oseledets space of the cocycle are $C^{k-1}$ in parameters. As applications, we obtain a linear response formula for the equivariant measure associated with random products of uniformly expanding maps, and we study the regularity of the Hausdorff dimension of a repeller associated with random compositions of one-dimensional cookie-cutters.

Appendices

A Real cone contraction theory

As the literature is somewhat disparate on the subject, we provide here a catalog of more or less standard results on cone contractions. Most of these results already occur in similar forms in e.g. [11, 21, 41]. In a few cases, we have added (short) proofs where appropriate. In the following, $(E,\Vert .\Vert )$ will denote a real Banach space.

Definition A.1

Let ${\mathcal {C}}\subset E$. We say that ${\mathcal {C}}$ is a proper closed convex cone (for short, a Birkhoff cone) if

• $\mathbb {R}_+{\mathcal {C}}={\mathcal {C}}$, i.e ${\mathcal {C}}$ is stable by multiplication with a positive scalar.

• ${\mathcal {C}}$ is a closed and convex subset of E.

• ${\mathcal {C}}\cap (-{\mathcal {C}})=\{0\}$ (the cone is proper).

Definition A.2

Let ${\mathcal {C}}\subset E$ be a Birkhoff cone. We say that ${\mathcal {C}}$ is

• 1. inner regular iff ${\mathcal {C}}$ has non-empty interior. Equivalently, there is $\rho >0$ so that:

  $$\begin{aligned} {\mathcal {C}}(\rho ) = \{ x\in {\mathcal {C}}: B_E(x,\rho \Vert x\Vert )\subset {\mathcal {C}}\} \end{aligned}$$

  is non-trivial (contains other points than the origin).

• 2. outer regular if there is $\ell \in E'$, $\Vert \ell \Vert =1$ and $K<+\infty$ such that for every $u\in {\mathcal {C}}$:

  $$\begin{aligned} \frac{1}{K} \Vert u\Vert \le \langle \ell ,u\rangle \le \Vert u\Vert . \end{aligned}$$

  (A.1)

We will say that ${\mathcal {C}}$ is regular if it is both inner and outer regular. We define:

$$\begin{aligned} {\mathcal {C}}_{\ell =1}(\rho )=\{u \in {\mathcal {C}}(\rho ): \langle \ell ,u\rangle =1\}. \end{aligned}$$

Definition A.3

Let ${\mathcal {C}}\subset E$ be a Birkhoff cone, and let $x,y\in {\mathcal {C}}^*$. We define the projective Hilbert metric:

$$\begin{aligned} \delta (x,y)&=\inf \{t>0,~tx-y\in {\mathcal {C}}\}\\ d_{\mathcal {C}}(x,y)&=\log (\delta (x,y)\delta (y,x)) \in [0,+\infty ]. \end{aligned}$$

Hilbert’s original (but equivalent) definition was through cross-ratios which may be formulated as follows: When $x,y\in {\mathcal {C}}^*$ are non-colinear, it is always possible to normalize x and y so that $I(x,y) = \{t\in \mathbb {R}: (1+t)x+(1-t)y \in {\mathcal {C}}\} = [t_1,t_2]$ is a bounded (closed) interval. with $t_1\le -1<1\le t_2$. One then has for their projective distance, relative to the cone:

$$\begin{aligned} d_{\mathcal {C}}(x,y) = \log \frac{t_2+1}{t_2-1} \frac{t_1-1}{t_1+1} \in [0,+\infty ] \end{aligned}$$

(A.2)

Lemma A.4

Let ${\mathcal {C}}\subset E$ be a regular Birkhoff cone, and let $d_{\mathcal {C}}$ be the associated Hilbert metric. Suppose that $B(x_1,r_1)\subset {\mathcal {C}}$ and $B(x_2,r_2)\subset {\mathcal {C}}$ for some $r_1,r_2>0$. Then

$$\begin{aligned} d_{{\mathcal {C}}}(x_1,x_2) \le \log \left( 1 + \frac{\Vert x_1-x_2\Vert }{r_1}\right) + \log \left( 1 + \frac{\Vert x_1-x_2\Vert }{r_2}\right) \le \left( \frac{1}{r_1} + \frac{1}{r_2} \right) \Vert x_1 - x_2\Vert . \end{aligned}$$

Proof

Inclusion of the two balls imply that $I(x_1,x_2) \supset \left[ -1 - \frac{2r_1}{\Vert x_1-x_2\Vert }, +1+ \frac{2r_2}{\Vert x_1-x_2\Vert } \right]$ which together with (A.2) yields the result. $\quad \square$

Corollary A.5

For $x,y\in {\mathcal {C}}_{\ell =1}(\rho )$: $\displaystyle d_{\mathcal {C}}(x,y) \le 2\log \left( 1 + \frac{\Vert x-y\Vert }{\rho }\right) \le \frac{2}{\rho }\Vert x-y\Vert$.

When $x\in {\mathcal {C}}_{\ell =1}(\rho )$ and $\Vert u\Vert <\rho$ we have: $\displaystyle d_{\mathcal {C}}(x,x+u) \le \frac{2 \Vert u\Vert }{\rho -\Vert u\Vert } = \frac{2}{\rho }\Vert u\Vert + o(\Vert u\Vert )$.

Lemma A.6

([21], Appendix A). Let ${\mathcal {C}}\subset E$ be an outer regular Birkhoff cone with $\ell \in E'$ as above. Then for all $x,y\in {\mathcal {C}}^*$:

$$\begin{aligned} \left\| \frac{x}{\langle \ell ,x\rangle }- \frac{y}{\langle \ell ,y\rangle }\right\| \le \frac{K}{2}d_{\mathcal {C}}(x,y) \end{aligned}$$

Theorem A.7

(Birkhoff’s theorem, [11]). Let ${\mathcal {C}}\subset E$ be a Birkhoff cone and let $\mathcal {L}\in L(E)$ be a contraction of ${\mathcal {C}}^*$, i.e. such that $\mathcal {L}({\mathcal {C}}^*)\subset {\mathcal {C}}^*$. Setting $\Delta =diam_{{\mathcal {C}}}(\mathcal {L}({\mathcal {C}}^*))\in [0,+\infty ]$ we have for $x,y\in {\mathcal {C}}^*$:

$$\begin{aligned} d_{{\mathcal {C}}}(\mathcal {L}x,\mathcal {L}y)\le \left( \tanh \frac{\Delta }{4}\right) d_{{\mathcal {C}}}(x,y) \end{aligned}$$

Corollary A.8

Let ${\mathcal {C}}$ be a regular cone and $n\in {\mathbb {N}}$. Let $(L_i)_{1\le i\le n}\in L(E)$ be cone contractions, i.e $L_i({\mathcal {C}}^*)\subset {\mathcal {C}}^*$ for any $1\le i\le n$ and set $\Delta _i=diam_{{\mathcal {C}}}(L_i({\mathcal {C}}^*))\in [0,+\infty ]$. Then for all $x,y\in {\mathcal {C}}^*$:

$$\begin{aligned} \left\| \frac{L_n\dots L_1 x}{\langle \ell ,L_n\dots L_1 x\rangle }- \frac{L_n\dots L_1 y}{\langle \ell ,L_n\dots L_1y\rangle }\right\| \le \frac{K}{2} \left( \prod _{i=1}^n\tanh \frac{\Delta _i}{4}\right) d_{\mathcal {C}}(x,y) \end{aligned}$$

(A.3)

Lemma A.9

Let ${\mathcal {C}}$ be a regular cone with associated $\ell$, $K<+\infty$ and $\rho >0$ as in Definition A.2. Let $\mathcal {L}\in L(E)$ with $L({\mathcal {C}})\subset {\mathcal {C}}$. Then for every $x \in {\mathcal {C}}(\rho )^*$:

$$\begin{aligned} \frac{\rho }{K} \Vert \mathcal {L}\Vert \; \Vert x\Vert \le \langle \ell , \mathcal {L}x \rangle \le \Vert \mathcal {L}\Vert \; \Vert x\Vert \ \ \text{ and } \ \ \frac{\rho }{K} \Vert \mathcal {L}\Vert \le \frac{\langle \ell , \mathcal {L}x \rangle }{\langle \ell , x \rangle } \le K \Vert \mathcal {L}\Vert . \end{aligned}$$

Proof

Suppose $x\in {\mathcal {C}}(\rho )$, $\Vert x\Vert =1$ and consider $\Vert u\Vert \le 1$. Then $x\pm \rho u \in {\mathcal {C}}$ so also $\mathcal {L}(x\pm \rho u) \in {\mathcal {C}}$. By outer regularity: $\langle \ell ,\mathcal {L}(x\pm \rho u) \rangle \ge \frac{1}{K}\Vert \mathcal {L}(x\pm \rho u)\Vert$. Therefore,

$$\begin{aligned} 2\rho \Vert \mathcal {L}u\Vert\le & {} \Vert \mathcal {L}(x+\rho u)\Vert +\Vert \mathcal {L}(x-\rho u)\Vert \\\le & {} K \langle \ell , \mathcal {L}(x+\rho u) + \mathcal {L}(x-\rho u) \rangle \\= & {} 2K \langle \ell , \mathcal {L}x \rangle \end{aligned}$$

Thus, $\frac{\rho }{K} \Vert \mathcal {L}\Vert \le \langle \ell ,\mathcal {L}x\rangle$ from which we deduce the left-most inequality. The rest follows from (A.1). $\quad \square$

Lemma A.10

Let ${\mathcal {C}}$ be a regular Birkhoff cone with associated linear functional $\ell \in E'$ and $K<+\infty$ as in Definition A.2. Let ${\mathcal {C}}_1\subset {\mathcal {C}}$ be a subcone of finite diameter, $\Delta =diam_{{\mathcal {C}}}({\mathcal {C}}_1^*)<+\infty$. Suppose that there is $x\in {\mathcal {C}}_1$, $\langle \ell ,x\rangle =1$ with $B(x,r)\subset {\mathcal {C}}_1$.

Then for every $y\in {\mathcal {C}}_1$,

$$\begin{aligned} B_E\left( y, \frac{1}{K} r e^{-\Delta } \Vert y\Vert \right) \subset {\mathcal {C}}. \end{aligned}$$

Proof

Pick $x,y\in {\mathcal {C}}_{\ell =1}$ and set $u_t=\frac{1}{2} ( (1+t)y+(1-t)x)$. Then $\{t\in \mathbb {R}: u_t \in {\mathcal {C}}\} = [t_1,t_2]$ is a bounded (closed) interval (definition A.3) with $t_1\le -1<1\le t_2$. Then $\Delta \ge d_{{\mathcal {C}}}(x,y) \ge \log \frac{t_2+1}{t_2-1}$ or $\frac{t_2-1}{t_2+1}\ge e^{-\Delta }$. For $|u|<1$ we have $x+r u\in {\mathcal {C}}$. Also $u_{t_2}\in {\mathcal {C}}$ so the following convex combination is also in ${\mathcal {C}}$:

$$\begin{aligned} \frac{(t_2-1) (x+ru)+2u_{t_2}}{t_2+1} = y+r\frac{t_2-1}{t_2+1} u \in {\mathcal {C}}. \end{aligned}$$

Here $\frac{1}{K}\Vert y\Vert \le \langle \ell ,y\rangle =1$ so $B(y,\rho \Vert y\Vert )\subset {\mathcal {C}}$ with $\rho = \frac{1}{K}re^{-\Delta }$. $\quad \square$

B Bochner and Strong Measurability

In our setup we need measurability of quantities related to sections and operators. It is close to standard Bochner measurability and strong measurability in the sense of e.g. [31, Appendix A] but not quite the same so we bring here a brief account of the notions we use.

In this appendix $(X,|\cdot |_X)$ denotes a Banach space and $(\Omega ,{{\mathcal {F}}})$ a non-empty space equipped with a $\sigma$-algebra.

Definition B.1

• 1. A map $\phi :\Omega \rightarrow X$ is said to be $\sigma$-simple if it has a countable image and is measurable (with respect to ${{\mathcal {F}}}$). We write $S_\sigma (\Omega ,X)$ for the set of such functions.

• 2. A map $\psi :\Omega \rightarrow X$ is said to be Bochner-measurable if it may be written as the uniform limit of a sequence of $\sigma$-simple functions. We write $L^\infty (\Omega ;X)$ for the set of Bochner measurable maps that are uniformly bounded. It is a Banach space under the uniform norm.

• 3. Let $A\subset X$ be a non-empty set. We say that $\mathscr {L}: \Omega \rightarrow L(X)$ is strongly Bochner measurable on A provided that $\omega \in \Omega \mapsto \mathscr {L}(\omega ) a \in X$ is Bochner-measurable for every $a\in A$.

• 4. We say that $\mathscr {L}$ in the previous definition is measurably bounded provided there is a measurable function $\rho :\Omega \mapsto [0,+\infty )$ so that $\Vert \mathscr {L}(\omega )\Vert _{L(X)} \le \rho (\omega )$ for every $\omega \in \Omega$.

  $\mathscr {L}$ is of course bounded if $\sup _{\omega \in \Omega } \Vert \mathscr {L}(\omega )\Vert _{L(X)} <+\infty$.

Remark B.2

A pointwise limit of measurable functions is measurable [39, VI,§ 1,M7], so the above definition a Bochner-measurable function is equivalent to saying that $\psi$ is measurable with image having a countable dense subset.

Proposition B.3

Let $\phi :\Omega \rightarrow A\subset X$ be Bochner measurable with values in A and let $\mathscr {L}:\Omega \rightarrow L(X)$ be measurably bounded and strongly Bochner measurable on A. Then

$$\begin{aligned} \omega \in \Omega \mapsto \mathscr {L}(\omega ) \phi (\omega )\in X \end{aligned}$$

is also Bochner measurable.

Proof

Let $\rho$ be as in the last part of Definition B.1. For $m=0,1,...$ we set $\Omega _m = \rho ^{-1} ([m,m+1))$ which provides a measurable partition of $\Omega$. Let $\epsilon >0$. By Bochner measurability, and for each $m\ge 0$ we may find sequences $x_{m,k}\in A\subset X$ and $E_{m,k}\in {{\mathcal {F}}}$, $k\ge 1$ so that $(E_{m,k})_{k\ge 1}$ form a measurable partition and $|\phi (\omega )-x_{m,k}|\le \frac{\epsilon }{2(m+1)}$. By strong measurability on A we have for each $m\ge 0, k\ge 1$ sequences $y_{m,k,\ell }\in X$ and $F_{m,k,\ell }\in {{\mathcal {F}}}$, $\ell \ge 1$ so that $(F_{m,k,\ell })_{\ell \ge 1}$ forms a measurable partition and $|\mathscr {L}(\omega )x_{m,k}-y_{m,k,\ell }|_X \le \epsilon /2$ for every $\omega \in F_{m,k,\ell }$. Then for $\omega \in G_{m,k,\ell }=\Omega _m\cap E_{m,k}\cap F_{m,k,\ell }$ we have

$$\begin{aligned} |\mathscr {L}(\omega )\phi (\omega ) -y_{m,k,\ell }|_X \le |\mathscr {L}(\omega )(\phi (\omega )-x_{m,k})| + |\mathscr {L}(\omega )x_{m,k} -y_{m,k,\ell }|_X \le \epsilon . \end{aligned}$$

The $G_{m,k,\ell }$-collection gives a measurable partition of $\Omega$ and the conclusion follows. $\quad \square$

C Differential Calculus

We provide a listing of some of the more or less standard results in differential calculus which we are using. Let U and V denote open convex sets in Banach spaces $B_U$ and $B_V$, respectively, and let Z be a fixed Banach space. We write $|\phi |_0$ (or $\Vert \phi \Vert _0$) to denote a uniform norm on the relevant domain of definition.

Let $r>0$ and $\delta _1 \in (0,1]$ be fixed number in this section. We write ${\underline{r}}=(k,\alpha )$, with $k\in \mathbb {N}_0$ and $\alpha \in (0,1]$ such that $r=k+\alpha$. $C^{\underline{r}}$-norms between open convex subsets of Banach spaces are defined in the standard way. By $c^{\underline{r}}(U;Z)$ we understand the closure of $C^{s}(U;Z)$ in $C^r(U;Z)$ with $s>r$ (the closure is independent of the choice of s).

Given $A\in L(E_1\times \cdots \times E_n; Z)$, a multilinear form with $E_1,...,E_n$ being Banach spaces, there is a natural isomorphism obtained by singling out the j’th Banach space, yielding: $A_j\in L(E_1\times \cdots \widehat{E}_j \cdots \times E_n; L(E_j;Z))$. In many places in this article we make use of the telescopic principle which asserts that

$$\begin{aligned}&\Vert A(M_1,\ldots ,M_n) - A(N_1,\ldots ,N_n) \Vert \nonumber \\&\quad \le \sum _{j=1}^n \Vert A_j(M_1,...,M_{j-1},N_{j+1},\ldots ,N_n)\Vert \; \Vert M_j - N_j\Vert . \end{aligned}$$

(C.1)

Proposition C.1

Let $\phi _1,\phi _2\in c^{\underline{r}}(V;\mathbb {C})$, $\phi \in c^{\underline{r}}(V;Z)$ and $\psi \in c^{\,{{\underline{t}}}}(U;V)$, $t=r\vee 1$. Then there are constants $C_{r,1},C_{r,2},C_{r,3},C_{4,r}$ depending only upon r and the chosen norms so that:

1.	$\phi _1\phi _2$	$\in$	$c^{\underline{r}}(V;\mathbb {C})$	and	$\Vert \phi _1\phi _2\Vert _r \le C_{1,r} \Vert \phi _1\Vert _r \Vert \phi _2\Vert _r$.
2.	$e^{\phi _1}$	$\in$	$c^{\underline{r}}(V;\mathbb {C})$	and	$\Vert e^{\phi _1}\Vert _r \le C_{2,r} |e^{\phi _1}|_0 \; (1 + \Vert \phi _1\Vert _{r})^r$
3.	$\phi \circ \psi$	$\in$	$c^{\underline{r}}(U; Z)$	and	$\Vert \phi \circ \psi \Vert _r \le C_{3,r} \Vert \phi _1\Vert _r (1 + \Vert D\psi \Vert _{t-1})^r$,

and also:

4.

$\Vert e^{\phi _1}-e^{\phi _2}\Vert _r \le C_{4,r} \; (|e^{\phi _1}|_0 \vee |e^{\phi _2}|_0)\; \;\Vert \phi _1-\phi _2\Vert _r \;(1 + \Vert \phi _1\Vert _{r}\vee \Vert \phi _2\Vert _r)^r$

Proof

1. For $C^k$-functions with k an integer, this is standard. When ${\underline{r}}=(k,\alpha )$ with $\alpha \in (0,1]$ we have $D^k(\phi _1\phi _2) = (D^k \phi _1) \phi _2 + \phi _1 (D^k \phi _2)+R_{k-1}$, where $R_{k-1}$ is a bilinear form in $\phi _1$ and $\phi _2$ involving derivatives of order at most $k-1$. The $\delta _1$-local Hölder constant may then be estimated using the MVT on the last term and the above-mentioned telescopic principle to obtain: $h_{\delta _1}^\alpha (D^k (\phi _1\phi _2)) \le h_{\delta _1}^\alpha (D^k \phi _1)|\phi _2|_0 + |\phi _1|_0 h_{\delta _1}^\alpha (D^k \phi _2) + |D R_{k-1}|_0$, The last term is bounded by the $C^k$ norm of $\phi _1\phi _2$.

2. Write $D^q (e^{\phi _1}) = e^{\phi _1} (D\phi _1 + D)^q \mathbf{1}$, $0\le q \le k$ and develop. For the Hölder constant for $D^k(e^{\phi _1})$ we use the same argument as above.

3. The so-called Faà di Bruno formula gives a combinatorial expression for $D^k (\phi \circ \psi )$. Exhibiting the k’th order derivative one has:

$$\begin{aligned} D^k (\phi \circ \psi ) = D^k \phi \circ \psi . \underbrace{(D\psi ,\ldots ,D\psi )}_{k\ \mathrm{times}} + D \phi \circ \psi . D^k \psi + R_{k-1}. \end{aligned}$$

Again $R_{k-1}$ involves only derivatives up to order $k-1$ and may be treated using the MVT. For the Hölder constant of the k’th derivative, let $K = \Vert D\psi \Vert _0$. If $y,y'\in U$ are at a distance at most $\delta _1$, then the $\psi$-image of this path has length (in V) at most $\widehat{\delta }_1=K\delta _1$. Lemma 4.14 then shows that $h_{\delta _1}^\alpha (D^k\phi \circ \psi ) \le \lceil K \rceil ^{1-\alpha } h_{\delta _1}^\alpha (D^k\phi )$.

4. For the last inequality, the MVT yields $|e^{\phi _1}-e^{\phi _2}|_0 \le L (|e^{\phi _1}|_0 \vee |e^{\phi _2}|_0)\; \;|\phi _1-\phi _2|_0$. Developing $D^q(e^{\phi _1}-e^{\phi _2})$ and using the telescopic principle yields the wanted bound. The above calculations were done for $C^r$ functions. But the uniform bounds implies that the result easily carries over to $c^{\underline{r}}$ functions as well. $\quad \square$

1.1 C.1 Regularity when extracting a parameter

In this section we will show how the regularity of a function of two variables behaves when extracting one variable as a parameter. The notation is as above. We have the following smoothness result when extracting a variable as a parameter:

Theorem C.2

We equip the product $B_U\times B_V$ with the max norm. Let $r,s,t \ge 0$ with $t=r-s>0$. We have the following canonical continuous injections:

$$\begin{aligned} \phi \in C^{r}(U\times V;Z)&\hookrightarrow&\widehat{\phi }\in C^{{{\,{{\underline{t}}}}}}(U; C^{s}(V;Z)) \end{aligned}$$

(C.2)

$$\begin{aligned} \phi \in c^{r}(U\times V;Z)&\hookrightarrow&\widehat{\phi }\in C^{{{\,{{\underline{t}}}}}}(U; c^{s}(V;Z)) \end{aligned}$$

(C.3)

under the natural identification: $\widehat{\phi }_u(x) := \phi (u,x)$, $u\in U, x\in V$.

Proof

Consider the first inclusion. Denote $Y_s= C^{s}(V;Z))$ and let $\phi \in C^{r}(U\times V;Z)$. Our claim is that $\widehat{\phi }_u\in Y_s$ and that the map $u\in U \mapsto \widehat{\phi }_u\in Y_s$ is $C^{{\,{{\underline{t}}}}}$ with $t=r-s>0$.

Case 1: We first show this for $0\le s < r\le 1$. Here $\Vert \phi \Vert _{r}= |\phi |_0 \vee h^r_{\delta _1}(\phi )$ is simply the local r-Hölder norm. For fixed $u\in U$, clearly $|\widehat{\phi }_u|_0\le |\phi |_0$ and $h^s_{\delta _1}(\widehat{\phi }_u) \le h^r_{\delta _1}(\widehat{\phi }_u)\le h^r_{\delta _1}(\phi )$ (since $0<\delta _1\le 1$). so $\widehat{\phi }_u\in Y_s$ with $\Vert \widehat{\phi }_u\Vert _{s}\le \Vert \phi \Vert _r$. To check the regularity w.r.t. u pick $u_1,u_0\in U$ with $0<|u_1-u_0|\le \delta _1$ and set $\Delta =\widehat{\phi }_{u_1}-\widehat{\phi }_{u_0}\in Y_s$. We have $| \Delta |_0 \le h^r_{\delta _1}(\phi ) |u_1-u_0|^{\,r}$.

To check Hölder regularity with respect to x consider $x_0,x_1\in V$ with $0<|x_1-x_0|\le \delta _1$. Then $\Delta (x_1)-\Delta (x_0)= \phi (u_1,x_1)- \phi (u_1,x_0)- \phi (u_0,x_1)+ \phi (u_0,x_0)$ may be estimated using Hölder-regularity either with respect to u or to x. This yields (the middle term is maximal when $|x_1-x_0|= |u_1-u_0|$):

$$\begin{aligned} \frac{|\Delta (x_1)-\Delta (x_0)|}{ |x_1-x_0|^{\,s}} \le 2h^r_{\delta _1}(\phi ) \frac{ |x_1-x_0|^{\,r} \wedge |u_1-u_0|^{\,r}}{ |x_1-x_0|^{\,s} } \le 2h^r_{\delta _1}(\phi ) |u_1-u_0|^{r- s}. \end{aligned}$$

Thus, $h_{s} (\Delta ) \le 2 h^r_{\delta _1} (\phi ) |u_1-u_0|^{r- s}$ from which: $\Vert \widehat{\phi }_{u_1} - \widehat{\phi }_{u_0} \Vert _{Y_s} \le 2\Vert \phi \Vert _r |u_1-u_0|^{r- s}$. So $\widehat{\phi }\in C^{(0,r-s)}(U;Y_s)$ with $\Vert \widehat{\phi }\Vert _{r-s}\le 2 \Vert \phi \Vert _r$.

Case 2: Consider now when $0 \le s\le 1< r \le 1+s$. Again for fixed u: $\widehat{\phi }_u\in Y_s$ with $\Vert \widehat{\phi }_u\Vert _s\le \Vert \phi \Vert _r$. With $\Delta$ as above we have $|\Delta |_0\le |D\phi |_0 \; |u_1-u_0| \le \Vert \phi \Vert _r\; |u_1-u_0|^s$.

Let $u(\tau )=\tau u_1+(1-\tau )u_0$, $0\le \tau \le 1$ be the segment joining $u(0)=u_0$ and $u(1)=u_1$ (included in U by convexity). Applying the MVT, one gets:

$$\begin{aligned} |\Delta (x_1) -\Delta (x_0)|&\le \int _0^1 \left| \frac{d}{d\tau } ( \widehat{\phi }_{u_\tau }(x_1)-\widehat{\phi }_{u_\tau }(x_0)) \right| d\tau \\&\le \int _0^1 |D\widehat{\phi }_{u_\tau }(x_1)-D\widehat{\phi }_{u_\tau }(x_0)) | \; |u'(\tau )| d\tau \\&\le h^{r-1}_{\delta _1}(D \phi ) |x_1-x_0|^{\; r-1} |u_1-u_0|. \end{aligned}$$

Interchanging the roles of $x_1,x_0$ and $u_1,u_0$ we also have:

$$\begin{aligned} |\Delta (x_1)-\Delta (x_0)| \le h^{r-1}_{\delta _1}(D \phi ) |u_1-u_0|^{r-1}|x_1-x_0|, \end{aligned}$$

and then (again the middle term is maximal for $|u_1-u_0| = |x_1-x_0|$):

$$\begin{aligned} \frac{ |\Delta (x_1) \!-\! \Delta (x_0)|}{|x_1-x_0|^{s}}&\le h^{r-1}_{\delta _1}(D\phi ) \frac{ |x_1 \!-\! x_0|^{\; r-1} |u_1 \!-\! u_0| \wedge |u_1 \!-\! u_0|^{\; r-1} |x_1 \!-\! x_0| }{|x_1-x_0|^{\; s}}\\&\le h^{r-1}_{\delta _1}(D\phi ) |u_1-u_0|^{\; r-s} \le \Vert \phi \Vert _r \; |u_1-u_0|^{\; r-s}. \end{aligned}$$

It follows that $\Vert \widehat{\phi }_{u_1}-\widehat{\phi }_{u_0}\Vert _{Y_s} \le \Vert \phi \Vert _r |u_1-u_0|^{\;r- s}$ and $\widehat{\phi }\in C^{(0,r-s)}(U;{Y_s})$ with $\Vert \widehat{\phi }\Vert _{r-s}\le \Vert \phi \Vert _r$. Note that when $r-s=1$ the conclusion is that

$$\begin{aligned} \phi \in C^{(1,s)}(U\times V;Z) \hookrightarrow \widehat{\phi }\in C^{(0,1)}(U; C^{s}(V;Z)), \end{aligned}$$

i.e. in general, $u\mapsto \widehat{\phi }_u$ need not be differentiable in this case, but it is Lipschitz continuous.

Higher order regularity can be reduced to the above two cases.

To see this let $0\le \beta < \alpha \le 1$ and $r=k+m+\alpha$, $s=m+\beta$ with $k,m\in \mathbb {N}_0$. For $\phi \in C^{r}(U\times V;Z)$ and fixed $u\in U$ it is clear that $\widehat{\phi }_u\in C^{(m,\beta )}(V;Z)$. For the regularity of the map $u\mapsto \widehat{\phi }_u$ we leave intermediate derivatives to the reader and consider only the highest order. Note that the first case treated above yields the following injection and identification (using natural isomorphisms for the linear maps involved):

$$\begin{aligned} \partial _u^k \partial _x^m \phi\in & {} C^{(0,\alpha )}(U\times V;L(B_U^k \times B_V^m; Z)) \\&\hookrightarrow&C^{ (0,\alpha -\beta )}(U; C^{(0,\beta )}(V;L(B_U^k \times B_V^m; Z))) \\\simeq & {} C^{ (0,\alpha -\beta )}(U; L(B_U^k ; C^{(0,\beta )}(V;L(B_V^m; Z)))). \\= & {} C^{ (0,\alpha -\beta )}(U; L(B_U^k ; W)) \end{aligned}$$

with $W=C^{(0,\beta )}( V;L(B_V^m; Z))$. We observe that the latter precisely gives the identification with $\partial _x^m ( ( \partial _u^k \widehat{\phi }_u))$ and implies that $\partial _u^k \widehat{\phi }_u \in C^{ (0,\alpha -\beta )}(U;L(B_U^k; C^{s} (V;Z)))$ whence that $\widehat{\phi }_u \in C^{ {\,{{\underline{t}}}}} (U;C^{s}(V;Z))$ with ${\,{{\underline{t}}}}=(k,\alpha -\beta )$ as we wished to show.

In the case $0<\alpha \le \beta \le 1$ and $r=k+m+1+\alpha$, $s=m+\beta$ we consider again $\partial _x^k \partial _u^m \phi$ which reduces the necessary injection to the second case treated above. In either case the norm increases at most by a factor of 2.

For the second injection, if $\phi$ is the $C^r$ limit of smooth functions then the induced function $\widehat{\phi }_u$ is the $C^s$ limit of smooth functions. The norm-estimates carry over from before. $\quad \square$

Remark C.3

For non-compact V and for integer values $n\ge k\ge 0$, there is in general no natural injection of $C^n (U \times V;\mathbb {R})$ into $C^{n-k}(U ; C^{k}(V;\mathbb {R}))$. An easy (counter) example is for $n=k=0$, $U=V=(0,1)$, where you may consider e.g. $\phi (u,x)=\sin (u/x)$.

Corollary C.4

Assume that $W\subset \mathcal {B}_W$ is an open convex subset of a Banach space. Let $r>1$ and $\beta \in (0,1]$. Let $\psi _0,\psi _1\in C^r(V;W)$ and $\phi \in C^{r+\beta }(W;Z)$. We write $M=\Vert \psi _0\Vert _r \vee \Vert \psi _1\Vert _r \vee 1$. Suppose that $\Vert \psi _0 -\psi _1 \Vert _r \le \delta _1$. Then we have

$$\begin{aligned} \Vert \phi \circ \psi _1 - \phi \circ \psi _0\Vert _{r} \le C_{r,\beta } \Vert \phi \Vert _{r+\beta } \; M^r \, \Vert \psi _0 -\psi _1 \Vert _r^\beta \end{aligned}$$

with $C_{r,\beta }$ a constant that depends only upon r, $\beta$ and the choice of norms.

Proof

We first show this under the additional assumption that $\psi _0,\psi _1\in C^{r+\beta }$. Let $a=\Vert \psi _0 -\psi _1 \Vert _r$. We may assume that $a>0$ or else there is nothing to show. We define for $(t,y) \in [0, a] \times V$ the following linear interpolation between $\psi _0$ and $\psi _1$ (allowed since W is convex):

$$\begin{aligned} \widehat{\psi }(t,y) = \frac{t}{a} \psi _1(y) + \left( 1-\frac{t}{a}\right) \psi _0(y) = \psi _0(y) + \frac{t}{a} \left( \psi _1(y) - \psi _0(y)\right) . \end{aligned}$$

One has $\widehat{\psi }\in C^{r+\beta }([0,a]\times V;W)$⁹. Note that $\partial _t \widehat{\psi }= \frac{1}{a}(\psi _1-\psi _0)$ has $(r+\beta -1)$-norm not greater than 1. It follows that $\Vert \widehat{\psi }\Vert _{r+\beta }\le \Vert \psi _0\Vert _{r+\beta }\vee \Vert \psi _1\Vert _{r+\beta }\vee 1$. It follows from Proposition C.1 that $F=\phi \circ \widehat{\psi }$ has $(r+\beta )$-norm bounded by $K=C'_{r,\beta }\Vert \phi \Vert _{r+\beta } M_{r+\beta }^r$.

By our parameter-extraction theorem C.2 we deduce that $t\in [0,a] \mapsto (F(t,\cdot ) \in C^{r}(U;Z))$ is $C^{(0,\beta )}$ with at most twice the indicated bound for the norm. But then the Hölder bound implies:

$$\begin{aligned} \Vert \phi \circ \psi _1-\phi \circ \psi _0\Vert _r = \Vert F(a,\cdot ) - F(0,\cdot )\Vert _r \le 2K |a-0|^\beta = 2K \Vert \psi _0 -\psi _1 \Vert _r^\beta \end{aligned}$$

as we wanted to show. Returning to the general case, consider the telescopic form for the derivative:

$$\begin{aligned} D(\phi \circ \psi _1)- D(\phi \circ \psi _0) = (D\phi \circ \psi _1 -D\phi \circ \psi _0).D\psi _1 + (D\phi \circ \psi _0).(D\psi _1-D\psi _0) \end{aligned}$$

Here, $D\phi \in C^{r+\beta -1}$ and $\psi _1\in C^{r}\subset C^{r+\beta -1}$ (since $\beta \le 1$). The first part then applies (with $r+\beta -1$ instead of $r+\beta$) and shows that $\Vert D\phi \circ \psi _1 -D\phi \circ \psi _0\Vert _{r-1}\le C_{r-1,\beta } \Vert D\phi \Vert _{r+\beta -1} \Vert \psi _1-\psi _0\Vert ^\beta$. The last term trivially verifies the same type of bound. From this we deduce the result for the difference without derivatives. $\quad \square$

Proposition C.5

Let us consider $r=k+\alpha$ with $k\in {\mathbb {N}}$ and $0<\alpha \le 1$, and ${\underline{r}}=(k,\alpha )$, $\mathcal {B},X,Y$ three Banach spaces and $U\subset \mathcal {B}$ an open subset. Under the natural identifications, we have the following injections:

$$\begin{aligned} L(X,C^{\underline{r}}(U,Y))&\hookrightarrow C^{\underline{r}}(U,L(X,Y)) \end{aligned}$$

Proof

We present a proof by induction on $k\in {\mathbb {N}}$.

For ${\underline{r}}=(0,\alpha )$, we assume that we have an operator $\mathcal {L}_u:X\rightarrow Y$, satisfying: there is a $C>0$, such that for any $\phi \in X$, any $u\not =v,~u,v\in U$,

$$\begin{aligned} \mathcal {L}_u\phi&\in C^{0,\alpha }(U,Y)\\ \Vert \mathcal {L}_u\phi \Vert _Y&\le C\Vert \phi \Vert _X\\ \Vert \mathcal {L}_u\phi -\mathcal {L}_v\phi \Vert _Y&\le C\Vert \phi \Vert _X\Vert u-v\Vert ^\alpha _\mathcal {B}\end{aligned}$$

Define ${\hat{\mathcal {L}}}$ as the map $u\in U\mapsto \mathcal {L}_u\in L(X,Y)$. Then it is easy to see that under the previous assumptions, ${\hat{\mathcal {L}}}$ is a $C^\alpha$ map

Let us assume that the wanted injection is established at rank $k-1$, and consider $r=k+\alpha$, an operator $\mathcal {L}_u:X\rightarrow Y$, with $\mathcal {L}_u\phi \in C^{\underline{r}}(U,Y)$ and $\Vert \mathcal {L}_u\phi \Vert _{C^r(U,Y)}\le C\Vert \phi \Vert _X$.

For any $\phi \in X$, we may consider the partial derivative (w.r.t u) of $\mathcal {L}_u\phi$, $\partial _u(\mathcal {L}_u\phi )\in L(\mathcal {B},Y)$, which is, by assumption, a $C^{(k-1,\alpha )}$ map w.r.t $u\in U$, with $\Vert \partial _u\mathcal {L}_u\phi \Vert _{C^{k-1,\alpha }}\le C\Vert \phi \Vert _X$. But by induction hypothesis, this means that the map ${\hat{\mathcal {L}}}:U\rightarrow L(X,Y)$ admits a derivative which is $C^{k-1,\alpha }$, i.e that ${\hat{\mathcal {L}}}$ is $C^{k,\alpha }$. $\quad \square$

1.2 C. 2 Bochner measurable smooth sections

Let $(\Omega ,\mathcal {F})$ be a measurable space. In this section there is no measure involved. We write

$$\begin{aligned} X_{r}(U;Z) := X_{(k,\alpha )}(U;Z) = L^\infty (\Omega ; c^{(k,\alpha )}(U;Z)) \end{aligned}$$

for a Bochner measurable map from $\Omega$ to the Banach space $Y=c^{(k,\alpha )}(U;Z)$, with $r=k+\alpha >0$, $\alpha \in (0,1]$, $k\in \mathbb {N}_0$. The norm of ${\varvec{\varphi } }\in X_r(U;Z)$ is the uniform norm: $\Vert {\varvec{\varphi } }\Vert _{X_r} = \sup _{\omega \in \Omega } \Vert {\varvec{\varphi } }_\omega \Vert _r$. Operations in the following proposition is understood to take place fiber-wise, e.g. for fixed $\omega \in \Omega$, $( {\varvec{\varphi } }\circ \varvec{\psi })_\omega := {\varvec{\varphi } }_\omega \circ \varvec{\psi }_\omega$. We have:

Proposition C.6

Let ${\varvec{\varphi } }_1, {\varvec{\varphi } }_2\in X_r(V;\mathbb {C})$, ${\varvec{\varphi } }\in X_r(V;Z)$ and $\varvec{\psi }\in X_s (U;V)$, $s=r\vee 1$. Then there are constants $C_{r,1},C_{r,2},C_{r,3}$ depending only upon r and the chosen norms so that:

1.	${\varvec{\varphi } }_1 {\varvec{\varphi } }_2$	$\in$	$X_r(V;\mathbb {C})$	and	$\Vert {\varvec{\varphi } }_1 {\varvec{\varphi } }_2\Vert _{X_r}$	$\le$	$C_{1,r} \Vert {\varvec{\varphi } }_1\Vert _{X_r} \Vert {\varvec{\varphi } }_2\Vert _{X_r}$.
2.	$e^{ {\varvec{\varphi } }_1}$	$\in$	$X_r(V;\mathbb {C})$	and	$\Vert e^{ {\varvec{\varphi } }_1}\Vert _{X_r}$	$\le$	$C_{2,r} |e^{ {\varvec{\varphi } }_1}|_0 \; (1 + \Vert D {\varvec{\varphi } }_1\Vert _{X_{r-1}})^r$
3.	${\varvec{\varphi } }\circ \varvec{\psi }$	$\in$	$X_r (U; Z)$	and	$\Vert {\varvec{\varphi } }\circ \varvec{\psi }\Vert _{X_r}$	$\le$	$C_{3,r} \Vert {\varvec{\varphi } }\Vert _{X_r} (1 + \Vert D\varvec{\psi }\Vert _{X_{s-1}})^r$, $(r>1)$.

Proof

As operations are fiber-wise we clearly have the stated bounds on the norms. The only issue is Bochner-measurability. We show this for the first case: Let $M > \Vert {\varvec{\varphi } }_1\Vert _{X_r} \vee \Vert {\varvec{\varphi } }_2\Vert _{X_r}$. Given $\epsilon >0$ we may find a countable measurable partition $(\Omega _m)_{m\in \mathbb {N}}$ so that for every $m\in \mathbb {N}$, $\omega ,\omega '\in \Omega _m$, $i=1,2$ we have: $\Vert ( {\varvec{\varphi } }_i)_\omega - ( {\varvec{\varphi } }_i)_{\omega '}\Vert _r \le \frac{\epsilon }{2 C_{1,r} M}$. Then by the telescopic principle and the above bounds:

$$\begin{aligned} \Vert ( {\varvec{\varphi } }_1 {\varvec{\varphi } }_2)_\omega - ( {\varvec{\varphi } }_1 {\varvec{\varphi } }_2)_{\omega '}\Vert _r \le 2 C_{1,r} \frac{\epsilon }{2 C_{1,r} M} M \le \epsilon , \end{aligned}$$

implying that ${\varvec{\varphi } }_1 {\varvec{\varphi } }_2$ is Bochner measurable in the sense of definition B.1. The other two statements follow in the same way. $\quad \square$

Lemma C.7

One has the following injections:

$$\begin{aligned}&{\varvec{\varphi } }\in L^\infty (\Omega ; C^{\underline{r}}(V; Z)) \hookrightarrow \varvec{\widehat{\phi }}\in C^{{\underline{r}}}(V; L^\infty (\Omega ;Z)),\\&{\varvec{\varphi } }\in L^\infty (\Omega ; c^{\underline{r}}(V; Z)) \hookrightarrow \varvec{\widehat{\phi }}\in c^{{\underline{r}}}(V; L^\infty (\Omega ;Z)), \end{aligned}$$

under the natural fiber-wise identification:

$$\begin{aligned} \varvec{\widehat{\phi }}(v)(\omega ) := {\varvec{\varphi } }(\omega )(v), \ \ \ \omega \in \Omega , v\in V. \end{aligned}$$

Proof

Given $\epsilon >0$ we find a countable measurable partition $(\Omega _i)_{i\in \mathbb {N}}$ so that for every $i\in \mathbb {N}$, $\omega ,\omega '\in \Omega _i$, we have: $\Vert ( {\varvec{\varphi } })_\omega - ( {\varvec{\varphi } })_{\omega '}\Vert _r \le {\epsilon }$. Pick also for each $i\ge 1$: $\omega _i\in \Omega _i$ and set $f_i= {\varvec{\varphi } }(\omega _i)$. Define:

$$\begin{aligned} \varvec{\widehat{f}}(v)(\omega ) =\varvec{f}(\omega )(v) :=\sum _i \mathbf{1}_{\Omega _i}(\omega ) f_i(v). \end{aligned}$$

Then $\varvec{f}$ is a $\sigma$-simple $\epsilon$-uniform approximation to ${\varvec{\varphi } }$. Clearly, $\varvec{\widehat{f}}$ takes values in $Y=L^\infty (\Omega ,Z)$. Also $\partial _v^q \varvec{\widehat{f}}(v) \in L(B_V^q; Y)$ and $\Vert \partial _v^q \varvec{\widehat{f}}- \partial _v^q \varvec{\widehat{\phi }}\Vert \le \epsilon$, $0\le q\le k$ and similarly for the $\alpha$-Hölder estimate for the k’th derivative. Thus $\varvec{\widehat{\phi }}\in c^{{\underline{r}}}(V; L^\infty (\Omega ;Z))$ and it has the same norm as ${\varvec{\varphi } }$. $\quad \square$

We conclude this section with a key ingredient for our applications section:

Proposition C.8

With the notation as in Theorem C.2 and this section, we have the following injection of norm at most 2:

$$\begin{aligned} {\varvec{\varphi } }\in X_r(U\times V; Z) \hookrightarrow \varvec{\widehat{\phi }}\in C^{{r-s}}(U; X_s(V;Z)), \end{aligned}$$

under the natural fiber-wise identification:

$$\begin{aligned} \varvec{\widehat{\phi }}(u)(\omega )(v) := {\varvec{\varphi } }(\omega )(u,v), \ \ \ \omega \in \Omega , u\in U, v\in V. \end{aligned}$$

Proof

Combining C.2 and C.7 we have the following injections:

$$\begin{aligned} X_r(U\times V; Z) \ \&= \ \ \ L^\infty (\Omega ; c^{\underline{r}}(U\times V; Z)) \\&\hookrightarrow \ \ L^\infty (\Omega ; C^{r-s}(U; c^{\,{{\underline{s}}}}(V; Z)) \\&\hookrightarrow \ \ C^{r-s}(U; L^\infty (\Omega ; c^{\,{{\underline{s}}}}(V;Z)))\\&= \ \ \ C^{r-s}(U; X_s(V;Z)). \end{aligned}$$

$\square$

D Graded Differential Calculus

An essential ingredient in differential calculus is the Leibnitz principle: When e.g. f, g are $C^r(\mathbb {R})$-functions for $r\ge 1$ (r not necessarily an integer) then so is their product and one has a formula for the derivative of the product $(f\cdot g)'=f' \cdot g + f \cdot g'$. The derivative is then $C^{r-1}$ and one may iterate the derivation formula when $r\ge 2$. The aim here is to develop a similar theory for graded differential calculus, in particular the Leibniz principle, when f, g are replaced by linear operators depending on a parameter u but where regularity with respect to the parameter only appears when downgrading the codomain (the image space) or upgrading the domain within a certain scale of Banach spaces. The upshot of this appendix is to show that the resulting regularity when performing algebraic operations on graded differential operators is as good as one could possibly hope for. In particular, we prove Lemma 3.8.

We will stick to the notation of Sect. 3. More precisely, let $\mathcal{X}=(X_t)_{t\in (0,r_0]}$ denote the scale of Banach spaces. By this we mean a parametrized family of Banach spaces coming with a family of bounded linear (downgrading) operators $j_{s,r}\in L(X_r;X_s)$, $0< s\le r\le r_0$. We assume that each operator is injective and has dense image and that the collection satisfies the transitivity condition: $j_{s,s}=\mathrm{Id}$ and $j_{s,c}j_{c,r}=j_{s,r}$ whenever $0< s\le c\le r\le r_0$.

Example D.1

An instructive example to have in mind is $X_t = C^t(S^1)$, $t \in (0,r_0]$ with $j_{s,r}: C^r(S^1) \rightarrow C^s(S^1)$ being the natural embedding for $0< s\le r\le r_0$.

We let $\mathcal {B}$ denote a Banach space and let $U\subset \mathcal {B}$ be a non-empty open convex subset.

Definition D.2

Let $n\in \mathbb {N}_0=\{0,1,\ldots \}$, $\gamma >0$ with $\gamma +n\le r_0$. We associate to the integer $n$ the following set of ordered pairs: $I_n= \{(s,r)\in (0,r_0]^2 : s + n\le r\}$. Consider a family $\mathcal {M}$ of bounded linear operators $M_{s,r}(u) \in L(\mathcal {B}^n; L(X_r,X_s))$, $(s,r)\in I_n$ and parametrized by $u\in U$. We say that the family $\mathcal {M}$ is (j-)equivariant and $(\gamma ,n)$-regular provided that:

• 1. For every $(s,r),(s',r')\in I_n$ with $s<s'$, $r<r'$ and $u\in U$: $\displaystyle j_{s,s'} M_{s',r'}(u) = M_{s,r}(u) j_{r,r'}$.

• 2. The map $u\in U \mapsto M_{s,r}(u)\in L(\mathcal {B}^n; L(X_r;X_s))$ is $C^t$ for all $(s,r)\in I_n$ and $0 \le t<\gamma \wedge (r-s-n)$.

Keeping the same notation as in the previous definition we define:

Definition D.3

Consider a family $\mathcal {N}$ of functions $N_{s}(u) \in L(\mathcal {B}^n; X_s)$, $0 \le s< r_0-n$, parametrized by $u\in U$. We say that $\mathcal {N}$ is left-equivariant and $(\gamma ,n)$-regular provided that for all $0\le s<r< r_0-n$, $u\in U$: $N_{s}(u) = j_{s,r} N_{r}(u)$, and the map $u\in U \mapsto N_{s}(u)\in L(\mathcal {B}^n;X_s)$ is $C^t$ for all $0 \le t<\gamma \wedge (r_0-n-s)$.

Lemma D.4

Let $\mathcal {M}$ be an equivariant $(\gamma ,n)$-regular family with $\gamma >1$. We define the derived family $\partial _u\mathcal {M}$ given by: $\partial _u M_{s,r}(u)\in L(\mathcal {B}; L(\mathcal {B}^n;L(X_s,X_r)))\equiv L(\mathcal {B}^{n+1}; L(X_s,X_r))$ for all $(s,r)\in I_{n+1}$. This derived family is equivariant and $(\gamma -1,n+1)$-regular. Conversely, suppose that $\mathcal {M}$ is $(1+\alpha ,n)$-regular with $\alpha >0$ and that derived family $\partial _u \mathcal {M}$ is $(\gamma ',n+1)$-regular, then setting $\gamma =\alpha \vee \gamma '+1$, we have that $\mathcal {M}$ is $(\gamma ,n)$-regular. A similar statement holds for a left-equivariant family $\mathcal {N}$.

Proof

The first statement is obvious from definitions. For the second, we may assume $\gamma '>\alpha$ or else it is trivial. Suppose that $\mathcal {M}$ is $(1+\alpha ,n)$-regular with $\alpha >0$ and let $\partial _u \mathcal {M}$ be the derived family. If $0<s<r\le r_0$ with $r-s\le r_0-n$ and $1< t< t_*=(\gamma '+1)\wedge (r-s-n)$. Then $u\mapsto \partial _u M_{s,r}(u)$ is $C^{t-1}$ and consequently $u\mapsto M_{s,r}(u)$ is $C^t$ as we wanted to show.

The main reason for introducing equivariant, $(\gamma ,n)$-regular families comes from the stability under products: $\quad \square$

Proposition D.5

Let $\mathcal {M}^1$ and $\mathcal {M}^2$ be two families of j-equivariant, $(\gamma _1,n_1)$-regular, respectively $(\gamma _2,n_2)$-regular, operators. Suppose that $n=n_1+n_2<r_0$ and set $\gamma =\gamma _1\wedge \gamma _2 \wedge (r_0-n)>0$. Then there is a well-defined product family $\mathcal {M}=\mathcal {M}^1\star \mathcal {M}^2$ obtained by declaring for $(s,r)\in I_n$:

$$\begin{aligned} M_{s,r}(u):= M^1_{s,c}(u) M^2_{c,r}(u)\in L(\mathcal {B}^{n_1}\times \mathcal {B}^{n_2}; L(X_r;X_s)) \simeq L(\mathcal {B}^{n}; L(X_r;X_s)) \end{aligned}$$

with c being any number in the non-empty interval $(s+n_1,r-n_2)$. This product family is equivariant and $(\gamma ,n)$-regular. Similarly, with $\mathcal {M}^1$ as above and $\mathcal {N}^2$ a left-equivariant and $(\gamma _2,n_2)$-regular family, the product $N_s(u):=M^1_{s,c} N^2_c(u)$ defines a family $\mathcal {N}=\mathcal {M}^1 \star \mathcal {N}^2$ which is left-equivariant and $(\gamma ,n)$-regular.

Proof

If $c<c'$ are two numbers in the above interval, then we have by equivariance (all operators being well-defined):

$$\begin{aligned}&M^1_{s,c}(u) M^2_{c,r}(u) = M^1_{s,c}(u) \left[ j_{c,c'} M^2_{c',r}(u) \right] = \left[ M^1_{s,c}(u) j_{c,c'} \right] M^2_{c',r}(u) \\&\quad = M^1_{s,c'}(u) M^2_{c',r}(u) , \end{aligned}$$

showing that the product does not depend upon the choice of c. Let $(s,r)\in I_n$ and $t_*=\gamma \wedge (r-s-n)>0$. Regularity will be shown by induction in $\gamma$. First, assume that $\gamma \in (0,1]$. Then, in particular, $t_*\le 1$. We will show that $u\mapsto M_{s,r}(u)\in L(X_r;X_s)$ is t-Hölder for every $0<t<t_*$. Let $0<\epsilon < t_*$ and set $s':=s+n_1+\epsilon /3 < r':=r-n_2-\epsilon /3$. We then have by equivariance:

$$\begin{aligned} M_{s,r}(u) = M^1_{s,s'}(u) j_{s',r'} M^2_{r',r}(u). \end{aligned}$$

Note that $r'-s-n_1=r-s'-n_2=r-s-n-\epsilon /3$. Also let $c=(s'+r')/2$. When $u,u+h\in U$, using a telescopic sum, equivariance and Hölder continuity we have the following identity:

$$\begin{aligned} M_{s,r}(u+h)&-M_{s,r}(u) = M^1_{s,s'}(u+h) j_{s',r'} M^2_{r',r}(u+h)- M^1_{s,s'}(u) j_{s',r'} M^2_{r',r}(u) \end{aligned}$$

(D.1)

$$\begin{aligned}&= \left( M^1_{s,r'}(u+h)-M^1_{s,r'}(u)\right) M^2_{r',r}(u)\nonumber \\&\quad + \left( M^1_{s,c}(u+h)-M^1_{s,c}(u)\right) \left( M^2_{c,r}(u+h)-M^2_{c,r}\right) \nonumber \\&\quad + M^1_{s,s'}(u) \ \left( M^2_{s',r}(u+h)-M^2_{s',r}(u) \right) \nonumber \\&\quad = \mathcal {O}(h^{t_*-\epsilon /3}) \mathcal {O}(1) + \mathcal {O}(h^{t_*/2-\epsilon /3}) \mathcal {O}(h^{t_*/2-\epsilon /3}) + \mathcal {O}(1) \mathcal {O}(h^{t_*-\epsilon /3})\nonumber \\&\quad = \mathcal {O}(h^{t_* - 2\epsilon /3}). \end{aligned}$$

(D.2)

We may here let $\epsilon \rightarrow 0$ and obtain the claim for this case.

For higher order regularity, let $k\ge 1$ be an integer and suppose that the proposition has been proven whenever $0<\gamma \le k$. For $k=1$ this was done above. So assume now that $\gamma \in (k,k+1]$. For $(s,r)\in I_n$ we set $t_*=\gamma \wedge (r-s-n)$. We may assume that $t_*\in (k,k+1]$ as well (or else there is nothing to show). We write $t_*=k+\alpha$ with $\alpha \in (0,1]$. We want to show that $u\mapsto M_{s,r}(u)$ is $C^t$ for all $0<t<t_*$. Set $s'=s+n_1+\alpha /3$ and $r'=r-n_2-\alpha /3$. Then as before $r'-s' =r-s-n-2\alpha /3 > k+\alpha /3$. Since $k\ge 1$ we obtain derived families $\partial _u \mathcal {M}^1$ and $\partial _u \mathcal {M}^2$ as described in Lemma D.4. We have e.g. $M^1_{s,s'}(u)j_{s',r'}= M^1_{s,r'}(u)$ so by the MVT we get:

$$\begin{aligned}&{\left| M^{1}_{s,r'}(u+h)-M^{1}_{s,r'}(u) - h \cdot \partial _u M^{1}_{s,r'}(u) \right| _{L(X_s;X_{r'})} }\\&\quad \le |h| \sup _{\tau \in [0,1]} \left| \partial _u M^{1}_{s,r'}(u+\tau h) -\partial _u M^{1}_{s,r'}(u) \right| _{L(X_s;X_{r'})} \\&\quad = O(h^1) O(h^{\alpha /3}) = O(h^{1+\alpha /3}) \ . \end{aligned}$$

With a similar expansion for $M^2$ and using Hölder estimates for the middle term we expand (D.2) to get:

$$\begin{aligned} M_{s,r}(u+h)-M_{s,r}(u)&=\! (h.\partial _uM^1_{s,r'})(u) M^2_{r',r}(u) \!+\! M^1_{s,s'}(u) \; (h.\partial _u M^2_{s',r})(u) \!+\! \mathcal {O}(h^{1\!+\!\alpha /3}), \end{aligned}$$

showing that $M_{s,r}$ is differentiable with derivative

$$\begin{aligned} \partial _u M_{s,r}(u) = \partial _u M^1_{s,r'}(u) M^2_{r',r}(u)+ M^1_{s,s'}(u) \partial _u M^2_{s',r}(u). \end{aligned}$$

Now, in this expression we may again use equivariance to write

$$\begin{aligned} \partial _u M_{s,r}(u) = \partial _u M^1_{s,c_1}(u) M^2_{c_1,r}(u)+ M^1_{s,c_2}(u) \partial _u M^2_{c_2,r}(u). \end{aligned}$$

with $c_1\in (s+1+n_1, r-n_2)$ and $c_2\in (s+n_1, r-1-n_2)$. The first term is the product of two j-equivariant families that are $(\gamma -1,n_1+1)$ and $(\gamma ,n_2)$ regular, respectively. Since $(\gamma -1)\wedge (r-s-n-1)=t_*-1\le k$ we may apply the induction hypothesis on this term to conclude that this first product is $(\gamma -1,n+1)$-regular. Similarly for the second term. Thus $u\mapsto \partial _u M_{s,r}(u)$ is $C^{t}$ for every $t<\gamma -1$, whence $u\mapsto M_{s,r}(u)$ is $C^{t}$ for every $t<\gamma$ as we wanted to show (see Lemma D.4). The proof in the left-equivariant case follows the same path. $\quad \square$

Lemma D.6

Let $(Q_{s,r}(u))$ be an equivariant, $(\gamma ,0)$-regular family with the additional property that $\mathbf{1}-Q_s(u)$ is invertible for all $0<s\le r_0$ and having a uniformly bounded inverse $R_s(u) = (\mathbf{1}-Q_{s}(u))^{-1}$ when $\epsilon <s\le r_0$ for any $\epsilon >0$. Then the family of operators $R_{s,r}(u)=\varvec{j}_{s,r} R_r(u)$, $0<s<r\le r_0$ is again equivariant and $(\gamma ,0)$-regular.

Proof

This boils down to the resolvent identity combined with equivariance. We have e.g. for $u,u+h\in U$:

$$\begin{aligned} R_{s,r}(u+h)-R_{s,r}(u) = R_{s}(u+h) \left( Q_{s,r}(u+h)-Q_{s,r}(u)\right) R_{r}(u). \end{aligned}$$

Hölder-continuity then follows using regularity of the middle term. When $t_*=\gamma \wedge (r-s-1)>1$ we may again develop the middle term and conclude that $R_{s,r}(u)$ is differentiable with derivative:

$$\begin{aligned} \partial _u R_{s,r}(u) = R_{s}(u) \left( \partial _u Q_{s,r}(u)\right) R_{r}(u) \in L(\mathcal {B};L(X_r;X_s)). \end{aligned}$$

Here we have a product of 3 operators being $(t_*,0)$, $(t_*-1,1)$ and $(t_*,0)$-regular, respectively. The product is then itself $(t_*-1,1)$-regular and therefore $R_{s,r}(u)$ is $(t_*,0)$-regular as we wanted to show. $\quad \square$

Proof of Lemma 3.8

First note that in Theorem 3.2 the collection of operators $L_{s,r}(u):=\mathscr {L}_{s,u}\varvec{j}_{s,r}$ with $0<s\le r\le r_0$, forms an equivariant family $\mathcal {L}$ of $(r_0,0)$-regular operators over $(0,r_0]$. The derived family $\left( \partial _u(\mathscr {L}_{s,u}\varvec{j}_{s,r_0})\right) _{s\in (0,r_0]}$ is then $(r_0-1,1)$-regular.

Under Hypothesis $\mathcal {H}(\gamma )$ the family of fixed fields $({{\mathbf {f}}}_s(u))_{s\in (0,r_0]}$ is left-equivariant and $(\gamma ,0)$-regular. From Proposition D.5 it follows that the family of products $\left( \partial _u(\mathscr {L}_{s,u}\varvec{j}_{s,r_0}) {{\mathbf {f}}}_{r_0}(u)\right) _{s\in (0,r_0]}$ is $(\gamma \wedge (r_0-1),1)$-regular. This is in fact the principal term in the definition 3.3 of $P_{s,r_0,u}({{\mathbf {f}}}_{r_0}(u))$ which is therefore also $(\gamma \wedge (r_0-1),1)$-regular: this shows the first claim in Lemma 3.8. In a similar way, using Proposition 3.4 and $\mathcal {H}(\gamma )$ we see that $M_{s,r}(u) = Q_{s,u}({{\mathbf {f}}}_{s}(u))\varvec{j}_{s,r}$ is equivariant and $(\gamma ,0)$-regular. This implies the second claim in the Lemma. $\quad \square$