Proof. The first inequality follows from a direct application of Hölder’s inequality. The second one is a straightforward application of Minkowski’s inequality,

$$ \begin{align*} \bigg(\int|{\widehat{\psi}}_{is}(\varphi)|^\gamma \, {d}\unicode{x3bb}_I\bigg)^{1/\gamma} &\leq \bigg(\int \widehat{\psi}(|\varphi|)^\gamma \, {d}\unicode{x3bb}_I\bigg)^{1/\gamma}\\&= \bigg(\int \bigg( \sum_{j=0}^{k-1} \frac{|\varphi|\circ \psi^{-1}_{j+1}}{|\psi'\circ \psi^{-1}_{j+1}|}\, \bigg)^{\gamma} {d}\unicode{x3bb}_I \bigg)^{1/\gamma} \\&\leq \sum_{j=0}^{k-1}\bigg( \int\bigg( \frac{|\varphi|\circ \psi^{-1}_{j+1}}{|\psi'\circ \psi^{-1}_{j+1}|}\bigg)^{\gamma}\, {d}\unicode{x3bb}_I \bigg)^{1/\gamma}\\&= \sum_{j=0}^{k-1} \bigg( \int\bigg(\frac{|\varphi|}{|\psi'|}\bigg)^{\gamma}\mathbf{1}_{[c_j,c_{j+1}]}|\psi'|\, {d}\unicode{x3bb}_I\bigg)^{1/\gamma} \\&\leq \frac{k}{\eta_{-}^{1-\gamma}}\bigg(\int |\varphi|^{\gamma}\,{d}\unicode{x3bb}_I\bigg)^{1/\gamma}. \end{align*} $$

Put $C_\gamma = k\cdot \eta _{-}^{\gamma -1}$ . Then,

$$ \begin{align*}\|{\widehat{\psi}}_{is}(\varphi)\|_{\gamma} \leq C_\gamma \|\varphi\|_\gamma .\\[-33pt] \end{align*} $$

Proof. Since $\widehat {\psi }$ is a bounded linear operator on each of the Banach spaces in equation (4.1) (in particular, due to the DFLY inequality below), the corollary follows from Lemmas A.11 and 4.1.

Proof of Lemma 4.3

Since $e^{i s\chi }$ is $2\pi $ periodic in s, we will estimate

$$ \begin{align*}\sup_{s\in [0,2\pi]}\sup_{{\varepsilon}\in (0,{\varepsilon}_0]}{\varepsilon}^{-\beta}\|R_{\alpha}\operatorname*{\mathrm{osc}}(e^{i s\chi}, B_{{\varepsilon}}(\cdot))\|_{\bar{\gamma}} .\end{align*} $$

Note that

$$ \begin{align*} &\sup_{{\varepsilon}\in (0,{\varepsilon}_0]}\|R_{\alpha}\operatorname*{\mathrm{osc}}(e^{i s\chi}, B_{{\varepsilon}}(\cdot))\|_{\bar{\gamma}}\cdot {\varepsilon}^{-\beta}\\ &\quad\leq \sup_{{\varepsilon}\in (0,{\varepsilon}_0]} \bigg(\int_0^{1/2} (R_{\alpha}\operatorname*{\mathrm{osc}}(e^{i s\chi}, B_{{\varepsilon}}(x)))^{\bar{\gamma}}\,{d}\unicode{x3bb}_I(x)\bigg)^{1/\bar{\gamma}}\cdot {\varepsilon}^{-\beta}\\ &\qquad+\sup_{{\varepsilon}\in (0,{\varepsilon}_0]} \bigg(\int_{1/2}^1 (R_{\alpha}\operatorname*{\mathrm{osc}}(e^{i s\chi}, B_{{\varepsilon}}(x)))^{\bar{\gamma}}\,{d}\unicode{x3bb}_I(x)\bigg)^{1/\bar{\gamma}}\cdot {\varepsilon}^{-\beta}. \end{align*} $$

We will only estimate the first summand as the estimation of the second follows analogously. Using the definition $\operatorname *{\mathrm {osc}}(h,A)=\operatorname *{\mathrm {osc}}(\Re h,A)+\operatorname *{\mathrm {osc}}(\Im h,A)$ and $|e^{it_1}-e^{it_2}|\leq \min \{2,|t_1-t_2| \}$ , we note that for any measurable set A, we have $\operatorname *{\mathrm {osc}}(e^{is\chi }, A)\leq \min \{4, 4s/\pi \operatorname *{\mathrm {osc}}(\chi , A)\}$ . Due to equation (4.2), there exists $C>0$ such that for all $s>0$ , for all ${\varepsilon }>0$ and all $x\in [{\varepsilon },1/2]$ , we have

$$ \begin{align*} \operatorname*{\mathrm{osc}}(e^{i s\chi}, B_{{\varepsilon}}(x)) &\leq \frac{8|s|{\varepsilon}}{\pi}\sup_{y\in B_{{\varepsilon}}(x)}\max\{|\chi'(y+)|, |\chi'(y-)|\} \leq \frac{8C|s|{\varepsilon}}{\pi} (x-{\varepsilon})^{-b}. \end{align*} $$

We have that $8C|s|{\varepsilon } (x-{\varepsilon })^{-b} /\pi \leq 4$ if and only if

$$ \begin{align*} x\geq \bigg(\frac{2C|s|{\varepsilon}}{\pi}\bigg)^{1/b}+{\varepsilon}=: K_{\varepsilon}> {\varepsilon}. \end{align*} $$

Since $K_{\varepsilon }> {\varepsilon }$ , on $[K_{\varepsilon }, 1/2]$ , we use $\frac {8C|s|{\varepsilon }}{\pi } (x-{\varepsilon })^{-b}$ and on $[0, K_{\varepsilon })$ , we use $4$ as upper bounds for $\operatorname *{\mathrm {osc}}(e^{i s\chi }, B_{{\varepsilon }}(x))$ , to obtain

(4.5) $$ \begin{align} \sup_{{\varepsilon}\in (0,{\varepsilon}_0]} \bigg(\int_0^{1/2} (R_{\alpha}\operatorname*{\mathrm{osc}}(e^{i s\chi}, B_{{\varepsilon}}(x)))^{\bar{\gamma}}\,{d}\unicode{x3bb}_I(x)\bigg)^{1/\bar{\gamma}}\cdot{\varepsilon}^{-\beta}\notag\\&\leq \sup_{{\varepsilon}\in (0,{\varepsilon}_0]} \bigg( 4{K_{\varepsilon}}\sup_{[0,K_{\varepsilon}]}R_\alpha\mathbf{1} \cdot {\varepsilon}^{-\beta}\nonumber\\&\qquad\quad\qquad+\bigg(\int_{K_{\varepsilon}}^{1/2} \bigg(\frac{8C|s|{\varepsilon}^{1-\beta}}{\pi} R_\alpha\mathbf{1} \cdot (x-{\varepsilon})^{-b}\bigg)^{\bar{\gamma}}{d}\unicode{x3bb}_{I}(x)\bigg)^{1/\bar{\gamma}}\bigg)\notag\\&\leq \sup_{{\varepsilon}\in (0,{\varepsilon}_0]} 4{K_{\varepsilon}}^{1+\alpha}{\varepsilon}^{-\beta}+\sup_{{\varepsilon}'\in (0,{\varepsilon}_0]}\frac{8C|s|{\varepsilon}^{1-\beta}}{\pi}\,\bigg(\int_{K_{\varepsilon}}^{1/2} ( x^{\alpha} (x-{\varepsilon})^{-b})^{\bar{\gamma}}{d}\unicode{x3bb}_{I}(x)\bigg)^{1/\bar{\gamma}}. \end{align} $$

For the first summand of equation (4.5), we have that there exists $\tilde C_{{\varepsilon }_0}>0$ such that

$$ \begin{align*} \sup_{{\varepsilon}\in (0,{\varepsilon}_0]} 4{K_{\varepsilon}}^{1+\alpha}{\varepsilon}^{-\beta} &\leq 8\sup_{{\varepsilon}\in (0,{\varepsilon}_0]} \max\bigg\{\bigg(\frac{2C|s|}{\pi}\bigg)^{(1+\alpha)/b}{\varepsilon}^{(1+\alpha)/b-\beta},{\varepsilon}^{1+\alpha-\beta} \bigg\}\\ &\leq \tilde C_{{\varepsilon}_0}(1+|s|^{(1+\alpha)/b}) < \infty, \end{align*} $$

which follows from the fact that $\beta < (1/\bar {\gamma }+\alpha )/b < (1+\alpha )/b$ and $\beta \leq 1$ .

For the second summand of equation (4.5), we use $\bar {\gamma } < 1/\alpha $ and $(x+{\varepsilon })^{\alpha \bar {\gamma }} \leq x^{\alpha \bar {\gamma }}+{\varepsilon }^{\alpha \bar {\gamma }}$ to compute

$$ \begin{align*} &\sup_{{\varepsilon}\in (0,{\varepsilon}_0]}\frac{8C|s|{\varepsilon}^{1-\beta}}{\pi}\,\bigg(\int_{K_{\varepsilon}}^{1/2} ( x^{\alpha} (x-{\varepsilon})^{-b})^{\bar{\gamma}}{d}\unicode{x3bb}_{I}(x)\bigg)^{1/\bar{\gamma}}\\&\quad\leq \frac{8C|s|}{\pi}\sup_{{\varepsilon}\in (0,{\varepsilon}_0]} {\varepsilon}^{1-\beta}\bigg(\int_{({2Cs{\varepsilon}}/{\pi})^{1/b}}^{1/2}(x+{\varepsilon})^{\alpha\bar{\gamma}} x^{-b\bar{\gamma}}{d}\unicode{x3bb}_{I}(x)\bigg)^{1/\bar{\gamma}}\\&\quad\leq \frac{8C|s|}{\pi}\sup_{{\varepsilon}\in (0,{\varepsilon}_0]} \bigg( {\varepsilon}^{1-\beta}\bigg(\int_{({2Cs{\varepsilon}}/{\pi})^{1/b}}^{1/2}x^{\bar{\gamma}(\alpha-b)}{d}\unicode{x3bb}_{I}(x)\bigg)^{1/\bar{\gamma}}\\&\qquad+{\varepsilon}^{1+\alpha-\beta}\bigg(\int_{({2Cs{\varepsilon}}/{\pi})^{1/b}}^{1/2}x^{-b\bar{\gamma}}{d}\unicode{x3bb}_{I}(x)\bigg)^{1/\bar{\gamma}}\bigg)\\&\quad\lesssim |s|\sup_{{\varepsilon}\in (0,{\varepsilon}_0]} \bigg( {\varepsilon}^{1-\beta} \max\bigg\{\frac{1}{2},\bigg(\frac{2Cs{\varepsilon}}{\pi}\bigg)^{1/b} \bigg\}^{1/\bar{\gamma}+\alpha-b}\\&\qquad+{\varepsilon}^{1+\alpha-\beta} \max\bigg\{\frac{1}{2},\bigg(\frac{2Cs{\varepsilon}}{\pi}\bigg)^{1/b} \bigg\}^{1/\bar{\gamma}-b}\bigg)\\&\quad\lesssim |s|\sup_{{\varepsilon}\in (0,{\varepsilon}_0]} (\max\{{\varepsilon}^{1-\beta},|s|^{1/(\bar{\gamma} b)+\alpha/b-1}{\varepsilon}^{1/(\bar{\gamma} b)+\alpha/b-\beta}\}\\&\qquad+\max\{{\varepsilon}^{1+\alpha-\beta},|s|^{1/(\bar{\gamma} b)-1}{\varepsilon}^{1/(\bar{\gamma} b)+\alpha-\beta}\} ) \\&\quad\leq \tilde{C}_{{\varepsilon}_0}|s|(1+|s|^{1/(\bar{\gamma} b)+\alpha/b-1}+|s|^{1/(\bar{\gamma} b)-1}) \end{align*} $$

for some constant $\tilde {C}_{{\varepsilon }_0}>0$ . This follows from the assumption that $1/(\bar {\gamma } b)+\alpha /b-\beta>0$ and $1/(\bar {\gamma } b)+\alpha -\beta>0$ .

Finally, combining this with the first step and using symmetry, we have that

$$ \begin{align*} &\sup_{s\in [0,2\pi]}\sup_{{\varepsilon}\in (0,{\varepsilon}_0]}\|R_{\alpha}\operatorname*{\mathrm{osc}}(e^{i s\chi}, B_{{\varepsilon}}(\cdot))\|_{\bar{\gamma}}\cdot {\varepsilon}^{-\beta}\\&\quad\leq \tilde{C}_{{\varepsilon}_0} \sup_{s\in [0,2\pi]}(1+|s|+|s|^{1/(\bar{\gamma} b)+\alpha/b}+|s|^{1/(\bar{\gamma} b)}+|s|^{(1+\alpha)/b}) \\&\quad\leq C_{{\varepsilon}_0} \end{align*} $$

for some $C_{{\varepsilon }_0}>0$ that is independent of $\bar {\gamma }\geq 1$ .

Proof. First, we notice that for all j,

(4.7) $$ \begin{align} \bar R_{j+1}(x) &=\frac{\psi_{j+1}(x)^\alpha(1-\psi_{j+1}(x))^\alpha}{x^\alpha(1-x)^\alpha} \leq \max\bigg\{\frac{(\psi_{j+1}(x)-0)^\alpha}{x^\alpha}, \frac{(1-\psi_{j+1}(x))^\alpha}{(1-x)^\alpha}\bigg\}\notag\\ &\leq \max\bigg\{\frac{((x-0)\eta_+)^{\alpha}}{x^{\alpha}},\frac{((1-x)\eta_+)^{\alpha}}{(1-x)^{\alpha}}\bigg\} \leq \eta_+^\alpha, \end{align} $$

where the first inequality holds true, because at most one of the arguments in the maximum can be larger than $1$ . Hence, for all j, $\bar R_{j+1}$ is bounded.

We know from part (1) in the proof of Lemma B.1 that $\bar {R}_{1}'$ is bounded at $0$ and $\bar {R}_{k-1}'$ is bounded at $1$ . We can infer from the representation in equation (B.2) that there exist $K_3',K_3>0$ such that

(4.8) $$ \begin{align} |\bar{R}^{\prime}_{j+1}(x)|\leq \frac{K_3'}{(\psi_{j+1}(x) (1-\psi_{j+1}(x)))^{1-\alpha}} \leq \frac{K_3}{((x-c_j) (c_{j+1}-x))^{1-\alpha}} \end{align} $$

for all $j\in \{1,\ldots ,k-2\}$ . This can be deduced as follows: We assume we are in the interval $[\delta _0, 1-\delta _0]$ with $\delta _0$ as in part (1) of the proof of Lemma B.1. Then, the subtrahend of equation (B.2) has to be bounded as it only has a pole at $0$ and $1$ . Furthermore, considering the minuend, it is easy to notice that the factor $\alpha \psi _{j+1}^{\prime }(x)(1-2\psi _{j+1}(x))/(x(1-x))^{\alpha }$ has to be bounded on $[\delta _0, 1-\delta _0]$ as well. This leaves the remaining factor as in the middle term of equation (4.8).

To verify the second inequality, we notice that $\psi _{j+1}(x)\in [\eta _-(x-c_j), \eta _+(x-c_j)]$ , which follows from the fact that $\lim _{{\varepsilon }\to 0}\psi _{j+1}(c_j+{\varepsilon })=0$ and from the bound on the derivative. With a similar argumentation, using that $\lim _{{\varepsilon }\to 0}\psi _{j+1}(c_{j+1}-{\varepsilon })=1$ , we obtain $1-\psi _{j+1}(x)\in [\eta _-(c_{j+1}-x), \eta _+(c_{j+1}-x)]$ .

In addition, from the proof of Lemma B.1,

$$ \begin{align*} |\bar{R}^{\prime}_{1}(x)|\leq \frac{K_3}{(c_{1}-x)^{1-\alpha}} \quad \text{and} \quad \bar{R}^{\prime}_{k}(x)\leq \frac{K_3}{(x-c_{k-1})^{1-\alpha}}. \end{align*} $$

Hence,

(4.9) $$ \begin{align} &\sup_{x \in [c_j+\delta+{\varepsilon}, c_{j+1}-\delta-{\varepsilon}]}{\Big(} (R_\alpha \mathbf{ 1})(x)\sup_{B_{{\varepsilon}}(x)}|\bar R^{\prime}_{j+1}| \Big)\nonumber\\&\quad\lesssim \begin{cases} \sup\dfrac{1}{[(x\pm \delta - c_j)(c_{j+1}-x\pm \delta)]^{1-\alpha}}, & j \notin\{ 0, k-1\},\\\sup\dfrac{1}{(c_{1}-x\pm \delta)^{1-\alpha}}, & j = 0, \\\sup\dfrac{1}{(x\pm \delta- c_{k-1})^{1-\alpha}}, & j = k-1, \end{cases} \,\\\nonumber &\quad\lesssim \delta^{\alpha-1}.\\[-37pt] \nonumber \end{align} $$

Proof of Lemma 4.6

Let $s\in \mathbb {R}$ and $h \in {{\mathsf V}_{\alpha ,\beta ,\gamma }}$ be $\mathbb {R}$ -valued. We estimate $|{\widehat {\psi }}_{is} h|_{\alpha ,\beta }$ :

$$ \begin{align*} \operatorname*{\mathrm{osc}}\big(R_\alpha ({\widehat{\psi}}_{is} h), B_{\varepsilon}(x)\big) &= \operatorname*{\mathrm{osc}}\bigg( R_\alpha \sum_{j=0}^{k-1} \bigg(\frac{e^{is\chi}\cdot h }{|\psi'|} \bigg)\circ \psi^{-1}_{j+1}\mathbf{ 1}_{\psi[c_j,c_{j+1}]}, B_{\varepsilon}(x) \bigg) \\&\leq \sum_{j=0}^{k-1} \operatorname*{\mathrm{osc}}\bigg( R_\alpha \bigg(\frac{e^{is\chi}\cdot h }{|\psi'|} \bigg)\circ \psi^{-1}_{j+1} , B_{\varepsilon}(x) \bigg) \\&\leq \sum_{j=0}^{k-1} \operatorname*{\mathrm{osc}}\bigg( \frac{ R_\alpha \mathbf{1}\circ \psi_{j+1} }{ R_\alpha \mathbf{1}} \cdot R_\alpha\frac{e^{is\chi } \cdot h }{|\psi'|}, \psi^{-1}_{j+1} B_{{\varepsilon}}(x) \cap [c_j,c_{j+1}]\bigg) \\&\leq \sum_{j=0}^{k-1} \operatorname*{\mathrm{osc}}\bigg(\! \frac{ R_\alpha \mathbf{1}\circ \psi_{j+1} }{ R_\alpha \mathbf{1}} \cdot R_\alpha\frac{e^{is\chi } \cdot h }{|\psi'|}, B_{{\varepsilon}/\eta_{-}}(\psi^{-1}_{j+1}x) \cap [c_j,c_{j+1}]\!\bigg) \\&= \sum_{j=0}^{k-1} \operatorname*{\mathrm{osc}}\bigg( \bar{R}_{j+1}\cdot \frac{e^{is\chi}}{|\psi'|} \cdot R_\alpha h, D_{j+1}(x,{\varepsilon}/\eta_{-})\bigg), \end{align*} $$

where $D_{j+1}(x,{\varepsilon }):=B_{{\varepsilon }}(\psi ^{-1}_{j+1}x) \cap [c_j,c_{j+1}]$ . So, by [Reference Saussol35, Proposition 3.2 (iii)], there exists $c>0$ such that

$$ \begin{align*} &\operatorname*{\mathrm{osc}}(R_\alpha ({\widehat{\psi}}_{is} h), B_{\varepsilon}(x))\\&\quad\leq \sum_{j=0}^{k-1} \operatorname*{\mathrm{osc}}( R_\alpha h, D_{j+1}(x,{\varepsilon}/\eta_{-})) \sup_{D_{j+1}(x,{\varepsilon}/\eta_{-})} \bigg|\bar{R}_{j+1}\cdot \frac{e^{is\chi}}{|\psi'|}\bigg| \\&\qquad+\sum_{j=0}^{k-1} \operatorname*{\mathrm{osc}}\bigg( \bigg|\bar{R}_{j+1}\cdot \frac{e^{is\chi}}{|\psi'|}\bigg| ,D_{j+1}(x,{\varepsilon}/\eta_{-})\bigg) \inf_{D_{j+1}(x,{\varepsilon}/\eta_{-})} |R_\alpha h|\\&\quad\leq (1+c({\varepsilon}\eta_{-}^{-1})^{\vartheta}) \sum_{j=0}^{k-1} \frac{\operatorname*{\mathrm{osc}}( R_\alpha h, B_{{\varepsilon}/\eta_{-}}(\psi^{-1}_{j+1}x))}{|\psi'|(\psi^{-1}_{j+1}x)} \sup_{D_{j+1}(x,{\varepsilon}/\eta_{-})} |\bar{R}_{j+1}| \\&\qquad+\sum_{j=0}^{k-1} \operatorname*{\mathrm{osc}}\bigg( \bigg|\bar{R}_{j+1}\cdot \frac{e^{is\chi}}{|\psi'|}\bigg|, D_{j+1}(x,{\varepsilon}/\eta_{-})\bigg) |R_\alpha h(\psi^{-1}_{j+1}x)|. \end{align*} $$

The last inequality follows from the fact that $\psi ^{-1}$ is $C^1$ and its derivative is uniformly ${\vartheta }$ -Hölder.

Hence, using the upper bound in equation (4.7) and then using the definition of the transfer operator $\widehat {\psi }$ , we have

(4.11) $$ \begin{align} \operatorname*{\mathrm{osc}}(R_\alpha ({\widehat{\psi}}_{is} h), B_{\varepsilon}(x)) &\leq (1+c({\varepsilon}\eta_{-}^{-1})^{\vartheta}){\eta^{\alpha}_{+}}\widehat{\psi}(\operatorname*{\mathrm{osc}}(R_\alpha h, B_{{\varepsilon}/\eta_{-}}(\,\cdot\,)))(x) \nonumber\\&\quad+\sum_{j=0}^{k-1} |R_\alpha h(\psi^{-1}_{j+1}x)| \operatorname*{\mathrm{osc}}\bigg( \bigg|\bar{R}_{j+1}\cdot \frac{e^{is\chi}}{|\psi'|}\bigg|, D_{j+1}(x,{\varepsilon}/\eta_{-})\bigg). \end{align} $$

Taking the integral over the first term in equation (4.11) and multiplying by ${\varepsilon }^{-\beta }$ , we obtain

(4.12) $$ \begin{align} {\varepsilon}^{-\beta}\int (1+c({\varepsilon}\eta_{-}^{-1})^{\vartheta}){\eta}^{ \alpha}_{+}\widehat{\psi}(\operatorname*{\mathrm{osc}}(R_\alpha h, B_{{\varepsilon}/\eta_{-}}(\,\cdot\,)))(x)\,{d}\unicode{x3bb}_I(x)\notag\\&\leq {\varepsilon}^{-\beta}(1+c({\varepsilon}\eta_{-}^{-1})^{\vartheta}){\eta}^{\alpha}_{+}\int \widehat{\psi}(\operatorname*{\mathrm{osc}}(R_\alpha h, B_{{\varepsilon} /\eta_{-}}(\,\cdot\,)))(x)\,{d}\unicode{x3bb}_I(x)\notag\\&= {\varepsilon}^{-\beta}(1+c({\varepsilon}\eta_{-}^{-1})^{\vartheta}){\eta}^{\alpha}_{+} \int \operatorname*{\mathrm{osc}}(R_\alpha h, B_{{\varepsilon}/\eta_{-}}(\,\cdot\,))(x)\,{d}\unicode{x3bb}_I(x)\notag\\&\leq(1+c({\varepsilon}\eta_{-}^{-1})^{\vartheta})\eta^{\alpha}_{+}\eta_-^{-\beta}|h|_{\alpha,\beta}\notag\\&\leq (1+c({\varepsilon}_0\eta_{-}^{-1})^{\vartheta})\kappa\|h\|_{\alpha,\beta,\gamma} \end{align} $$

for all $\gamma \geq 1$ . Next, we analyse the second term in equation (4.11). Again, by [Reference Saussol35, Proposition 3.2 (iii)], we have

(4.13) $$ \begin{align} &\operatorname*{\mathrm{osc}}\bigg( \bigg|\bar{R}_{j+1}\cdot \frac{e^{is\chi}}{|\psi'|}\bigg|, D_{j+1}(x,{\varepsilon}/\eta_{-})\bigg)\nonumber \\ &\quad\leq \operatorname*{\mathrm{osc}}\bigg( \frac{1}{|\psi'|}, B_{{\varepsilon}/\eta_{-}}(\psi^{-1}_{j+1}x)\bigg)\Big(\operatorname*{\mbox{ess sup}}_{B_{{\varepsilon}/\eta_{-}}(\psi^{-1}_{j+1}x)} |\Re\bar{R}_{j+1}e^{is\chi}|+ \operatorname*{\mbox{ess sup}}_{B_{{\varepsilon}/\eta_{-}}(\psi^{-1}_{j+1}x)} |\Im\bar{R}_{j+1}e^{is\chi}|\Big) \nonumber\\ &\qquad+\operatorname*{\mathrm{osc}}( \bar{R}_{j+1}e^{is\chi}, D_{j+1}(x,{\varepsilon}/\eta_{-}))\inf_{D_{j+1}(x,{\varepsilon}/\eta_{-})} \frac{1}{|\psi'|} \nonumber\\ &\quad\leq c({\varepsilon}\eta_{-}^{-1})^{\vartheta}\eta^{\alpha}_{+} \frac{1}{|\psi'|(\psi^{-1}_{j+1}x)}+(1+c({\varepsilon}\eta_{-}^{-1})^{\vartheta})\frac{\operatorname*{\mathrm{osc}}(\bar{R}_{j+1}e^{is\chi},D_{j+1}(x,{\varepsilon}/\eta_{-}))}{|\psi'|(\psi^{-1}_{j+1}x)}. \end{align} $$

Note that

(4.14) $$ \begin{align} {\varepsilon}^{-\beta}c({\varepsilon}\eta_{-}^{-1})^\eta\eta^{\alpha}_{+} \int \sum_{j=0}^{k-1} \frac{|R_\alpha h(\psi^{-1}_{j+1}x)|}{|\psi'|(\psi^{-1}_{j+1}x)}\, {d}\unicode{x3bb}_I(x) \nonumber &={\varepsilon}^{-\beta} c({\varepsilon}\eta_{-}^{-1})^{\vartheta}\eta^{\alpha}_{+} \int\widehat{\psi}(|R_\alpha h|)\, {d}\unicode{x3bb}_I(x) \\ &={\varepsilon}^{-\beta} c({\varepsilon}\eta_{-}^{-1})^{\vartheta}\eta^{\alpha}_{+} \int |R_\alpha h|\, {d}\unicode{x3bb}_I(x) \nonumber \\ &\leq K_1 {\varepsilon}^{{\vartheta} - \beta}\|R_\alpha\mathbf{ 1}\|_{\gamma_1}\|h\|_{\bar{\gamma}}, \end{align} $$

where $\gamma ^{-1}_1+\bar {\gamma }^{-1}=1$ , $K_1:= c\eta _{-}^{-{\vartheta }}\eta ^{\alpha }_{+}\|R_\alpha \mathbf { 1}\|_{\bar {\gamma }}$ and $\beta < {\vartheta }$ . So, the contribution from the first summand of equations (4.13)–(4.11) is under control.

To estimate the contribution from the second summand of equations (4.13)–(4.11), we note that for all j and for all $A \subset [c_j,c_{j+1}]$ , we have

$$ \begin{align*}\operatorname*{\mathrm{osc}}( \bar{R}_{j+1}e^{is\chi},A )= \operatorname*{\mathrm{osc}}\bigg( \sum_{j=0}^{k-1}\bar{R}_{j+1}e^{is\chi}\mathbf{ 1}_{[c_j,c_{j+1})},A\bigg), \end{align*} $$

and therefore, we can bound this contribution by

(4.15) $$ \begin{align} &(1+c({\varepsilon}\eta_{-}^{-1})^{\vartheta})\sum_{j=0}^{k-1} \frac{|R_\alpha h(\psi^{-1}_{j+1}x)|}{|\psi'|(\psi^{-1}_{j+1}x)}\operatorname*{\mathrm{osc}}( F,\, B_{{\varepsilon}/\eta_{-}}(\psi^{-1}_{j+1}x) )\nonumber \\ &\quad=(1+c({\varepsilon}\eta_{-}^{-1})^{\vartheta}) \widehat{\psi} (|R_\alpha h| \operatorname*{\mathrm{osc}}( F,\, B_{{\varepsilon}/\eta_{-}}(\,\cdot\,))), \end{align} $$

where

$$ \begin{align*}F(x) = e^{is\chi(x)} \sum_{j=0}^{k-1} \bar R_{j+1}(x) \mathbf{ 1}_{[c_j,c_{j+1})}(x)=e^{is\chi(x)} \sum_{j=0}^{k-1} \frac{R_{\alpha } \mathbf{1} \circ \psi_{j+1}(x)}{R_{\alpha } \mathbf{1} (x)} \mathbf{1}_{[c_j,c_{j+1})}(x).\end{align*} $$

This is bounded by

(4.16) $$ \begin{align} & (1+c({\varepsilon}\eta_{-}^{-1})^{\vartheta}) \int \widehat{\psi} (|R_\alpha h| \operatorname*{\mathrm{osc}}( F,\, B_{{\varepsilon}/\eta_{-}}(\,\cdot\,)))(x)\, {d}\unicode{x3bb}_I(x) \nonumber \\ &\quad= (1+c({\varepsilon}\eta_{-}^{-1})^{\vartheta})\int|R_\alpha h|(x) \operatorname*{\mathrm{osc}}( F,\, B_{{\varepsilon}/\eta_{-}}(x))\, {d}\unicode{x3bb}_I(x) \nonumber\\ &\quad= (1+c({\varepsilon}\eta_{-}^{-1})^{\vartheta})\int|h (x)| \cdot (R_\alpha\operatorname*{\mathrm{osc}}( F,\, B_{{\varepsilon}/\eta_{-}}(x)))\, {d}\unicode{x3bb}_I(x). \end{align} $$

To estimate the integral, we split it as follows.

$$ \begin{align*} &\int|h (x)| \cdot (R_\alpha\operatorname*{\mathrm{osc}}( F,\, B_{{\varepsilon}/\eta_{-}}(x)))\, {d}\unicode{x3bb}_I(x) \\ &\quad= \bigg( \sum_{ j=1}^{k-1} \int_{c_j-{\varepsilon}^\iota-{\varepsilon}}^{ c_j+{\varepsilon}^\iota+{\varepsilon}} + \sum_{ j=1}^{k} \int_{ c_{j-1}+{\varepsilon}^\iota+{\varepsilon}}^{c_{j}-{\varepsilon}-{\varepsilon}^\iota} + \int_{0}^{ {\varepsilon}+{\varepsilon}^\iota } + \int_{ 1-{\varepsilon}-{\varepsilon}^\iota}^{1}\bigg) |h (x)|\\ &\qquad\cdot (R_\alpha\operatorname*{\mathrm{osc}}( F,\, B_{{\varepsilon}/\eta_{-}}(x)))\, {d}\unicode{x3bb}_I(x), \end{align*} $$

where we choose for $\iota $ any number fulfilling

(4.17) $$ \begin{align} \frac{\beta}{1-\alpha}<\iota< \frac{1-\beta}{1-\alpha}. \end{align} $$

Because $\beta <1/2$ , such a choice is possible. Note that for $j=1,\ldots k-1$ , $x \in { [c_j-{\varepsilon }^\iota -{\varepsilon },c_j+{\varepsilon }^\iota +{\varepsilon }]},$

$$ \begin{align*} \operatorname*{\mathrm{osc}}( F,\, B_{{\varepsilon}/\eta_{-}}(x)) \leq 2 (\sup \bar R_{j}+\sup \bar R_{j+1}) \leq 4K \end{align*} $$

and for $x \in { [0,{\varepsilon }+{\varepsilon }^\iota ) \cup (1-{\varepsilon }-{\varepsilon }^\iota , 1]},$

$$ \begin{align*} \operatorname*{\mathrm{osc}}( F,\, B_{{\varepsilon}/\eta_{-}}(x)) \leq 2 (\sup \bar R_{0}+\sup \bar R_{ k})\, \leq 4K, \end{align*} $$

where $K:= \sup _j \sup R_{j+1} < \infty $ . So,

(4.18) $$ \begin{align} &\sum_{j=0}^{k} \int_{ (c_j-{\varepsilon}^\iota-{\varepsilon}) \vee 0}^{ (c_j+{\varepsilon}^\iota+{\varepsilon}) \wedge 1 } |h (x)| \cdot (R_\alpha\operatorname*{\mathrm{osc}}( F,\, B_{{\varepsilon}/\eta_{-}}(x)))\, {d}\unicode{x3bb}_I(x) \notag\\ &\quad\leq \| h\|_{\bar{\gamma}}\sum_{j=0}^{k} \bigg(\int_{ (c_j-{\varepsilon}^\iota-{\varepsilon}) \vee 0}^{(c_j+{\varepsilon}^\iota+{\varepsilon}) \wedge 1 } (R_\alpha\operatorname*{\mathrm{osc}}( F,\, B_{{\varepsilon}/\eta_{-}}(x)))^{\gamma_1}\, {d}\unicode{x3bb}_I(x)\bigg)^{1/\gamma_1}\notag \\ &\quad\leq K_{\alpha}{\varepsilon}^{\iota/\gamma_1}\|h\|_{\bar{\gamma}}, \end{align} $$

where $\gamma ^{-1}_1+\bar {\gamma }^{-1}=1$ and $K_{\alpha }= 4^{\iota /\gamma _1}2^{-2\alpha }K$ . Here, we choose $\bar {\gamma }$ such that

(4.19) $$ \begin{align} \max\bigg\{\gamma,\, \frac{\iota}{\iota-\beta},\, \frac{1}{1-b\beta+\alpha},\frac{1}{1-b(\beta-\alpha)} \bigg\}<\bar{\gamma}<\frac{1}{\alpha}. \end{align} $$

We will see later in the proof why these restrictions on $\bar {\gamma }$ are needed.

Now, we show that such a choice is possible. Since we were assuming that $\iota>\beta /(1-\alpha )$ , we have $\iota /(\iota -\beta )<1/\alpha $ . We note that when $b \leq 1$ , $1-b\beta +\alpha \geq 1-b(\beta -\alpha )$ , and it is enough to see whether $\alpha < 1-b(\beta -\alpha )$ . In fact, this is true because $b(\beta -\alpha )< \beta -\alpha < 1 - \alpha $ . In contrast, when $b> 1$ , we have $1-b\beta +\alpha < 1-b(\beta -\alpha )$ , and it is enough to see whether $\alpha < 1-b\beta +\alpha $ . This is true because $\beta < 1/b$ .

To estimate the remaining terms we note, using equation (4.7) and [Reference Saussol35, Proposition 3.2(iii)], that for all $j=0,\ldots , k-1,$ for all $x \in [c_j+{\varepsilon }^\iota +{\varepsilon }, c_{j+1}-{\varepsilon }^\iota -{\varepsilon }]$ ,

$$ \begin{align*} \operatorname*{\mathrm{osc}}( F,\, B_{{\varepsilon}/\eta_{-}}(x))\\ &= \operatorname*{\mathrm{osc}}( e^{is\chi}\bar R_{j+1},\, B_{{\varepsilon}/\eta_{-}}(x))\\ &\leq \sup_{B_{{\varepsilon}}(x)} (\Re |e^{is\chi}|+\Im |e^{is\chi}|) \operatorname*{\mathrm{osc}}(\bar R_{j+1},\, B_{{\varepsilon}/\eta_{-}}(x))\\ &\quad+ \operatorname*{\mathrm{osc}}( e^{is\chi},\, B_{{\varepsilon}/\eta_{-}}(x)) \inf_{B_{{\varepsilon}}(x)} \bar R_{j+1}\\ &\leq 2 \sup_{B_{{\varepsilon}}(x)}|\bar R^{\prime}_{j+1}|\, \frac{{\varepsilon}}{\eta_-} + \operatorname*{\mathrm{osc}}( e^{is\chi},\, B_{{\varepsilon}/\eta_{-}}(x))\, \eta_+^{\alpha}, \end{align*} $$

and thus,

$$ \begin{align*} &\sum_{j=0}^{k-1}\int_{c_j+{\varepsilon}^\iota+{\varepsilon}}^{c_{j+1}-{\varepsilon}^\iota-{\varepsilon}} |h (x)| \cdot (R_\alpha\operatorname*{\mathrm{osc}}( F,\, B_{{\varepsilon}/\eta_{-}}(x)))\, {d}\unicode{x3bb}_I(x) \nonumber \\& \quad\leq \frac{2{\varepsilon}}{\eta_-}\bigg\||h| \sum_{j=0}^{k-1}\mathbf{ 1}_{[c_j+{\varepsilon}^\iota+{\varepsilon},c_{j+1}-{\varepsilon}^\iota-{\varepsilon}]}\bigg\|_{1} \sup_{x \in { [c_j+{\varepsilon}^\iota+{\varepsilon}, c_{j+1}-{\varepsilon}^\iota-{\varepsilon}]}}R_{\alpha}\sup_{B_{{\varepsilon}}(x)}|\bar R^{\prime}_{j+1}|\\&\qquad+\eta_+^{\alpha}\bigg\| \sum_{j=0}^{k-1}\mathbf{1}_{[c_j+{\varepsilon}^\iota+{\varepsilon}, c_{j+1}-{\varepsilon}-{\varepsilon}^\iota]}\cdot |h| (R_\alpha \operatorname*{\mathrm{osc}}( e^{is\chi},\, B_{{\varepsilon}/\eta_{-}}(\cdot)))\bigg\|_{1}\\&\quad\leq \frac{2{\varepsilon}}{\eta_-}\|h\|_{1} \sup_{x \in { [c_j+{\varepsilon}^\iota+{\varepsilon}, c_{j+1}-{\varepsilon}^\iota-{\varepsilon}]}}R_{\alpha}\sup_{B_{{\varepsilon}}(x)}|\bar R^{\prime}_{j+1}|+\eta_+^{\alpha}\|h\|_{\bar{\gamma}}\|R_\alpha \operatorname*{\mathrm{osc}}( e^{is\chi},\, B_{{\varepsilon}/\eta_{-}}(\cdot))\|_{\bar{\gamma}}. \end{align*} $$

Now, to estimate the first summand taking the maximum over j of the supremum in equation (4.6) above with $\delta ={\varepsilon }^\iota $ yields that the outer supremum above is bounded by $C{\varepsilon }^{\iota (\alpha -1)}$ for some constant $C>0$ . For the second summand, from equation (4.19), we have that ${\bar {\gamma }}^{-1} < 1 - b\beta +\alpha $ , which implies that $b\beta < 1- {\bar {\gamma }}^{-1}+\alpha = \bar {\gamma }^{-1}+\alpha $ , and hence, when $b>1$ , we have the condition in equation (4.3). Also from equation (4.19), ${\bar {\gamma }}^{-1} < 1 - b(\beta -\alpha )$ , which implies that $b\beta < \bar {\gamma }^{-1}+b\alpha $ , and hence, when $b \leq 1$ , we have equation (4.3). Therefore, we can apply Lemma 4.3 with $\alpha $ , $\beta $ , b, $\bar {\gamma }$ , ${\varepsilon }/\eta _{-}$ to conclude

$$ \begin{align*}\|R_\alpha \operatorname*{\mathrm{osc}}( e^{is\chi},\, B_{{\varepsilon}/\eta_{-}}(\cdot))\|_{\bar{\gamma}} \leq C_{{\varepsilon}_0} {\varepsilon}^\beta \eta^{-\beta}_{-},\end{align*} $$

where $C_{{\varepsilon }_0}$ is independent of $\bar {\gamma }$ . Therefore, for all $s \neq 0$ ,

(4.20) $$ \begin{align} &\sum_{j=0}^{k-1} \int_{ c_j+{\varepsilon}^\iota+{\varepsilon}}^{ c_{j+1}-{\varepsilon}^\iota-{\varepsilon}} |h (x)| \cdot ({ R_{\alpha}}\operatorname*{\mathrm{osc}}( F,\, B_{{\varepsilon}/\eta_{-}}(x)))\, {d}\unicode{x3bb}_I(x)\notag\\ &\quad\leq \bar C_{{\varepsilon}_0} {\varepsilon}^{\min\{1-\iota(1-\alpha), \beta\}}\|h\|_{\bar{\gamma}}. \end{align} $$

Finally, combining equations (4.18) and (4.20), we estimate equation (4.16) multiplied by ${\varepsilon }^{-\beta }$ by

(4.21) $$ \begin{align} &{\varepsilon}^{-\beta}(1+c({\varepsilon}\eta_{-}^{-1})^{\vartheta}) \int \widehat{\psi} (|R_\alpha h| \operatorname*{\mathrm{osc}}( F,\, B_{{\varepsilon}/\eta_{-}}(\,\cdot\,)))(x)\, {d}\unicode{x3bb}_I(x) \nonumber\\ &\quad\leq {\varepsilon}^{ (\iota /\bar{\gamma}\wedge (1-\iota (1-\alpha))\wedge \beta)-\beta}C_{{\varepsilon}_0} \|h\|_{\bar{\gamma}} \leq C_{{\varepsilon}_0} \|h\|_{\bar{\gamma}}. \end{align} $$

To justify the last inequality, we analyse the exponent of ${\varepsilon }$ . By equation (4.19) and the relation $\gamma _1^{-1}+\bar {\gamma }^{-1}=1$ , we have $\iota /\gamma _1>\iota (1-{\bar {\gamma }}^{-1})>\iota (1-(\iota -\beta )/\iota ))=\beta $ . Furthermore, the second inequality of equation (4.17) implies that $1-\iota (1-\alpha )>\beta $ .

Combining equations (4.11), (4.12), (4.14) and (4.21), we have

$$ \begin{align*} | {\widehat{\psi}}_{is} h|_{\alpha, \beta}&=\sup_{{\varepsilon}\in (0,{\varepsilon}_0)}\int \frac{\operatorname*{\mathrm{osc}}(R_\alpha ({\widehat{\psi}}_{is} h), B_{\varepsilon}(x))}{{\varepsilon}^{\beta}}\,{d}\unicode{x3bb}_I(x)\\ &\leq (1+c({\varepsilon}_0\eta_{-}^{-1})^{\vartheta})\kappa\|h\|_{\alpha,\beta,\gamma}+C_{{\varepsilon}_0}\|h\|_{\bar{\gamma}} \end{align*} $$

for all $\gamma \geq 1$ . Therefore, for all $\bar {\gamma }$ chosen appropriately,

$$ \begin{align*} \| {\widehat{\psi}}_{is} h\|_{\alpha, \beta, \gamma} &= | {\widehat{\psi}}_{is} h|_{\alpha,\beta} + \|{\widehat{\psi}}_{is} h\|_{\gamma}\\ &\leq (1+c({\varepsilon}_0\eta_{-}^{-1})^{\vartheta})\kappa\|h\|_{\alpha,\beta,\gamma}+C_{{\varepsilon}_0}{\|h\|_{\bar{\gamma}}}+ C_\gamma \|h\|_\gamma\, \\ &\leq \bar\kappa\|h\|_{\alpha,\beta,\gamma}\bar C \|h\|_{\bar{\gamma}}, \end{align*} $$

where $\bar \kappa = (1+c({\varepsilon }_0\eta _{-}^{-1})^\eta )\kappa < 1$ for sufficiently small ${\varepsilon }_0$ , and $\bar C=C_{{\varepsilon }_0}+C_{\gamma }$ , where $C_{\gamma }$ is given in Lemma 4.1.

Iterating, we obtain the following DFLY inequality: for all $h\in {{\mathsf V}_{\alpha ,\beta ,\gamma }}$ ,

$$ \begin{align*} \sup_{s}\Vert {\widehat{\psi}}^n_{is} h\Vert_{\alpha,\beta,\gamma} &\leq \kappa\Vert {\widehat{\psi}}^{n-1}_{is}h\Vert_{\alpha,\beta,\gamma}+\bar C\| {\widehat{\psi}}^{n-1}_{is}h\|_{\bar{\gamma}}\,\\ &\leq \kappa^2\Vert {\widehat{\psi}}^{n-2}_{is}h\Vert_{\alpha,\beta,\gamma}+ \kappa \bar C\|{\widehat{\psi}}^{n-2}_{is}h\|_{\bar{\gamma}}+\bar C \bar C^{n-1}\| h\|_{\bar{\gamma}}\,\\ &\leq \kappa^n\Vert h\Vert_{\alpha,\beta,\gamma}+ \bar C \|h\|_{\bar{\gamma}} \sum_{j=0}^{n-1} \kappa^j \bar C^{n-1-j}\,\\ &\leq \kappa^n\Vert h\Vert_{\alpha,\beta,\gamma}+ C \bar C^{n+1} \|h\|_{\bar{\gamma}}\, \end{align*} $$

for some $C>0$ .

In the proof above, we assumed that h is $\mathbb {R}$ -valued. When $h=h_1+ih_2$ , where $h_j,\, j=1,2$ are $\mathbb {R}$ -valued, using linearity of the operator

$$ \begin{align*}\|{\widehat{\psi}}_{is}^n h\|_{\alpha,\beta,\gamma} \leq \|{\widehat{\psi}^n}_{is} h_1\|_{\alpha,\beta,\gamma}+ \|{\widehat{\psi}^n}_{is} h_2\|_{\alpha,\beta,\gamma},\end{align*} $$

and also, $\| h_j\|_{\alpha ,\beta ,\gamma } \leq \| h\|_{\alpha ,\beta ,\gamma }$ and $\|h_j\|_{\bar {\gamma }} \leq \|h\|_{\bar {\gamma }}$ for all $j=1,2$ . So, applying the DLFY inequality proven above in the $\mathbb {R}$ -valued case to $h_1$ and $h_2$ , we conclude that DFLY in the general case of h holds up to a constant multiple.

Proof of Theorem 2.5

From equation (2.7), we obtain that there exist $\alpha , \beta $ fulfilling

(5.1) $$ \begin{align} a&< \alpha < \beta \cdot \min\bigg\{1,\frac{\log\eta_-}{\log \eta_+}\bigg\}< \min\bigg\{\vartheta,\frac{1}{b},\frac{1}{2}\bigg\}\cdot \min\bigg\{1,\frac{\log\eta_-}{\log \eta_+}\bigg\}. \end{align} $$

Furthermore, since $\alpha>a$ , the inequality $\beta <1/b$ , which we can deduce immediately from equation (5.1) that ${1}/{b}<{1}/({b-a})$ . So, by Lemma A.18, we obtain $|\chi |_{\alpha ,\beta }<\infty $ and also $\chi \in {{\mathsf V}}_{\alpha ,\beta ,2} \hookrightarrow L^2$ . Furthermore, from the second inequality of equation (5.1), we obtain $\eta _+^{\alpha }/\eta _-^{\beta }< 1$ .

Since $\psi $ is a piecewise $C^2$ uniformly expanding and covering map of the interval, it has a unique absolutely continuous invariant mixing probability (acip) with a bounded invariant density; see [Reference Liverani27]. Let us call this acip $\pi $ . Then, $L^2 \hookrightarrow L^2(\pi )$ because

$$ \begin{align*} \int |h|^2\, {d}\pi = \int |h|^2 \frac{{d}\pi}{{d}\unicode{x3bb}_I} \, {d}\unicode{x3bb}_I \leq \bigg\|\frac{{d}\pi}{{d}\unicode{x3bb}_I}\bigg\|_{\infty} \int |h|^2\, {d}\unicode{x3bb}_I.\end{align*} $$

We claim that $\widehat {\psi }$ has a spectral gap in ${{\mathsf V}_{\alpha ,\beta ,\gamma }}$ with $\gamma =2$ . In Appendix A.2, we show that ${{\mathsf V}}_{\alpha ,\beta ,2}$ is continuously embedded in $L^{\bar \gamma }$ , where $\bar {\gamma } \in (2, 1/\alpha )$ , and that the unit ball of ${{\mathsf V}}_{\alpha ,\beta ,2}$ is relatively compact in $L^{\bar {\gamma }}$ . A suitable $\bar {\gamma }$ exists by the condition $\alpha <1/2$ from equation (5.1). So, the claim follows from [Reference Demers, Kiamari and Liverani7, Lemma B.15] due to the DFLY inequality in equation (4.10) with $s=0$ and Remark A.9.

Now, the CLT (in the stationary case) follows directly from Proposition 3.1 applied to $\chi - \pi (\chi )$ . That is, from equation (3.1), we have

$$ \begin{align*} {\mathbb{P}}_{\pi}\bigg(\frac{S_{n}(\chi) - n\, \pi(\chi)}{\sigma\sqrt{n}} \leq x\bigg) - \mathfrak{N}(x) = o(1)\quad\text{as}\; n\to \infty \end{align*} $$

with $\sigma ^2>0$ because $\chi $ is not a coboundary.

Proof of Theorem 2.8

Equation (2.10) implies that there exist $\alpha , \beta $ such that $\alpha>a$ and

$$ \begin{align*} 3\bar{\alpha}:&=3\min\{2\alpha, \max\{\alpha,\alpha+b-2\}\}\\ &<\beta\cdot\min\bigg\{1,\frac{\log\eta_+}{\log \eta_-}\bigg\}< \min\bigg\{\vartheta, \frac{1}{b},\frac{1}{2}\bigg\}\cdot\min\bigg\{1,\frac{\log\eta_+}{\log \eta_-}\bigg\}. \end{align*} $$

Since either $b<a+1$ or $1/b<(1+\alpha -a)/(b-a)$ , we obtain by Lemma A.18 that $|\chi |_{\alpha ,\beta }<\infty $ and additionally, we obtain by the last inequality that:

1. (a) $0< 3\bar \alpha < \beta < \min \{{\vartheta },1/b, 1/2\};$

2. (b) $\eta _+^{3\bar \alpha }<\eta _-^{\beta }$ .

Hence, under our assumptions, we have the following.

1. (1) The second inequality in equation (2.6) and $|\chi |_{\alpha , \beta } < \infty $ imply that $|e^{is\chi }|_{0,\beta }<\infty $ for all $s>0$ (see Remark A.13). So, due to Corollary 4.2(1), we have $\widehat {\psi }_{is} \in \mathcal {L}({{\mathsf V}}_{\tilde {\alpha },\beta ,\tilde {\gamma }})$ for all $0<\tilde {\alpha }< \beta $ and $\tilde {\gamma } \geq 1$ .

2. (2) Since $|\chi |_{\alpha ,\beta }<\infty $ , from Remark A.15, for all $\alpha ^{*}>0\,$ close to $0$ ,

   $$ \begin{align*}\lim_{s\to 0}|1-e^{is\chi}|_{\alpha^{*},\beta}=0.\end{align*} $$
   Along with Corollary 4.2(2), this yields that for all $0 \leq \alpha _0 < \beta $ , $\gamma _0 \geq 1$ ,
   $$ \begin{align*}s \mapsto \widehat{\psi}_{is} \in \mathcal{L}({{\mathsf V}}_{\alpha_0,\beta,\gamma_0}, {{\mathsf V}}_{\alpha_1,\beta,\gamma_1})\end{align*} $$
   is continuous for $\alpha _1=\alpha ^{*}+\alpha _0$ and $1 \leq \gamma _1\leq \gamma _0.$

3. (3) From the second inequality in equation (2.6) and $|\chi |_{\alpha , \beta } < \infty $ , for all $\alpha ^{**}>\min \{2\alpha , \max \{\alpha +b-2,\alpha \}\},$

   $$ \begin{align*}\lim_{s\to 0}\bigg|\frac{e^{is\chi}-1-is\chi}{s}\bigg|_{\alpha^{**}, \beta}=0\,\end{align*} $$
   due to Remark A.17. Then, we have that for all $0 \leq \alpha _1 <\beta $ and $\gamma _1 \geq 1,$
   $$ \begin{align*}s \mapsto \widehat{\psi}_{is} \in \mathcal{L}({{\mathsf V}}_{\alpha_1,\beta, \gamma_1},{{\mathsf V}}_{\alpha_2,\beta,\gamma_2} )\end{align*} $$
   is continuously differentiable for all ${\alpha _2 = \alpha ^{*}+\max \{\alpha ^{*},\alpha ^{**}\}+\alpha _1}$ and $1 \leq \gamma _2 \leq (\gamma ^{-1}_1+\gamma ^{-1})^{-1}$ due to Corollary 4.2(2) and (3).

Next, we define the following chain of spaces to invoke Proposition 3.3 with $r=1$ :

$$ \begin{align*} {{\mathsf V}}_{\alpha_0,\beta,\gamma_0 } &\hookrightarrow {{\mathsf V}}_{\alpha_1,\beta,\gamma_1} \hookrightarrow {{\mathsf V}}_{\alpha_2,\beta,\gamma_2} \hookrightarrow {{\mathsf V}}_{\alpha_3,\beta,\gamma_3} \hookrightarrow {{\mathsf V}}_{\alpha_4,\beta,\gamma_4}\\ &\quad\hookrightarrow{{\mathsf V}}_{\alpha_5,\beta,\gamma_5} \hookrightarrow{{\mathsf V}}_{\alpha_6,\beta,\gamma_6}\hookrightarrow{{\mathsf V}}_{\alpha_7,\beta,\gamma_7}, \end{align*} $$

where $\alpha _0=0$ , $\alpha _{2j} - \alpha _{2j-1}\geq \min \{2\alpha , \max \{\alpha + b - 2, \alpha \}\}$ for $j=1,2,3$ , $\alpha _{2j+1}> \alpha _{2j}$ for $j=0,1,2,3$ , and $\alpha _7<\beta $ . By assumption (a), such a choice is possible. Furthermore, we assume that the $\gamma _j$ values are chosen such that $\gamma _0= M \gg 1$ sufficiently large, ${\gamma _{2j+1} = \gamma _{2j}}$ and $\gamma _{2j}< (\alpha ^{-1}+\gamma _{2j-1}^{-1})^{-1}$ .

Now, to prove the theorem, we verify the conditions in Proposition 3.3 for the above sequence of Banach spaces. We notice that if for some observable $\varphi $ it holds that ${|\varphi |_{\alpha ,\beta }<\infty }$ , then $\|\varphi \|_{\alpha ,\beta ,\gamma }<\infty $ as long as $\gamma < 1/\alpha $ . We next verify that it is possible to construct valid spaces with the above choice of parameters. First, we notice that by assumption (a), it is possible to construct $\alpha _0\leq \cdots \leq \alpha _7$ with the above properties that $\alpha _7<\beta $ and thus $\alpha _j<\beta $ for all j. Furthermore, by assumption (a), we have $\alpha <1/3$ . Thus, it is possible that $1 \leq \gamma _{2j} \leq (\gamma ^{-1} + \gamma ^{-1}_{2j-1})$ holds together with $1/\gamma _j>\alpha _j$ . Moreover, under assumption (b), we have that $\eta _+^{\alpha _j}/\eta _-^{\beta }<1$ holds for all j.

With that, it becomes immediate from applying the conditions of this theorem on the parameters in the Banach spaces and from the calculations in parts (1)–(3) applied to all indices j that conditions (I)–(III) of Proposition 3.3 are satisfied.

For each j, we apply Lemma 4.6 with $\gamma =\gamma _j$ and we choose $\bar {\gamma } = \bar {\gamma }_j$ as in the proof of the lemma. In Appendix A.2, we show that ${{\mathsf V}}_{\alpha _j,\beta ,\gamma _j}$ is continuously embedded in $L^{\bar {\gamma }_j}$ and that the unit ball of ${{\mathsf V}}_{\alpha _j,\beta ,\gamma _j}$ is relatively compact in $L^{\bar {\gamma }_j}$ . Also, we recall from Lemma 4.1 that for all $h \in L^{\bar {\gamma }_j}$ , $\|\widehat {\psi }_{is} (h)\|_{\bar {\gamma }_j} \leq C_{\bar {\gamma }_j} \|h\|_{\bar {\gamma }_j}$ , where $ C_{\bar {\gamma }_j}>1$ . Therefore, $\|\widehat {\psi }^n_{is} (h)\|_{\bar {\gamma }_j} \leq C_{\bar {\gamma }_j} \|\widehat {\psi }^{n-1}_{is} (h)\| \leq C_{\bar {\gamma }_j}^n \|h\|_{\bar {\gamma }_j}$ , which gives us $\|\widehat {\psi }^n_{is}\|_{L^{\bar {\gamma }_j} \to L^{\bar {\gamma }_j}} \leq C_{\bar {\gamma }_j}^n.$ Choose $\kappa = \max \limits _{0 \leq j \leq 7} \eta ^{\alpha _j}_+ \eta ^{-\beta }_{-} <1$ . Also, by our previous constructions, we have that $\gamma _j < 1/\alpha _j$ for all j. So, due to Lemma 4.6, we have the DFLY inequality: for all $h\in {{\mathsf V}}_{\alpha _j,\beta ,\gamma _j}$ ,

$$ \begin{align*}\Vert {\widehat{\psi}^n}_{is} h\Vert_{\alpha_j,\beta,\gamma_j} \le \widetilde C (\kappa^n\Vert h\Vert_{\alpha_j,\beta,\gamma_j}+C^n\| h\|_{\bar{\gamma}_j})\, \end{align*} $$

for some $\gamma _j< \bar {\gamma }_j < 1/\alpha _j$ and C uniform in j and s. Therefore, we have the first conclusion, [Reference Keller and Liverani22, Theorem 1, equation (8)] uniformly over all spaces. That is, there exist v and w such that

$$ \begin{align*}\sup_{z \in D_{\kappa}}\|(z\mathsf{Id} - \widehat{\psi}_{is})^{-1} h\|_{{{\mathsf V}}_{\alpha_j,\beta,\gamma_j} \to {{\mathsf V}}_{\alpha_j,\beta,\gamma_j}} \leq v\|h\|_{\alpha_j,\beta, \gamma_j} + w\|h\|_{\bar{\gamma}_j}\end{align*} $$

for all space pairs ${{\mathsf V}}_{\alpha _j,\beta ,\gamma _j} \hookrightarrow L^{\bar {\gamma }_j}$ and $s \in \mathbb {R}$ . This gives condition (IV) of Proposition 3.3.

Conditions (V)–(VII) of Proposition 3.3 are equivalent to [Reference Fernando and Pène11, §I.1.2, Assumption (B)] for a single dynamical system, that is, when [Reference Fernando and Pène11, §I.1.2, Assumptions (0) and (A)(1)] are trivially true. Moreover, as discussed in [Reference Fernando and Pène11], [Reference Fernando and Pène11, Lemma 4.5] implies Assumption (B). Therefore, we verify the conditions (with a slight modification) in [Reference Fernando and Pène11, Lemma 4.5] to establish conditions (V)–(VII).

• • We have assumed that $\chi $ is non-arithmetic.

• • Due to Remark A.9 and the DFLY inequality in equation (4.10), we can apply [Reference Demers, Kiamari and Liverani7, Lemma B.15] to conclude that for all s, the essential spectral radius of $\widehat {\psi }_{is}$ on ${{\mathsf V}}_{\alpha _j,\beta ,\gamma _j}$ is at most $\kappa $ . This is precisely the conclusion of [Reference Fernando and Pène11, Proposition 4.3].

• • We know that ${{\mathsf V}}_{\alpha _j,\beta ,\gamma _j}\hookrightarrow L^1$ for all j, and that $\|\widehat {\psi }_{is} h\|_1 \leq \|\widehat {\psi } h \|_1 \leq \|h\|_1$ for all ${h\in L^1}$ . So, the spectral radius of $\widehat {\psi }_{is}$ on $L^1,$ and hence, on ${{\mathsf V}}_{\alpha _j,\beta ,\gamma _j}$ for all j, is at most $1$ .

• • Since $\psi $ is a uniformly expanding, piecewise $C^2$ and a full branch map with finitely many branches, $\psi $ is exact (cf. [Reference Hofbauer and Keller17, Theorem 3]) and $\psi ^{-1}x$ is finite for all x.

• • [Reference Fernando and Pène11, Assumption (A)(1)] is trivially true because there is only a single dynamical system in [Reference Fernando and Pène11, Figure 2].

Hence, conditions (V) and (VI) are true due to the first part of [Reference Fernando and Pène11, Lemma 4.5]. To establish condition (VII), we need a slight modification of the second part of [Reference Fernando and Pène11, Lemma 4.5]. First, we note that $\chi \in {{\mathsf V}_{\alpha ,\beta ,\gamma }} \hookrightarrow L^2$ for $\gamma \geq 3,$ and $\widehat {\psi }$ has a spectral gap on ${{\mathsf V}_{\alpha ,\beta ,\gamma }}$ . So, we can repeat the argument in the first part of the proof of [Reference Fernando and Pène11, Lemma 4.5] to conclude that $\sum ^{n-1}_{k=0}\bar \chi \circ \psi ^k$ is $L^2$ -bounded. So, it has an $L^2$ -weakly convergent subsequence. This establishes condition (VII).

Finally, the non-arithmeticity of $\chi $ implies that $\chi $ is not cohomologous to a constant, and hence, we have condition (VIII) of Proposition 3.3.

Proof of Theorem 2.6

To prove this theorem, we use Proposition 3.2. By Theorem 2.5, we immediately obtain condition (V) of Proposition 3.2.

Next, we define the following chain of spaces:

$$ \begin{align*} {{\mathsf V}}_{\alpha_0,\beta,M} &\hookrightarrow {{\mathsf V}}_{\alpha_1,\beta,M} \hookrightarrow L^p \hookrightarrow L^1(\pi) \end{align*} $$

with $p \leq M$ , where the choices correspond to $0\leq \alpha _0<\alpha _1<\beta $ and $\gamma _0=\gamma _1=M \geq 1$ in the proof of Theorem 2.8. Then, the conditions (I)–(IV) and (VI) of Proposition 3.2 follow as in the proof of Theorem 2.8 due to Corollary 4.2(2) and [Reference Fernando and Pène11, Lemma 4.5].

Proofs of the results in Example 2.10

We first note that

$$ \begin{align*} |\chi'(x)|\lesssim x^{-c}(1-x)^{-c} \end{align*} $$

and

$$ \begin{align*} |\chi'(x)| =\bigg|-cx^{-c-1}\sin\bigg(\frac{1}{x}\bigg)-x^{-c-2}\cos\bigg(\frac{1}{x}\bigg)\bigg| \lesssim x^{-c-2}(1-x)^{-c-2}. \end{align*} $$

So, we obtain $a=c$ and $b=c+2$ in the notation of Theorems 2.5, 2.6 and 2.8. To prove condition (1) we note that equation (2.7) then simplifies to

$$ \begin{align*} c<\min\bigg\{\vartheta, \frac{1}{2+c}\bigg\}\min\bigg\{1, \frac{\log\eta_-}{\log \eta_+}\bigg\}. \end{align*} $$

So, on the one hand, we have the requirement $c<\vartheta \tilde {\eta }$ and, on the other hand, we have the condition $c< \tilde {\eta }/(c+2)$ which, given that we assume $c\geq 0$ , is equivalent to $c<\sqrt {1+\tilde {\eta }}-1$ giving condition (1). Furthermore, in the doubling map case, we have $\vartheta =2$ and $\tilde {\eta }=1$ implying condition (1a).

Next, we notice that equation (2.10) in our case simplifies to

$$ \begin{align*} 3c<\min \bigg\{{\vartheta}, \frac{1}{2+c}\bigg\}\min\bigg\{1,\frac{\log\eta_+}{\log \eta_-}\bigg\}. \end{align*} $$

With a similar calculation as above, applying Theorem 2.8 gives condition (2) and, as above, we get condition (2a).

Proof. It is enough to show that ${\mathbb {E}}_{\mu }(\widetilde S^{r}_n(h)) = {\mathbb {E}}_{\unicode{x3bb} _I}(S^{r}_n(\chi ))$ for all r. In fact, due to equation (2.13),

$$ \begin{align*} {\mathbb{E}}_{\mu}( h \circ \phi^{j_1}\, h \circ \phi^{j_2}\, \cdots \, h \circ \phi^{j_k}) &= {\mathbb{E}}_{\unicode{x3bb}_I}( h \circ \xi \circ \psi^{j_1}\, h \circ \xi \circ \psi^{j_2}\, \cdots \, h \circ \xi \circ \psi^{j_k}) \\ &={\mathbb{E}}_{\unicode{x3bb}_I}( \chi \circ \psi^{j_1}\, \chi \circ \psi^{j_2}\, \ldots \, \chi \circ \psi^{j_k}) \end{align*} $$

for all $j_1, \ldots , j_k \in \mathbb {N}_0$ such that $j_1+\cdots +j_k={r}$ .

Proof. We will apply Lemma A.18. First, we note that

$$ \begin{align*}\lim_{x\to 0}\xi(x)x=1/\pi\quad\text{and}\quad\lim_{x\to 1}\xi(x)(1-x)=1/\pi.\end{align*} $$

This and equation (2.14) imply

(5.3) $$ \begin{align} |\chi(x)|\lesssim x^{-u}(1-x)^{-u} \end{align} $$

and, in particular, $\chi \in L^\gamma $ with $1 \leq \gamma <1/u$ .

For simplicity, we assume $\chi $ is differentiable. Otherwise, at a point where $\chi $ is not differentiable, both one-sided derivatives will exist and the following estimates do hold for them.

Note that we have $|{ h}'(\xi (x))|\lesssim x^{-v} (1-x)^{-v}$ . Using the chain rule, $|\chi '(x) |= |{ h}'(\xi (x))||\xi '(x)|.$ Since $\xi '(x)=-\pi /\sin ^2(\pi x),$ we have that

(5.4) $$ \begin{align} |\chi'(x)|\lesssim x^{-2-v}(1-x)^{-2-v}. \end{align} $$

So, we have $ |\chi '(x)|\lesssim x^{-b}(1-x)^{-b}$ with $b=2+v> 2$ . The lemma then follows immediately by applying Lemma A.18.

Proof of Proposition 2.11

To prove the statement, it is enough to prove its counterpart for $S_n(\chi ,\psi )$ , where $\chi = h\circ \xi $ and $\psi $ is the doubling map.

From Lemma 5.2, we have

$$ \begin{align*} |\chi(x)|\lesssim x^{-u}(1-x)^{-u}\quad\!\text{and}\!\quad \max\{|\chi'(x+)|, |\chi'(x-)|\}\lesssim x^{-b}(1-x)^{-b},\quad b=2+v.\end{align*} $$

Now, we invoke Theorem 2.5 with $\psi $ , $\eta _+=\eta _-=2$ and $\log \eta _-/\log \eta _+ =1$ . Since $\psi $ is linear, $\vartheta =1$ . Hence, equation (2.7) simplifies to $u<1/(2+v)$ . Also, the assumption that h is not an $L^2(\mu )$ coboundary implies that $\chi $ is not an $L^2(\unicode{x3bb} )$ coboundary.

Therefore, $\chi $ and $\psi $ satisfy the conditions of Theorem 2.5 and, hence, satisfy the CLT given by equation (2.8) with

$$ \begin{align*}\sigma^2 ={\mathbb{E}}_{\unicode{x3bb}}(\chi^2)+ 2\sum_{k=1}^\infty {\mathbb{E}}_\unicode{x3bb}(\chi\cdot \chi\circ {\psi}^k) \in (0,\infty).\end{align*} $$

From Lemma 5.1, $\widetilde {\sigma }^2 = {\sigma }^2 $ and ${\mathbb {E}}_{\mu }(h)={\mathbb {E}}_{\unicode{x3bb} _I}(\chi )$ . As a direct consequence of equation (2.13), we obtain the required CLT given by equation (2.15).

Proof of Proposition 2.12

Our assumption allows us to apply Theorem 2.6 to the Birkhoff sum $S_n(\chi )=\sum _{k=0}^{n-1}\chi \circ \psi ^k$ with $\chi = h\circ \xi $ and $\psi $ the doubling map, and conclude

$$ \begin{align*} &\sup_{\ell \in \mathbb{R} }\bigg|\sigma\sqrt{2\pi n}\,{\mathbb{E}}_{\xi_* {\overline{m}}}(U \circ \xi \circ \psi^n \,V(S_n(\overline\chi)-\ell)\, W \circ \xi)\\&\quad\qquad- e^{-{\ell^2}/{2n\sigma^2}} \,{\mathbb{E}}_\pi(U\circ \xi)\,{\mathbb{E}}_{\xi_* {\overline{m}} }(W \circ \xi )\int V(x) \, {d}x \bigg|= o(1). \end{align*} $$

From Lemma 5.1 and the fact that $\xi $ is a conjugacy, we have

$$ \begin{align*} &\sup_{\ell \in \mathbb{R} }\bigg|\widetilde\sigma\sqrt{2\pi n}\,{\mathbb{E}}_{{\overline{m}}}(U \circ \phi^n \,V(\widetilde S_n(\overline{h})-\ell)\, W )\\&\quad\qquad - e^{-{\ell^2}/{2n\widetilde \sigma^2}} \,{\mathbb{E}}_\mu(U)\,{\mathbb{E}}_{{\overline{m}}}(W )\int V(x) \, {d}x \bigg|= o(1). \end{align*} $$

This is because the two left-hand sides are exactly the same.

Proof of Corollary 2.15

To apply Proposition 2.11, we have to show the existence of $u,v$ as in equation (2.14). It is well known that for any $s \in (0,1)$ , for any $\delta> 0$ ,

(5.5) $$ \begin{align} \max\{|\zeta|(s+ix), |\zeta '|(s+ix)\}\lesssim |x|^{(1-s)/2 + \delta}; \end{align} $$

see, for example, [Reference Titchmarsh39].

So, we pick $u=v=(1-s)/2 + \delta $ and this is possible when $((1-s)/2 + \delta ) ((1-s)/ 2 + \delta +2) < 1$ and such $\delta>0$ exists if and only if $(1-s)(5-s)<4$ if and only if $s \in (3 - 2 \sqrt {2}, 1)$ . So, for such choices of s, we can apply Proposition 2.11 and obtain the CLT provided that h is not $\phi $ -cohomologous to a constant. The MLCLT follows from Proposition 2.12 analogously, when $\phi $ is non-arithmetic.

Proof of Corollary 2.17

To apply Proposition 2.11, we have to show the existence of $u,v$ as in equation (2.14). We assume $a \geq 1$ and set $\tilde {h}(x)=h(x)^{1/a}$ . Note that $h'(x) = a \tilde {h}(x)^{a-1} \tilde {h}'(x).$ Since we restrict ourselves to the critical line, $s=1/2$ , $|\tilde h(x) | \lesssim |x|^{13/84+\delta } $ and $|\tilde h'(x) | \lesssim |x|^{13/84+\delta }$ for all $\delta>0$ , due to equation (5.5). So, we can take $u=13a/84+\delta $ and $v= 13(a-1)/84 + 13/84+ \delta = 13a/84 + \delta ,$ and the condition in Proposition 2.11 for $u,v$ reduces to $(13a/84)(13a/84+2) < 1.$ This is equivalent to $1 \leq a < 84/13 (\sqrt {2}-1)$ . So, for such choices of a, we can apply Proposition 2.11 and obtain the CLT provided that h is not $\phi $ -cohomologous to a constant. The MLCLT follows from Proposition 2.12 analogously, when $\phi $ is non-arithmetic.

Proof of Proposition 2.13

We follow the proof of Proposition 2.11 and invoke Theorem 2.8.

Consider $S_n(\chi ,\psi )$ , where $\chi = \xi \circ h$ and $\psi $ is the doubling map. Remember that from Lemma 5.2, we have

$$ \begin{align*} |\chi(x)|\lesssim x^{-u}(1-x)^{-u}\quad\!\text{and}\!\quad \max\{|\chi'(x+)|, |\chi'(x-)|\}\lesssim x^{-b}(1-x)^{-b},\quad b=2+v. \end{align*} $$

Next, to apply Theorem 2.8, we observe that $\eta _+=\eta _-=2$ and $\log \eta _-/\log \eta ^+=1$ and since $\psi $ is linear $\vartheta =1$ . Hence, equation (2.10) simplifies to equation (2.17). Also, the assumption that h is not an $L^2(\mu )$ coboundary implies that $\chi $ is not an $L^2(\unicode{x3bb} )$ coboundary.