Posted by: matheuscmss | May 20, 2015

## Decay of correlations for flows and Dolgopyat’s estimate II.b

Last time, we reduced the proof of the exponential mixing property for expanding semiflows to the following Dolgopyat-like estimate:

Proposition 1 Let ${T}$ be an uniformly expanding Markov map on ${\Delta:=(0,1)}$ and let ${r:\Delta\rightarrow\mathbb{R}^+}$ be a good roof function with exponential tails.Then, there exist ${\sigma_0'>0}$, ${T_0\geq 1}$, ${\beta<1}$ and ${C>0}$ such that the iterates ${L_s^k}$ of the weighted transfer operator ${L_su(x):=\sum\limits_{T(y)=x} e^{-sr(y)}\frac{1}{|T'(y)|}u(y)}$ satisfy

$\displaystyle \|L_s^k u\|_{L^2}\leq C\beta^k\left(\|u\|_{C^0}+\frac{1}{\max\{1,|t|\}}\|Du\|_{C^0}\right):= C\beta^k \|u\|_{1,t}$

for all ${k\in\mathbb{N}}$ and ${s=\sigma+it}$ with ${|\sigma|\leq\sigma_0'}$, ${|t|\geq T_0}$.

Remark 1 We use the same terminology from the previous post of this series.

Roughly speaking, the basic idea behind the exponential contraction property in Proposition 1 is that “oscillations produce cancellations”. In particular, the analysis of the “size” of ${L_s^k u}$ is divided into two regimes:

• (A) If ${u}$ exhibits a high oscillation at scale ${\frac{1}{|t|}}$ (in the sense that ${\|Du\|_{C^0}\gg |t|\|u\|_{C^0}}$), then we will have a “cancelation” (significant reduction of the size of ${L_s^k u}$) thanks to classical methods (Lasota-Yorke inequality);
• (B) If the oscillation of ${u}$ at scale ${\frac{1}{|t|}}$ is not high, then we will have a “cancelation” thanks to Dolgopyat’s mechanism, i.e., a combination of high oscillations of Birkhoff sums ${r^{(n)}(x)}$ of the roof function ${r}$ (coming from the fact that ${r}$ is not a ${C^1}$-coboundary) and the big phases ${e^{-itr^{(n)}(y)}}$, ${|t|\geq T_0}$, of the terms ${e^{-sr^{(n)}(y)}}$ in the formula defining ${L_s^n u(x)}$.

In the remainder of this post, we will formalize this outline of proof of Proposition 1. More precisely, the next section contains a discussion of Lasota-Yorke inequality and the regime (A), and the last section is devoted to Dolgopyat’s cancelation mechanism and the regime (B).

1. Lasota-Yorke inequality and regime (A)

Before attacking Proposition 1, let us warm up with a digression on the spectral properties of the weighted transfer operators ${L_s}$.

The usual transfer operator ${L_0}$ acts on the space of ${C^1}$ functions. This action has a simple isolated eigenvalue at ${1}$. The eigenfunction ${f_0}$ associated to the eigenvalue ${1}$ and normalized so that ${\int_{\Delta} f_0 \, d Leb=1}$ is the density ${f_0=d\mu/dLeb}$ of the unique absolutely continuous invariant probability measure of ${T}$. Furthermore, the essential spectral radius of ${L_0}$ is ${<1}$ and ${L_s}$ has no eigenvalues ${\lambda_0}$ of modulus ${|\lambda_0|=1}$ except for ${\lambda_0=1}$. (See Aaronson’s book for more explanations.)

For ${\sigma\in\mathbb{R}}$ close to ${0}$, the operator ${L_{\sigma}}$ is a small perturbation of ${L_0}$. In particular, ${L_{\sigma}}$ has an eigenfunction ${f_{\sigma}}$ associated to its unique eigenvalue ${\lambda_{\sigma}}$ close to ${\lambda_0=1}$ such that ${\int f_{\sigma} = 1}$ and ${f_{\sigma}}$ converges to ${f_0}$ in the ${C^1}$-topology as ${\sigma\rightarrow 0}$.

From now on, let us fix ${0<\sigma_1<\min\{\sigma_0, 1\}}$ such that ${f_{\sigma}}$ is well-defined and bounded away from zero for ${|\sigma|\leq \sigma_1}$. (Here, ${\sigma_0>0}$ is the constant appearing in the exponential tails condition ${\int e^{\sigma_0 r} dLeb<\infty}$ for the roof function ${r}$.)

From the technical point of view, it is convenient to “uniformize” this spectral picture by normalizing the operators ${L_s}$ (for ${s=\sigma+it}$ with ${|\sigma|\leq\sigma_1}$) as follows:

$\displaystyle \widetilde{L}_s(u) = \frac{L_s(f_{\sigma}u)}{\lambda_{\sigma} f_{\sigma}} \ \ \ \ \ (1)$

The normalized weighted transfer operator ${\widetilde{L}_s}$ satisfy ${\widetilde{L}_{\sigma}(1)=1}$ and ${|\widetilde{L}_s u|\leq \widetilde{L}_{\sigma}|u|}$. In other words, if we replace ${L_s}$ by ${\widetilde{L}_s}$, we normalize both the eigenvalue ${\lambda_{\sigma}}$ and the eigenfunction ${f_{\sigma}}$ to ${1}$. Moreover, the proof of Proposition 1 can reduced to the analogous statement for the normalized operators:

Proposition 2 Suppose that there exist ${\sigma_0''>0}$, ${T_0\geq 1}$, ${\beta_0<1}$, ${C>0}$ and ${n\in \mathbb{N}}$ such that

$\displaystyle \|\widetilde{L}_s^{2mn} u\|_{L^2(\mu)}\leq C\beta_0^m\|u\|_{1,t}$

for all ${m\in\mathbb{N}}$ and ${s=\sigma+it}$ with ${|\sigma|\leq\sigma_0''}$ and ${|t|\geq T_0}$. Then, the conclusion of Proposition 1 is valid.

The proof of this proposition is based on Lasota-Yorke inequality (cf. Lemmas 7.8 and 7.9 of AGY paper):

Lemma 3 There exists a constant ${C_3>5}$ such that, for all ${k\in\mathbb{N}}$ and ${s=\sigma+it}$ with ${|\sigma|\leq\sigma_1}$, ${t\in\mathbb{R}}$, we have

• ${\|D(\widetilde{L}_s^k u)(x)\|\leq C_3(1+|t|)\widetilde{L}_{\sigma}^k(|u|)(x) + \kappa^{-k} \widetilde{L}_{\sigma}^k(\|Du\|)(x)}$ for all ${x\in\Delta}$;
• if ${|t|\geq 1}$, then ${\|\widetilde{L}_s^ku\|_{1,t}\leq C_3\|u\|_{C^0} + \frac{\kappa^{-k}}{|t|} \|Du\|_{C^0}}$;
• if ${|t|\geq 1}$, then ${\|\widetilde{L}_s^k u\|_{1,t}\leq C_3\|u\|_{1,t}}$,

where ${\kappa:=\inf |T'|>1}$ is the expansion constant of ${T}$.

Before proving this lemma, let us use it to show Proposition 2. Recall that the assumption of this proposition is that

$\displaystyle \|\widetilde{L}_s^{2mn} u\|_{L^2(\mu)}\leq C\beta_0^m\|u\|_{1,t}$

for some fixed ${\beta_0<1}$, ${n\in\mathbb{N}}$, and for all ${m\in\mathbb{N}}$ and ${s=\sigma+it}$ with ${|\sigma|\leq\sigma_0''}$, ${|t|\geq T_0}$, and our task is to prove the analogous statement in Proposition 1 for ${L_s^k}$.

The idea is very simple: from the spectral discussion above, it is not hard to see that we introduce a factor of the order of ${\lambda_{\sigma}^k}$ when replacing ${L_s^k u}$ by ${\widetilde{L}_s^k u}$; since ${\lambda_{\sigma}}$ is close to ${1}$, this factor does not significantly affect definite contraction on the size of ${\widetilde{L}_s^k u}$ provided by the hypothesis of Proposition 2. Let us now turn into the details.

Given ${k\in\mathbb{N}}$, we write ${k=2mn+r}$ with ${0\leq r < 2n}$. Since ${L_s^ku = \lambda_{\sigma}^k f_{\sigma} \widetilde{L}_s^k(u/f_{\sigma})}$ and ${f_{\sigma}}$ is uniformly ${C^1}$ close to ${f_0=d\mu/dLeb}$, the previous estimate for the normalized operator ${\widetilde{L}_s}$ gives that

$\displaystyle \|L_s^k u\|_{L^2(Leb)} \leq C \lambda_{\sigma}^k \|\widetilde{L}_s^k (u/f_{\sigma})\|_{L^2(\mu)} \\ \leq C \lambda_{\sigma}^k \beta_0^m \|\widetilde{L}_s^r (u/f_{\sigma})\|_{1,t}$

for all ${s=\sigma+it}$ with ${|\sigma|\leq\sigma_0''}$, ${|t|\geq T_0}$. By the last item of Lemma 3, it follows that

$\displaystyle \begin{array}{rcl} \|L_s^k u\|_{L^2(Leb)} &\leq& C \lambda_{\sigma}^k \beta_0^m \|\widetilde{L}_s^r (u/f_{\sigma)}\|_{1,t} \\ &\leq& C \lambda_{\sigma}^k\beta_0^m\|u/f_{\sigma}\|_{1,t} \\ &\leq& C\lambda_{\sigma}^k\beta_0^m\|u\|_{1,t} \end{array}$

Because ${\lambda_{\sigma}}$ is close to ${1}$ for ${|\sigma|}$ small, we can choose ${\sigma_0'>0}$ such that

$\displaystyle \sup\limits_{|\sigma|\leq\sigma_0'}\lambda_{\sigma}(\beta_0)^{1/2n}:=\beta < 1,$

we deduce from the previous estimate that

$\displaystyle \|L_s^k u\|_{L^2(Leb)}\leq C\beta^k\|u\|_{1,t}$

for all ${k\in\mathbb{N}}$ and ${s=\sigma+it}$ with ${|\sigma|\leq\sigma_0'}$, ${|t|\geq T_0}$. This completes the proof of Proposition 2 modulo Lemma 3.

1.1. Proof of Lasota-Yorke inequality

Let us now prove Lemma 3. The first item follows from a computation similar to the proof of Lemma 8 of the previous post, and its half-page proof is given in Lemma 7.8 of AGY paper.

For the sake of convenience, we provide just a sketch of proof. We write

$\displaystyle \widetilde{L}_s^k u(x) = \sum\limits_{h\in\mathcal{H}^{(k)}} \frac{e^{-sr^{(k)}(hx)} J^{(k)}(hx) (f_{\sigma}u)(hx)}{\lambda_{\sigma}^k f_{\sigma}(x)}$

where ${\mathcal{H}^{(k)}}$ is the set of inverse branches of ${T}$, ${r^{(k)}(x) = \sum\limits_{n=0}^{k-1} r(T^nx)}$ and ${J^{(k)}(x) = \prod\limits_{n=0}^{k-1} J(T^nx)}$ (with ${J(z):=1/|T'(z)|}$). By taking the derivative ${D\widetilde{L}_s^k u(x)}$, we obtain five terms ${(I)}$, ${(II)}$, ${(III)}$, ${(IV)}$, ${(V)}$ depending by differentiating ${e^{-sr^{(k)}(hx)}}$, ${J^{(k)}(hx)}$, ${f_{\sigma}(hx)}$, ${u(hx)}$ or ${1/f_{\sigma}(x)}$.

The terms ${(III)}$ and ${(V)}$ are easy to deal with: the uniform ${C^1}$ bounds on ${f_{\sigma}}$ and ${1/f_{\sigma}}$, and the contraction of inverse branches of ${T}$ imply that ${\|D(f_{\sigma}\circ h)(x)\|\leq C f_{\sigma}(x)}$ and ${\|D(1/f_{\sigma})(x)\|\leq C/f_{\sigma}(x)}$. Thus, ${|(III)|, |(V)|\leq C\widetilde{L}_{\sigma}^k(|u|)(x)}$.

Similarly, the distortion bound (Renyi condition) on ${T}$ (see the previous post) implies that ${\|D(J^{(k)}\circ h)(x)\|\leq C J^{(k)}(hx)}$, so that ${|(II)|\leq C\widetilde{L}_{\sigma}^k(|u|)(x)}$.

Since ${D(e^{-sr^{(k)}(hx)}) = -s D(r^{(k)}\circ h)(x) e^{-sr^{(k)}(hx)}}$ and ${\|D(r^{(k)}\circ h)(x)\|\leq C}$ (because ${r}$ is a good roof function and the inverse branches of ${T}$ contract exponentially), we see that ${\|D(e^{-sr^{(k)}(hx)})\|\leq C|s|e^{-\sigma r^{(k)}(hx)}}$ and, a fortiori, ${|(I)|\leq C|s|\widetilde{L}_{\sigma}^k(|u|)(x) \leq C(1+|t|) \widetilde{L}_{\sigma}^k(|u|)(x)}$.

Finally, the exponential contraction of inverse branches of ${T}$ says that ${\|D(u\circ h)(x)\|\leq \kappa^{-k}\|Du(hx)\|}$, so that ${|(V)|\leq \kappa^{-k}\widetilde{L}_{\sigma}^k(\|Du\|)}$. This proves the first item of Lemma 3.

The second item is an immediate consequence of the estimate ${\|\widetilde{L}_s^k u\|_{C^0}\leq \|u\|_{C^0}\widetilde{L}_{\sigma}^k1=\|u\|_{C^0}}$ and the first item just proved. Indeed, if ${|t|\geq 1}$, then

$\displaystyle \begin{array}{rcl} \|\widetilde{L}_s^k u\|_{1,t}&:=& \|\widetilde{L}_s^k u\|_{C^0} + \frac{\|D(\widetilde{L}_s^k u)\|_{C^0}}{|t|} \\ &\leq& \|u\|_{C^0} + \frac{1}{|t|}\left(2C|t|\|u\|_{C^0} + \kappa^{-k}\|Du\|_{C^0}\right) \\ &\leq& C\|u\|_{C^0} + \frac{\kappa^{-k}}{|t|}\|Du\|_{C^0} \end{array}$

Closing the proof of Lemma 3, we observe that the third item is a direct corollary of the second item.

1.2. First part of Proposition 1: study of regime (A)

By Proposition 2, one can establish Proposition 1 by showing a contraction property for the normalized weighted transfer operators ${\widetilde{L}_s}$.

As we already mentioned, the first step in this direction is the treatment of the regime (A) of ${C^1}$ functions displaying high oscillations via Lasota-Yorke inequality:

Lemma 4 There exists ${N_0\in\mathbb{N}_0}$ such that any ${n\geq N_0}$ has the following property.Let ${s=\sigma+it}$ with ${|\sigma|\leq\sigma_1}$ and ${|t|\geq 10}$ and suppose that ${v\in C^1(\Delta)}$ exhibits a high oscillation at scale ${1/|t|}$ in the sense that ${\|Dv\|_{C^0}\geq 2C_3|t|\|v\|_{C^0}}$ (where ${C_3>5}$ is the constant from Lemma 3). Then,

$\displaystyle \|\widetilde{L}_s^n v\|_{1,t}\leq \frac{9}{10} \|v\|_{1,t}$

Proof: The facts that ${\widetilde{L}_s}$ is normalized and ${v}$ has high oscillations imply that

$\displaystyle \|\widetilde{L}_s^n v\|_{C^0}\leq \|v\|_{C^0}\leq\frac{1}{2C_3|t|}\|Dv\|_{C^0}\leq\frac{1}{2C_3}\|v\|_{1,t}$

Furthermore, the Lasota-Yorke inequality in the first item of Lemma 3 says that

$\displaystyle \begin{array}{rcl} \|D(\widetilde{L}_s^n v)(x)\| &\leq& C_3(1+|t|)\widetilde{L}_{\sigma}^n(|v|)(x) + \kappa^{-n} \widetilde{L}_{\sigma}^n(\|Dv\|)(x) \\ &\leq& C_3(1+|t|)\|v\|_{C^0} + \kappa^{-n}\|Dv\|_{C^0} \\ &\leq& \left(\frac{1+|t|}{2} + \kappa^{-n}|t|\right)\|v\|_{1,t} \end{array}$

Since ${C_3>5}$ and ${|t|\geq 10}$, we get that

$\displaystyle \begin{array}{rcl} \|\widetilde{L}_s^n v\|_{1,t} &=& \|\widetilde{L}_s^n v\|_{C^0} + \frac{1}{|t|}\|D(\widetilde{L}_s^n v)\|_{C^0} \\ &\leq& \left(\frac{1}{2C_3} + \frac{1+|t|}{2|t|}+\kappa^{-n}\right)\|v\|_{1,t} \\ &\leq& \frac{9}{10}\|v\|_{1,t} \end{array}$

for ${n\geq N_0}$ and ${N_0}$ sufficiently large (so that ${\kappa^{-N_0}\leq 1/5}$). $\Box$

Of course, we can iterate this lemma to establish the contraction property in Proposition 2 while the high oscillation property is not destroyed by ${\widetilde{L}_s^k}$:

Corollary 5 Fix ${n\geq N_0}$ where ${N_0}$ is the integer provided by Lemma 4. Suppose that ${u}$ is a ${C^1}$ function on ${\Delta}$ and ${m\in\mathbb{N}}$ is an integer such that the high oscillation property is not destroyed by the first ${m-1}$ iterates of ${\widetilde{L}_s^{n}u}$, i.e.,

$\displaystyle \|D(\widetilde{L}_s^{pn}u)\|_{C^0}\geq 2C_3|t|\|u\|_{C^0}$

for all ${0\leq p and for some ${s=\sigma+it}$ with ${|\sigma|\leq\sigma_1}$, ${|t|\geq 10}$. Then,

$\displaystyle \|\widetilde{L}_s^{2mn} v\|_{L^2(\mu)}\leq C_3\left(\frac{9}{10}\right)^m \|v\|_{1,t}$

Proof: By the third item of Lemma 3, we have that

$\displaystyle \|\widetilde{L}_s^{2mn} v\|_{L^2(\mu)}\leq \|\widetilde{L}_s^{2mn} v\|_{C^0} \leq \|\widetilde{L}_s^{2mn} v\|_{1,t}\leq C_3\|\widetilde{L}_s^{mn} v\|_{1,t}$

By iterating Lemma 4, it follows that

$\displaystyle \|\widetilde{L}_s^{2mn} v\|_{L^2(\mu)}\leq \|\widetilde{L}_s^{mn} v\|_{1,t}\leq C_3\left(\frac{9}{10}\right)^m\|v\|_{1,t}$

This proves the corollary. $\Box$

Remark 2 The proof of the corollary shows that a strong (pointwise/${C^0}$) form of cancellation ${\|\widetilde{L}_s^{2mn} v\|_{1,t}\leq C_3\left(\frac{9}{10}\right)^m\|v\|_{1,t}}$ occurs in the high oscillation case (regime (A)). As we are going to see in the next section, one has a much weaker (${L^2}$) form of cancellation in the low oscillation case (regime (B)).

2. Dolgopyat’s mechanism and regime (B)

After our success in dealing with the regime (A) (cf. Corollary 5), let us analyze the regime (B) of ${C^1}$ functions ${v}$ whose oscillation is not high.

2.1. UNI condition

As we told in the introduction, the cancellation mechanism in regime (B) originates from the oscillations of the Birkhoff sums ${r^{(k)}(y)}$ of the roof function ${r}$. Here, by “oscillations of Birkhoff sums” we mean the following:

Proposition 6 Let ${T:\bigcup\Delta^{(l)}\rightarrow\Delta:=(0,1)}$ be an uniformly expanding Markov map. Let ${r:\Delta\rightarrow\mathbb{R}}$ be a ${C^1}$ function such that

$\displaystyle \sup\limits_{h\in\mathcal{H}} \|D(r\circ h)\|_{C^0}<\infty$

(where ${\mathcal{H}}$ is the set of inverse branches of ${T}$.)Then, the following conditions are equivalent:

1. ${r}$ is not a ${C^1}$ coboundary: it is not possible to write ${r=\psi + \phi\circ T - \phi}$ with ${\psi:\bigcup\Delta^{(l)}\rightarrow\mathbb{R}}$ which is constant on each ${\Delta^{(l)}}$ and ${\phi\in C^1(\Delta).}$.
2. Uniform non-integrability (UNI) condition (“oscillation of Birkhoff sums”): there exists a constant ${C>0}$ such that

$\displaystyle |D(r^{(n)}\circ h)(x) - D(r^{(n)}\circ k)(x)|>C$

for some ${n}$ arbitrarily large and some inverse branches ${h, k\in\mathcal{H}^{(n)}}$ of ${T^n}$.

This proposition is (contained in) Proposition 7.4 of AGY paper and we refer to it for the one page proof of this result.

Remark 3 The main point of this proposition is that a qualitative property (“not a ${C^1}$ coboundary”) in our definition of good roof function in the previous post turns out to be equivalent to a quantitative property (“definite oscillation of Birkhoff sums”). The nomenclature “uniform non-integrability” (UNI) comes from the fact that this is a (uniform) quantitative property issued from the fact that the suspension flow ${T_t}$ associated to ${T}$ and ${r}$ is not integrable (conjugated to a suspension flow with piecewise constant roof function) when ${r}$ is not a ${C^1}$ coboundary.

2.2. Dolgopyat’s cancellation mechanism

Let us now use UNI condition to produce non-trivial cancellations in some regions of the phase space ${\Delta}$ for ${C^1}$ functions with “low oscillations”. In other terms, we want to study ${\widetilde{L}_s^k v}$, ${s=\sigma+it}$, for ${v}$ such that ${\|Dv\|_{C^0}\leq 2C_3|t|\|v\|_{C^0}}$.

For technical reasons, we will keep track of ${v}$ and an appropriate ${C^0}$-bound ${u}$ for ${v}$:

Definition 7 Given ${t\in\mathbb{R}}$, we say that ${(u,v)\in\mathcal{E}_t}$ if ${v:\Delta\rightarrow\mathbb{C}}$ is ${C^1}$, ${u:\Delta\rightarrow\mathbb{R}^+}$ is a ${C^1}$ function providing an adequate ${C^0}$ bound for ${v}$ in the sense that

• ${0\leq |v(x)|\leq u(x)}$, and
• ${\max\{\|Du(x)\|,\|Dv(x)\|\}\leq 2C_3|t|\cdot u(x)}$

for all ${x\in\Delta}$.

Remark 4 ${\mathcal{E}_t}$ is a cone and we are going to show in Lemma 10 that ${\widetilde{L}_s^n}$ sends ${\mathcal{E}_t}$ “strictly inside itself” in a certain sense.

Moreover, we need to anticipate the fact in Remark 3 that the cancellations in regime (B) occur only at certain spots of the phase space. In particular, the following “localization tool” (cf. Lemma 7.12 in AGY paper) will be helpful in the subsequent cancellation discussion.

Lemma 8 There exists an integer ${N_1\in\mathbb{N}}$ such that any ${n\geq N_1}$ has the following property. Let ${s=\sigma+it}$ with ${|\sigma|\leq\sigma_1}$, ${|t|\geq 10}$, let ${(u,v)\in\mathcal{E}_t}$, and let ${\chi\in C^1(\Delta)}$ with ${\|D\chi\|\leq |t|}$, ${3/4\leq\chi\leq 1}$. Suppose that, for all ${x\in\Delta}$, we have

$\displaystyle |\widetilde{L}_s^n v(x)| \leq \widetilde{L}_{\sigma}^n(\chi u)(x)$

Then, ${(\widetilde{L}_{\sigma}^n(\chi u), \widetilde{L}_s^n v)\in\mathcal{E}_t}$.

This lemma is a consequence of the Lasota-Yorke inequality in Lemma 3 and its half-page proof can be found after the statement of Lemma 7.12 in AGY paper.

Remark 5 We call this lemma a “localization tool” for the following reason. We think of ${1-\chi}$ as a ${C^1}$ bump function supported on disjoint intervals of size ${\sim 1/|t|}$. Next, we use ${\chi}$ to “localize” an appropriate ${C^0}$ bound ${u}$ in a pair ${(u,v)\in\mathcal{E}_t}$ to the support of ${\chi}$ by considering ${\chi u}$. In this setting, the lemma says that if the iterate ${\widetilde{L}_s^n(\chi u)}$ of this localization gives a ${C^0}$ bound to ${\widetilde{L}_s^n v}$, then ${\widetilde{L}_s^n(\chi u)}$ is actually an appropriate ${C^0}$ bound for ${\widetilde{L}_s^n v}$ (i.e., ${\widetilde{L}_s^n(\chi u), \widetilde{L}_s^n v)\in\mathcal{E}_t}$).

From now on, we use UNI condition (cf. Proposition 6) to fix ${n\geq \max\{N_0, N_1\}}$ and inverse branches ${h,k\in\mathcal{H}^{(n)}}$ such that

$\displaystyle |D(r^{(n)}\circ h)(x) - D(r^{(n)}\circ k)(x)|>9C_3\max\{\|Dh(x)\|, \|Dk(x)\|\} \quad \forall\,x\in\Delta, \ \ \ \ \ (2)$

where ${C_3}$, ${N_0}$ and ${N_1}$ are the constants in Lemmas 3 and 8. (This is possible because ${\max\{\|Dh(x)\|, \|Dk(x)\|\}\leq\kappa^{-n}}$.)

In this language, Dolgopyat’s cancellation mechanism can be stated as:

Lemma 9 There exist a small constant ${\delta>0}$ and a large constant ${\zeta>0}$ with the following property. Let ${s=\sigma+it}$ with ${|\sigma|\leq\sigma_1}$, ${|t|\geq 10}$, and ${(u,v)\in\mathcal{E}_t}$.Then, for every interval ${I=[x_0-\frac{(\zeta+\delta)}{|t|}, x_0+\frac{(\zeta+\delta)}{|t|}]\subset\Delta}$, we can find a point ${|x_1-x_0|\leq\zeta/|t|}$ such that one of the following possibilities holds:

• Cancellation of type ${h}$: for all ${x\in I'=[x_1-\frac{\delta}{|t|}, x_1+\frac{\delta}{|t|}]}$, we have

$\displaystyle \begin{array}{rcl} & & \left|e^{-sr^{(n)}(hx)} J(hx) (f_{\sigma}v)(hx) + e^{-sr^{(n)}(kx)} J(kx) (f_{\sigma}v)(kx)\right| \\ &\leq& \frac{3}{4}e^{-\sigma r^{(n)}(hx)} J(hx) (f_{\sigma} u)(hx) + e^{-\sigma r^{(n)}(kx)} J(kx) (f_{\sigma} u)(kx) \end{array}$

• Cancellation of type ${k}$: for all ${x\in I'=[x_1-\frac{\delta}{|t|}, x_1+\frac{\delta}{|t|}]}$, we have

$\displaystyle \begin{array}{rcl} & & \left|e^{-sr^{(n)}(hx)} J(hx) (f_{\sigma}v)(hx) + e^{-sr^{(n)}(kx)} J(kx) (f_{\sigma}v)(kx)\right| \\ &\leq& e^{-\sigma r^{(n)}(hx)} J(hx) (f_{\sigma} u)(hx) + \frac{3}{4} e^{-\sigma r^{(n)}(kx)} J(kx) (f_{\sigma} u)(kx) \end{array}$

In other words, this lemma says that, for each interval of ${I}$ of size of order ${1/|t|}$, we can find a subinterval ${I'}$ of size of the same order where the two terms of

$\displaystyle \widetilde{L}_s^n v(x) = \frac{1}{\lambda_{\sigma}^n f_{\sigma}(x)} \sum\limits_{g\in\mathcal{H}^{(n)}} e^{-s r^{(n)}(gx)} J(gx) (f_{\sigma} v)(gx)$

associated to the inverse branches ${h}$ and ${k}$ fixed above exhibits a significant cancellation w.r.t. the trivial bound (i.e., one gets a factor of ${3/4}$ instead of ${1}$).

This lemma is exactly Lemma 7.13 in AGY paper and we provide a sketch of its proof in the sequel.

Proof: Let ${\delta>0}$, resp. ${\zeta>0}$ be a small, resp. large constant. Consider ${(u,v)\in\mathcal{E}_t}$ and ${I=[x_0-\frac{(\zeta+\delta)}{|t|}, x_0+\frac{(\zeta+\delta)}{|t|}]\subset\Delta}$. Our task is to find ${x_1}$ with ${|x_1-x_0|\leq\zeta/|t|}$ so that the conclusion of the lemma is valid.

The argument is divided into two cases. In the first (easy) case, we assume that the appropriate ${C^0}$ bound ${u}$ for ${v}$ is not “tight”, i.e., there exists ${|x_1-x_0|\leq\zeta/|t|}$ such that ${|v(hx_1)|\leq u(hx_1)/2}$ or ${|v(kx_1)|\leq u(kx_1)/2}$. In this situation, we want to take advantage of the non-tightness (and the fact that ${v}$ does not have high oscillations [as ${(u,v)\in\mathcal{E}_t}$]) to get some cancellation in an interval of size ${\delta/|t|}$ centered at ${x_1}$.

For this sake, up to exchanging the roles of ${h}$ and ${k}$, we can assume that ${|v(hx_1)|\leq u(hx_1)}$. Since ${(u,v)\in\mathcal{E}_t}$, we know that ${\max\{\|Du(x)\|, \|Dv(x)\|\}\leq 2C_3|t|u(x)}$. Thus, ${\max\{\|D(u\circ h)(x), \|D(v\circ h)(x)\|\}\leq 2C_3|t|u(hx)}$ because ${h}$ is a contraction. By Gronwall’s inequality, it follows that ${u(hx')\leq e^{2C_3|t|\cdot |x'-x|}u(hx)}$. Therefore,

$\displaystyle \|D(v\circ h)(x)\|\leq 2C_3|t|u(hx)\leq 2C_3|t|e^{2C_3|t|\delta/|t|}u(hx_1) = 2C_3|t|e^{2C_3\delta}u(hx_1)$

for all ${|x-x_1|\leq\delta/|t|}$. By integrating this estimate, we see that

$\displaystyle |v(hx)-v(hx_1)|\leq 2C_3|t|e^{2C_3\delta} u(hx_1)\delta/|t| = 2C_3 \delta e^{2C_3\delta} u(hx_1)$

for all ${|x-x_1|\leq\delta/|t|}$. Since ${|v(hx_1)|\leq u(hx_1)/2}$ (by “non-tightness” assumption), we obtain from the estimates above that, if ${\delta>0}$ is small enough, then

$\displaystyle \begin{array}{rcl} |v(hx)|&\leq& |v(hx_1)| + |v(hx)-v(hx_1)|\leq \left(\frac{1}{2}+2C_3\delta e^{2C_3\delta}\right)u(hx_1) \\ &\leq& \left(\frac{1}{2}+2C_3\delta e^{2C_3\delta}\right)e^{2C_3\delta}u(hx)\leq \frac{3}{4}u(hx) \end{array}$

for all ${|x-x_1|\leq\delta/|t|}$. This proves the lemma in this “non-tight” case.

In the second case, we assume that the ${C^0}$ bound ${u}$ for ${v}$ is “tight”, i.e., ${|v(hx)|>u(hx)/2}$ and ${|v(kx)|>u(kx)/2}$ for all ${|x-x_0|\leq\zeta/|t|}$. In this context, we want to find ${|x_1-x_0|\leq\zeta/|t|}$ such that the complex numbers

$\displaystyle e^{-sr^{(n)}(hx_1)} J^{(n)}(hx_1) (f_{\sigma} v)(hx_1) \quad \textrm{and} \quad e^{-sr^{(n)}(kx_1)} J^{(n)}(kx_1) (f_{\sigma} v)(kx_1)$

have opposite phases.

For this sake, we put ${x^{\tau}:=x_0+\tau}$ for ${\tau\in[0,\zeta/|t|]}$ and we denote by ${\gamma(\tau)}$ the difference of the phases (arguments) of the complex numbers

$\displaystyle F(x^{\tau}):=e^{-sr^{(n)}(hx^{\tau})} J^{(n)}(hx^{\tau}) (f_{\sigma} v)(hx^{\tau}) \,\, \textrm{and} \,\, G(x^{\tau}):=e^{-sr^{(n)}(kx^{\tau})} J^{(n)}(kx^{\tau}) (f_{\sigma} v)(kx^{\tau})$

Using UNI condition (2), the fact that ${v}$ does not have high oscillations (i.e., ${(u,v)\in\mathcal{E}_t}$) and the “tightness” of the ${C^0}$ bound ${u}$ for ${v}$, it is possible to prove with a short calculation that

$\displaystyle |\gamma'(\tau)|\geq C_3|t|\gamma_0$

where ${\gamma_0:=\min\limits_{x\in\Delta}\{\|Dh(x), \|Dk(x)\|\}}$. (See page 192 of AGY paper.)

It follows that if ${\zeta}$ is sufficiently large (e.g., ${\zeta=16\pi/(C_3\gamma_0)}$), then there exists ${\tau_0\in[0,\zeta/(8|t|)]}$ such that ${F(x^{\tau_0})}$ and ${G(x^{\tau_0})}$ have opposite phases.

We set ${x_1=x^{\tau_0}}$. By exchanging the roles of ${h}$ and ${k}$ if necessary, we can assume that ${|F(x_1)|\geq |G(x_1)|}$. By exploiting the fact that ${v}$ does not have high oscillations and the tightness of the ${C^0}$ bound ${u}$ for ${v}$ (via Gronwall inequality), it is possible to show (through another short calculation, cf. page 193 in AGY paper) that if ${\delta>0}$ is sufficiently small, then for all ${|x-x_1|\leq\delta/|t|}$ one has

$\displaystyle |F(x)|\geq |G(x)|/2$

and

$\displaystyle |\Gamma_F(x) - \Gamma_G(x) - \pi|\leq\pi/6$

where ${\Gamma_F(x)}$ and ${\Gamma_G(x)}$ denote the phases of the complex numbers ${z=F(x)}$ and ${z'=G(x)}$. In other terms, for any ${x}$ in the interval ${[x_1-\delta/|t|, x_1+\delta/|t|]}$,

• the size of ${z=F(x)}$ does not drop too much in comparison with the size of ${z'=G(x)}$: except for a factor ${1/2}$, one still has the same comparison ${|F(x_1)|\geq |G(x_1)|}$ from the case ${x=x_1}$;
• ${z=F(x)}$ and ${z'=G(x)}$ have almost opposite phases.

On the other hand, an elementary trigonometry lemma (cf. Lemma 7.14 in AGY paper) says that two complex numbers ${z=re^{i\theta}}$ and ${z'=r'e^{i\theta'}}$ such that

$\displaystyle |z|=r\geq r'/2=|z'|/2$

and

$\displaystyle |\theta-\theta'-\pi|\leq\pi/6$

verify the “cancellation estimate”

$\displaystyle |z+z'|\leq r+\frac{r'}{2} = |z|+\frac{|z'|}{2}$

By applying this lemma to ${F(x)}$ and ${G(x)}$ for ${|x-x_1|\leq\delta/|t|}$, we deduce that if ${|F(x_1)|\geq|G(x_1)|}$, then

$\displaystyle |F(x)+G(x)|\leq |F(x)|+\frac{|G(x)|}{2} \quad \forall \, |x-x_1|\leq\delta/|t|,$

that is, one has a cancellation of type ${k}$.

This completes the sketch of proof of Lemma 9. $\Box$

2.3. Second part of Proposition 1: study of regime (B)

By Proposition 2 and Corollary 5, one can establish Proposition 1 by proving a ${L^2}$-contraction property in regime (B) for the normalized weighted transfer operators.

Evidently, the key tool to get this ${L^2}$-contraction property is Dolgopyat’s cancellation lemma 9 because it says that, given any interval ${I}$ of size ${\sim 1/|t|}$, a non-trivial amount of cancellation for ${\widetilde{L}_s^n u}$ must happen inside a subinterval ${I'}$ of comparable size ${\sim 1/|t|}$. In particular, for the regime (B), even though we do not get cancellation everywhere in phase space, we still have cancellation in large chunks of the phase space. Thus, it is reasonable to expect a contraction property for ${\widetilde{L}_s^n u}$ in the ${L^2}$ norm (but not for the ${C^0}$ norm) in the regime (B).

Let us now try to formalize this heuristic argument. Fix ${T_0\geq 10}$ such that ${T_0>4(\zeta+\delta)}$ where ${\zeta>0}$ and ${\delta>0}$ are the constants in Lemma 9.

Lemma 10 There exists ${\beta_0<1}$ and ${0<\sigma_2<\sigma_1}$ with the following property. Let ${s=\sigma+it}$ with ${|\sigma|\leq \sigma_2}$, ${|t|\geq T_0}$, and let ${(u,v)\in\mathcal{E}_t}$. Then, there exists ${\widetilde{u}:\Delta\rightarrow\mathbb{R}}$ such that ${(\widetilde{u}, \widetilde{L}_s^n v)\in\mathcal{E}_t}$ and

$\displaystyle \int \widetilde{u}^2\,d\mu\leq\beta_0\int u^2\,d\mu$

In other words, this lemma says that for any ${v}$ in regime (B) (i.e., ${(u,v)\in\mathcal{E}_t}$), we can find an appropriate ${C^0}$ bound ${\widetilde{u}}$ for ${\widetilde{L}_s^n v}$ whose ${L^2}$-norm got contracted by a definite factor ${\sqrt{\beta_0}<1}$ in comparison with the ${L^2}$-norm of an appropriate ${C^0}$-bound ${u}$ for ${v}$.

This lemma is exactly Lemma 7.15 in AGY paper. For the sake of convenience, let us now sketch its proof.

Proof: We take a maximal set of points ${x_1,\dots, x_k\in\Delta=(0,1)}$ such that the intervals ${I_j=[x_j-2(\zeta+\delta)/|t|, x_j+2(\zeta+\delta)/|t|]}$ are pairwise disjoint and compactly contained in ${\Delta}$. Note that the intervals ${[x_j-4(\zeta+\delta)/|t|, x_j+4(\zeta+\delta)/|t|]}$ cover ${\Delta}$.

By Dolgopyat’s cancellation lemma 9, we know that each ${I_j}$ contains a subinterval ${I_j' = [x_j'-\delta/|t|, x_j' + \delta/|t|]}$ such that a cancellation of type ${h}$ or ${k}$ occurs for the pair ${(u,v)\in\mathcal{E}_t}$ in regime (B). We say that ${I_j'}$ has type ${h}$, resp. type ${k}$ depending on the type of cancellation occurring in ${I_j'}$.

We want to use our knowledge of the cancellation on each ${I_j'}$ to modify the trivial ${C^0}$ bound ${\widetilde{L}_{\sigma}^n u}$ for ${\widetilde{L}_s^n v}$: roughly speaking, we want to insert a factor ${3/4}$ in front of the terms of ${\widetilde{L}_{\sigma}^n u}$ associated to ${h}$ or ${k}$ whenever ${x\in I_j'}$. At this point, the localization tool in Lemma 8 will prove itself useful.

From the technical point of view, we implement the idea from the previous paragraph as follows. We consider a bump function ${\rho_j}$ localized on each ${I_j'}$: we impose ${\rho_j=1}$ on ${I_j'' = [x_j - \delta/(C|t|), x_j + \delta/(C|t|)]}$, ${\rho_j=0}$ outside ${I_j'}$ and ${\|\rho_j\|_{C^1}\leq C|t|/\delta}$ (for some universal constant ${C>0}$). Next, we construct ${\rho}$ on ${\Delta}$ by putting

$\displaystyle \rho = \left\{\begin{array}{cl} \left(\sum\limits_{I_j' \textrm{ of type } h}\rho_j\right)\circ T^n & \textrm{ on } h(\Delta), \\ \left(\sum\limits_{I_j' \textrm{ of type } k}\rho_j\right)\circ T^n & \textrm{ on } k(\Delta), \\ 0 & \textrm{ on } \Delta-(h(\Delta)\cup k(\Delta)). \end{array}\right.$

Note that ${\|\rho\|_{C^1}\leq|t|/\eta_0}$ for some constant ${0<\eta_0<1/4}$ depending only on the fixed objects ${n\in\mathbb{N}}$, ${h, k\in\mathcal{H}^{(n)}}$ and ${\delta>0}$. In particular, ${\eta_0}$ is independent of ${s=\sigma+it}$ and the pair ${(u,v)\in\mathcal{E}_t}$. The function ${\rho}$ was built up so that ${\chi:=1-\eta_0\rho}$ is a ${C^1}$ function taking values in ${[3/4,1]}$ such that ${\|D\chi\|\leq |t|}$ and

$\displaystyle \lambda_{\sigma}^n f_{\sigma}(x) \widetilde{L}_{\sigma}^n(\chi u)(x) = \sum\limits_{g\in\mathcal{H}^{(n)}} \chi(gx) e^{-\sigma r^{(n)}(gx)} J^{(n)}(gx) (f_{\sigma} u)(gx)$

has a definite contraction factor ${3/4<\chi(gx)=(1-\eta_0)<1}$ in front of the term associated to ${g=h}$ or ${g=k}$ whenever ${x\in I_j''}$ for some ${j=1,\dots, k}$.

In summary, we manipulated some bump functions associated to ${X:=\bigcup\limits_{j=1}^k I_j''}$ to get a function ${\chi}$ such that:

• ${3/4\leq \chi\leq 1}$, ${\|D\chi\|_{C^1}\leq |t|}$,
• ${3/4<\chi(gx)=1-\eta_0<1}$ for ${x\in I_j''}$ and ${g\in\{h,k\}}$ given by the type of ${I_j''}$,
• ${\chi(gx)=1}$ for ${x\in Y:=\Delta-X}$ or ${x\in I_j''}$ and ${g\in\mathcal{H}^{(n)}}$ not of the type of ${I_j''}$.

From the properties in the last two items, we see that the statement of Dolgopyat’s cancellation lemma 9 implies that

$\displaystyle |\widetilde{L}_s^n v|\leq \widetilde{L}_{\sigma}(\chi u),$

that is, ${\widetilde{u}:=\widetilde{L}_{\sigma}(\chi u)}$ is a ${C^0}$ bound for ${\widetilde{L}_s^n v}$. By combining this fact with the property in the first item and the localization tool in Lemma 8, we have that ${\widetilde{u}}$ is actually an appropriate ${C^0}$ bound for ${\widetilde{L}_s^n v}$, i.e.,

$\displaystyle (\widetilde{u}, \widetilde{L}_s^n v)\in\mathcal{E}_t$

Therefore, our task is reduced to prove that ${\int \widetilde{u}^2\,d\mu\leq\beta_0\int u^2 \,d\mu}$ for some constant ${\beta_0<1}$ (independent of ${u}$, ${v}$ and ${s=\sigma+it}$). In this direction, a short (half-page) calculation (cf. page 193 in AGY paper) exploiting the cancellation mechanism for any ${x\in I_j''}$ shows that

$\displaystyle \widetilde{u}(x)^2\leq \eta_1\xi(\sigma) \widetilde{L}_0^n(u^2)(x) \ \ \ \ \ (3)$

for all ${x\in X=\bigcup I_j''}$, and

$\displaystyle \widetilde{u}^2(x)\leq \xi(\sigma) \widetilde{L}_0^n(u^2)(x) \ \ \ \ \ (4)$

for all ${x\in Y=\Delta-X}$, where

$\displaystyle 1-\eta_1:= (1-(1-\eta_0)^2)\cdot\sup\limits_{\substack{|\sigma|\leq\sigma_1, \\ g\in\{h,k\}, \\ x\in\Delta}}\left(e^{-2\sigma r^{(n)(gx)}} J(gx)\frac{f_{2\sigma}(gx)}{\lambda_{2\sigma}^n f_{2\sigma}(x)}\right)>0$

and

$\displaystyle \xi(\sigma):=\left(\sup\limits_{\Delta}\frac{\lambda_{2\sigma}^n f_0(x) f_{2\sigma}(x)}{\lambda_{\sigma}^{2n}f_{\sigma}^2(x)}\right) \left(\sup\limits_{\Delta}\frac{f_{\sigma}}{f_0}\right) \left(\sup\limits_{\Delta}\frac{f_{\sigma}}{f_{2\sigma}}\right) \rightarrow 1 \textrm{ as } \sigma\rightarrow 0.$

Note that the estimate (3) goes in the good direction: it implies that

$\displaystyle \int_X \widetilde{u}^2\,d\mu\leq\eta_1\xi(\sigma)\int_X \widetilde{L}_0^n(u^2)\,d\mu \leq \eta_1\xi(\sigma)\int_{\Delta} \widetilde{L}_0^{n}(u^2)\,d\mu = \eta_1\xi(\sigma)\int_{\Delta} u^2\,d\mu$

by the definition of the normalized transfer operator ${\widetilde{L}_0}$ (and the fact that ${f_0=d\mu/d Leb}$), and we know that ${\eta_1\xi(\sigma)<1}$ for ${|\sigma|}$ sufficiently small.

On the other hand, the estimate (4) by itself is not sufficient to control ${\int_Y \widetilde{u}^2\,d\mu}$ in the desired way.

Fortunately, we know that ${X}$ covers “most” of the phase space and ${v}$ does not oscillate too much: more precisely, we have that ${X}$ is formed of intervals ${I_j''}$ of size ${2\delta/|t|}$ contained in some intervals ${\widehat{I}_j:=[x_j-4(\zeta+\delta)/|t|, x_j+4(\zeta+\delta)/|t|]}$ covering the whole phase space ${\Delta}$ (as it was said in the beginning of the proof) and ${(u,v)\in\mathcal{E}_t}$.

By combining this facts with Gronwall’s inequality, it is possible to prove (after a short calculation, cf. pages 195 and 196 of AGY paper) that there exists a constant ${C>0}$ such that

$\displaystyle \frac{\int_{\widehat{I}_j}\widetilde{L}_0^n(u^2)\,d\mu}{\mu(\widehat{I}_j)}\leq C\frac{\int_{I_j''} \widetilde{L}_0^n(u^2)\,d\mu}{\mu(I_j'')}.$

Because the intervals ${I_j''}$ are pairwise disjoint, the intervals ${\widehat{I}_j}$ cover ${\Delta}$, the density ${f_0=d\mu/d Leb}$ is bounded away from ${0}$ and ${\infty}$, and ${Leb(\widehat{I}_j)/Leb(I_j'')}$ is uniformly bounded (since the size of both ${\widehat{I}_j}$ and ${I_j''}$ is ${\sim1/|t|}$), the estimate above implies that there exists a constant ${C'>0}$ such that

$\displaystyle \int_Y \widetilde{L}_0^n(u^2)\,d\mu\leq C'\int_X \widetilde{L}_0^n(u^2)\,d\mu \ \ \ \ \ (5)$

This new information (coming from the low oscillations of ${v}$ and the fact that ${X}$ covers almost all of ${\Delta}$) permits to get the desired ${L^2}$-contraction for ${\widetilde{u}^2}$. Indeed, this is a consequence of the following simple computation (resembling to “Peter-Paul inequality”). Denote by ${w:=\widetilde{L}_0^n(u^2)}$. By (3) and (4), we have

$\displaystyle \begin{array}{rcl} \int_{\Delta}\widetilde{u}^2\,d\mu &=& \int_X \widetilde{u}^2\,d\mu + \int_Y \widetilde{u}^2\,d\mu \\ &\leq& \eta_1\xi(\sigma)\int_X \widetilde{L}_0^n(u^2)\,d\mu + \xi(\sigma)\int_Y \widetilde{L}_0^n(u^2)\,d\mu \\ &=& \eta_1\xi(\sigma)\int_X w\,d\mu + \alpha\xi(\sigma)\int_Y w\,d\mu + (1-\alpha)\xi(\sigma)\int_Y w\,d\mu \end{array}$

for any parameter ${\alpha\in[0,1]}$. By applying (5) to the right-hand side of the previous inequality, we deduce that

$\displaystyle \int_{\Delta}\widetilde{u}^2\,d\mu\leq \eta_1\xi(\sigma)\int_X w\,d\mu +C'\alpha\xi(\sigma)\int_X w\,d\mu + (1-\alpha)\xi(\sigma)\int_Y w\,d\mu$

for each ${\alpha\in[0,1]}$. By taking ${\alpha_0>0}$ so that ${\eta_1+C'\alpha_0=\sqrt{\eta_1}<1}$, we see that the estimate above implies that

$\displaystyle \int_{\Delta}\widetilde{u}^2\,d\mu\leq \sqrt{\eta_1}\xi(\sigma)\int_X w\,d\mu + (1-\alpha_0)\xi(\sigma)\int_Y w\,d\mu\leq \eta_1'\xi(\sigma)\int_{\Delta} w\,d\mu$

where ${\eta_1':=\max\{\sqrt{\eta_1},(1-\alpha_0)\}<1}$.

Therefore, if we take ${0<\sigma_2<\sigma_1}$ small enough so that ${\beta_0:=\sup\limits_{|\sigma|\leq\sigma_2}\eta_1'\xi(\sigma)<1}$, then

$\displaystyle \int_{\Delta}\widetilde{u}^2\,d\mu\leq \beta_0\int_{\Delta} w\,d\mu:=\beta_0\int_{\Delta}\widetilde{L}_0^n(u^2)d\mu = \beta_0\int_{\Delta}u^2\,d\mu$

This completes the proof of the lemma. $\Box$

Evidently, the main point of Lemma 10 is that one can iterate it to obtain the contraction property in Proposition 2 in regime (B):

Corollary 11 Consider the integer ${n\geq \max\{N_0, N_1\}}$ fixed above. Let ${0<\sigma_2<\sigma_1}$, ${T_0\geq 10}$ and ${\beta_0<1}$ be the constants provided by Lemma 10, and denote by ${s=\sigma+it}$ with ${|\sigma|\leq\sigma_2}$, ${|t|\geq T_0}$.Let ${u\in C^1(\Delta)}$ and ${m\in\mathbb{N}}$. Suppose that ${0\leq p is the first time such that ${\widetilde{L}_s^{kn} u}$ does not exhibit high oscillation at scale ${1/|t|}$, i.e.,

$\displaystyle \|D(\widetilde{L}_s^{pn} u)\|_{C^0}\leq 2C_3|t|\|\widetilde{L}_s^{pn}u\|_{C^0}$

Then,

$\displaystyle \|\widetilde{L}_s^{2mn} u\|_{L^2(\mu)}\leq \beta_0^{(2m-p)/2}\|u\|_{C^0}\leq \beta_0^{m/2}\|u\|_{C^0}$

Proof: We set ${v:=\widetilde{L}_s^{pn}u}$. By definition, ${(u_0,v)\in\mathcal{E}_t}$ where ${u_0}$ is the constant function ${u_0:=\|v\|_{C^0}}$. By successively applying Lemma 10, we obtain a sequence of pairs ${(u_k,\widetilde{L}_s^{kn}v)=(u_k, \widetilde{L}_s^{(p+k)n}u)\in\mathcal{E}_t}$ such that

$\displaystyle \int_{\Delta} |\widetilde{L}_s^{(p+k)n}u|^2\,d\mu\leq \int_{\Delta} u_k^2\,d\mu\leq\beta_0^k\int_{\Delta} u_0^2\,d\mu=\beta_0^k\|v\|_{C^0}^2$

By setting ${k=2m-p}$ and by recalling that ${\|\widetilde{L}_s^l u\|_{C^0}\leq\|u\|_{C_0}}$ for all ${l\in\mathbb{N}}$, we conclude from the previous estimate that

$\displaystyle \int_{\Delta} |\widetilde{L}_s^{2mn}u|^2\,d\mu\leq \beta_0^{2m-p}\|v\|_{C^0}^2\leq \beta^{2m-p}\|u\|_{C^0}^2.$

This proves the corollary. $\Box$

At this point, the proof of Proposition 1 is complete. Indeed, this is so because the hypothesis of Proposition 2 is always satisfied in both regimes (A) (by Corollary 5) and (B) (by Corollary 11).