1 Correction to: J Stat Phys (2017) 166:525–557 https://doi.org/10.1007/s10955-016-1663-0

We provide corrections and complements to [1, §4]: Formulas (29) and, especially, (31) must be amended, as explained in Sects. 3 and 4 below. An argument to replace the \(C^r\) norm of the weight by a supremum was missing at the end of the proof of Theorem 4.1 (see Sect. 2 below). Sections 1, 5, and 6 below give minor clarifications. None of the main statements are changed, except that:

  • In Lemma 4.2 and Theorem 4.1, the condition \(-(r-1)<s<-t<0\) must be replaced by \(t-(r-1)<s<-t<0\). (See Sect. 4 below.)

  • The bound (25) on the essential spectral radius in Theorem 4.1 is only proved up to an (arbitrarily small) \(\epsilon >0\) and for a “zoomed” version of the space \(\mathcal {U}^{t,s}_1(R)\) for some large \(R(\epsilon )\). (See (1) in Sect. 2 below.)

2 Details for the Leafwise Young Inequality (38)

To prove (38) on p. 542, notice that, for \(\ell \ge 1\) and \(x\in \mathbb {R}^{d_s}\), Fubini implies

$$\begin{aligned}&\big (\psi _\ell ^{(d_s)}\big )^{Op}\left[ (\varphi *\hat{\psi }) \circ \pi _{\Gamma }^{-1}\right] (x)\\&\qquad = \left[ \big (\psi _\ell ^{(d_s)}\big )^{Op}\int \hat{\psi }(z) \varphi \big (\pi _{\Gamma }^{-1}(\cdot )-z\big ) \mathrm {d} z \right] (x)\\&\qquad = \left[ \big (\psi _\ell ^{(d_s)}\big )^{Op}\int \hat{\psi }(z) \varphi \big (\pi _{\Gamma -z}^{-1}(\cdot )\big ) \mathrm {d} z \right] (x)\\&\qquad = \int _{\mathbb {R}^n}\hat{\psi }(z) \big (\psi _\ell ^{(d_s)}\big )^{Op} \bigl [\varphi \circ \pi _{\Gamma -z}^{-1} \bigr ] ( x ) \mathrm {d}z . \end{aligned}$$

Since \(\Vert \int _{\mathbb {R}^n} \hat{\psi }(z) \Phi _z(\cdot ) \mathrm {d} z \Vert _{L_p(\mathbb {R}^{d_s})} \le \Vert \hat{\psi }\Vert _{L_1(\mathbb {R}^d)} \sup _z \Vert \Phi _z(\cdot )\Vert _{L_p(\mathbb {R}^{d_s})}\) by the Minkowski integral inequality, we find,

$$\begin{aligned} \left\| \left( \psi _\ell ^{(d_s)}\right) ^{Op}\bigl [(\varphi *\hat{\psi }) \circ \pi _{\Gamma }^{-1}\bigr ] \right\| _{L_p(\mathbb {R}^{d_s})} \leqslant \Vert \hat{\psi }\Vert _{L_1(\mathbb {R}^d)} \sup _{\tilde{\Gamma } \in \mathcal {F} } \bigl \Vert \big (\psi _\ell ^{(d_s)}\big )^{Op} \big [\varphi \circ \pi _{\tilde{\Gamma }}^{-1} \big ] \bigr \Vert _{L_p(\mathbb {R}^{d_s})} \, . \end{aligned}$$

3 Proof of Theorem 4.1

The bound (43) is correct as stated, but to prove Theorem 4.1 we would need it for weights \(\tilde{f}_m(x)=\prod _{j=0}^{m-1} (\theta ^{(m)}_{\vec \omega } g (T^{-j}(x)) \), where \(\sum _{\vec \omega } \theta ^{(m)}_{\vec \omega }\equiv 1\) is a smooth partition of unity adapted to \(T^m\) (independently of \(\Gamma \)), taking into account the fact that minimal sub-covers are needed for thermodynamic estimates. The bound (43) does not seem to always hold for such \(\tilde{f}_m\). Hence, Theorem 4.1 is only proved if we add (arbitrarily small) \(\epsilon >0\) to the expression (25), and if we replace \(\mathcal {U}^{t,s}_1\) by its R-zoomed version \(\mathcal {U}^{t,s}_1(R)\)(as in [3, §2.3]), for R depending on \(\epsilon \):

We recall the construction in [3, §2.3]. Our charts are \(\kappa _\omega : U_\omega \rightarrow V_\omega \subset M\) with \(U_\omega \in \mathbb {R}^d\) and \(\omega \in \Omega \). Fix a \(C^\infty \) function \(\alpha :\mathbb {R}^d\rightarrow [0,1]\) with \(\alpha (z)=0\) if \(|z| \ge d\), and \(\sum _{m\in \mathbb {Z}^d} \alpha (z-m)=1\). For \(R \ge 1\) and

$$\begin{aligned} (\omega ,m)\in \mathcal {Z}(R) :=\{ (\omega ,m) \mid \omega \in \Omega , \, m\in (R \cdot U_{\omega }) \cap \mathbb {Z}^d\} , \end{aligned}$$

define \(\hat{\alpha }^R_{\omega , m} : M \rightarrow [0,1]\) by \(\hat{\alpha }^R_{\omega ,m}(x)=0\) if \(x \notin V_{\omega }\), and

$$\begin{aligned} \hat{\alpha }^R_{\omega , m}(x)=\alpha \big [R \cdot \big (\kappa _\omega ^{-1}(x)\big ) -m\big ] , \quad \forall x \in V_\omega . \end{aligned}$$

This gives a partition of unity in the sense that \( \sum _{m\in (R \cdot U_{\omega }) \cap \mathbb {Z}^d} \hat{\alpha }^R_{\omega ,m}(x)= 1\) for all \(x \in V_{\omega }\). (The intersection multiplicity of this partition of unity is bounded, uniformly in R.) Finally, fixing \(s,t\in \mathbb {R}\) and \(R\ge 1\), we write \(\kappa _\omega ^R(z)=\kappa _\omega (z/R)\), and we set for \(\varphi \in L^\infty (M)\),Footnote 1

$$\begin{aligned} \Vert \varphi \Vert _{\mathcal {U}_{1}^{t,s}(R)}= \sum _{(\omega , m)\in \mathcal {Z}(R)}\big \Vert \big (\hat{\alpha }^R_{\omega , m} \cdot \varphi \big ) \circ \kappa ^R_\omega \big \Vert _ {\mathcal {U}_1^{t,s}} \le \infty . \end{aligned}$$
(1)

The space \(\mathcal {U}_{p}^{t,s}(R)\) is the closure of \(L^\infty (M)\) for the norm \(\Vert \varphi \Vert _{\mathcal {U}_{p}^{t,s}(R)}\).

Since \(\Vert \tilde{f}_m^R \Vert _{C^r}\le 2 \sup |\tilde{f}^R_m|\), for \(\tilde{f}_m(z)=\tilde{f}_m(z/R)\), if \(R\ge 1\) is large enough (depending on m), we may then prove Theorem 4.1 up to adding \(\epsilon >0\) to (25) and replacing \(\mathcal {U}^{t,s}_1\) by \(\mathcal {U}^{t,s}_1(R)\) for suitable \(R(\epsilon )\).

Finally, the last sentence in the proof of Theorem 4.1 on p. 543 must be shortened to: “We just mention here that, in the present case, the “fragmentation lemma” (used to expand along a partition of unity) is just the triangle inequality, while the “reconstitution lemma” (used to regroup the terms from a partition of unity) is the trivial inequality \(\sum | a_k e_k| \le (\sum |a_k|) \sup |e_k|\).” Footnote 18 on the same page must be suppressed.

4 Details for (40) in Sublemma 4.4 and Correcting (29)

We explain how to bound \(\Vert f(\varphi \circ F)\Vert ^s_{p,\Gamma }\), by duality, giving the proof of (40) on p. 543: Since \(B^s_{p,\infty }\) is the dual of \(b^{|s|}_{p',1}\) (with \(1/p'=1-1/p\)), setting \(F_\Gamma = \pi _{F(\Gamma )} \circ F\circ \pi _\Gamma ^{-1}\), it suffices to estimate \(\Vert ((fh) \circ F_\Gamma ^{-1}) |\det ( DF_\Gamma )^{-1}|\Vert _{B^{|s|}_{p',1}(\mathbb {R}^{d_s})}\) for \(C^\infty \) functions h. First note that \(\Vert ((fh) \circ F_\Gamma ^{-1}) |\det ( DF_\Gamma )^{-1}|\Vert _{L_{p'}(\mathbb {R}^{d_s})} \le \sup \frac{| f |}{|\det DF_\Gamma |^{1/p}}\). The \(B^{|s|}_{p',1}(\mathbb {R}^{d_s})\) norm of v is equivalent toFootnote 2

$$\begin{aligned} \Vert v\Vert _{W^{ \lceil |s|-1\rceil }_{p'}(\mathbb {R}^{d_s})}+ \sum _{|\beta | = \lceil |s|-1\rceil } \int _{\mathbb {R}^{d_s}} \frac{1}{|y|^{d_s} } \frac{ \Vert Z(D^\beta v,\cdot , y) \Vert _{L_{p'}(\mathbb {R}^{d_s})} }{|y|^{ |s|- \lceil |s|-1 \rceil }} \mathrm {d}y , \end{aligned}$$

where \(\Vert v\Vert _{W^k_{p'}}= \sum _{0\le |\beta | \le k} \Vert D^\beta v\Vert _{L_{p'}}\) and \(Z(w,x,y)= w(x+y)+w(x-y)-2 w(x)\). Thus, since \(\inf |DF_\Gamma |\ge \Vert F\Vert _-\ge 1\), and using the“Zygmund derivation” inFootnote 3 [4, §2]

$$\begin{aligned} Z(f h, x,y)&=f(x) Z(h, x,y) + h(x) Z(f, x,y)\\&\qquad +\Delta _+ (f,x,y) \Delta _+(h,x,y) +\Delta _-(f,x,y)\Delta _-(h,x,y) , \end{aligned}$$

where \(\Delta _+(\upsilon ,x,y)=(\upsilon (x+y)-\upsilon (x))\) and \(\Delta _-(\upsilon ,x,y)=(\upsilon (x)-\upsilon (x-y))\), and recalling that for any noninteger \(\sigma >0\) [6, Props 2.1.2, 2.2.1]

$$\begin{aligned} \Vert v\Vert _{\mathcal {W}^{\sigma }_{p',p'}}\le C (p',\sigma ) \Vert v\Vert _{B^{\sigma }_{p',p'}}\le C^2 (p',|s|) \Vert v\Vert _{B^{\sigma }_{p',1}} , \end{aligned}$$

(with \(\mathcal {W}^{\sigma }_{p',p'}\) the Slobodeckij norm), we find for any \(\epsilon >0\) constants \(C(\mathcal {F})\) and \(C(\mathcal {F},\epsilon )\) so that

figure a

Finally, using \(\inf | \det (DF |_{(C'_+)^\perp })| \ge C | \det (D (F |_{\Gamma })|\), we get (40), up to slightly amending (29) as follows (\(d_s\) and p are fixed):

figure b

Similarly, in (42) one should replace \(\sup _\Gamma \Vert f \circ F^{-1}\Vert _{C^{r-1}(F(\Gamma ))}\) by

$$\begin{aligned} \sup _\Gamma \left\| f \circ F^{-1}\left\| _{C^{r-1}(F(\Gamma ))} \right\| \left| \det \left( D \left( F_{(C'_+)^\perp }\right) ^{-1}\right) \right| \right\| _{C^{r-1}(F(\Gamma ))} . \end{aligned}$$

5 Bounding \(\Vert H^{\ell , \tau }_{n,\sigma }(v)\Vert ^s_{p,\Gamma }\) (Proof of Lemma 4.2): Correcting (31)

Since \(\Vert \cdot \Vert ^s_{p,\Gamma }\) is not an \(L_p\) norm, (38) does not suffice to deduce from (54) a bound on \(\Vert H^{\ell , \tau }_{n,\sigma }(v)\Vert ^s_{p,\Gamma }\). For any compact \(K\subset \mathbb {R}^d\) and any \(\delta >0\), there exists \(C_0\ge 2\) so that for all \(C_0'\ge C_0\) there exists \(\widetilde{C}_0\) so that for all v supported in K,

$$\begin{aligned}&\left\| \big (\psi _\ell ^{Op} v\big ) \circ \pi _{\widetilde{\Gamma }}^{-1} \right\| _{L_p(\mathbb {R}^{d_s})}\\&\quad \le \widetilde{C}_0 2^{\ell (-s+\delta )} \sum _{j=0}^{\ell +[C_0']} 2^{j (s-\delta )} \left\| \big (\psi _{j}^{d_s}\big )^{Op}\big (\big (\psi _\ell ^{Op} v\big ) \circ \pi _{\widetilde{\Gamma }}^{-1} \big )\right\| _{L_p(\mathbb {R}^{d_s})}\\&\qquad + C_0 \sum _{j=\ell +[C_0']+1}^{\infty } 2^{-j r} \sum _{m=\ell -2}^{\ell +2} \sup _{\widehat{\Gamma }} \left\| \big (\psi _{m}^{Op} v\big )\circ \pi _{\widehat{\Gamma }}^{-1} \right\| _{L_p(\mathbb {R}^{d_s})} , \, \forall \ell , \quad \forall \widetilde{\Gamma }. \end{aligned}$$

(This is clear if \(\widetilde{\Gamma }\) isFootnote 4 affine, otherwise, proceed as in [2, Lemma 3.5], using the \(L_p\) version of the leafwise Young inequality [5, Lemma 4.2], to obtain the above estimate in the sum over \(j > \ell +C_0'\), after decomposing \(v= \sum _m \psi _m^{Op}v\) and using almost orthogonality.) Therefore, since \(|s|< r\) and \(a \le b + \epsilon a\) implies \(a \le (1-\epsilon )^{-1}b\) if \(a>0\), \(b>0\), and \(\epsilon <1\), for each \(\delta >0\) there is C so that

figure c

Then, applying \(\Vert \phi \Vert _{B^s_{p,\infty }} \le C \Vert \phi \Vert _{L_p}\) to \(\phi = (H_{n,\sigma } ^{\ell ,\tau } (\sum _{i=-2}^2\psi _{\ell +i}^{Op} v))\circ \pi _{\Gamma }^{-1}\), and using (54) and the \(L_p\) version of [5, Lemma 4.2], one obtains

$$\begin{aligned} 2^{\ell t}\Vert H^{\ell , \tau }_{n,\sigma }( v)\Vert ^s_{p,\Gamma }=2^{\ell t}\left\| H^{\ell , \tau }_{n,\sigma }\left( \sum _{i=-2}^2\psi _{\ell +i}^{Op} v\right) \right\| ^s_{p,\Gamma } \le C_{F,f} 2^{-(r-1)\max \{n ,\ell \}} 2^{(-s+\delta ) \ell } \Vert v\Vert _{\mathcal {U}_p^{t,s}} . \end{aligned}$$

This replaces the stronger bound stated two lines above (52) on p. 546 and gives the following weakening of (31):

figure d

Therefore, one must replace \(-(r-1)<s<-t<0\) by \(t-(r-1)<s<-t<0\) in Lemma 4.2, and thus inFootnote 5 Theorem 4.1.

6 Fixing the End of the Proof of Sublemma 4.4

The formulas for some kernels in the proof of Sublemma 4.4 (p. 548) are garbled. The corrections are detailed below. The statement of the sublemma is unchanged.

Lines 9–14 and the footnote of p. 548 must be replaced by “Recalling the functions \(b_m\) from (53), we claim that there exists a constant \(C_0 >1\) depending only on \(C_\mathcal {F}\) and \(\mathbf {C}_\pm \) so that, for any \(\Gamma \in \mathcal {F}(\mathbf {C}_+)\) and all n, \(n_s\), the kernels \(V^{n,-}_{n_s, \Gamma }(w,y)\) defined for \(w \in \Gamma \) and \(y\in \mathbb {R}^d\) by

$$\begin{aligned} \int _{\mathbb {R}^d} \mathbb {F}^{-1}(\psi _{\Theta ',n,-})(-x)&(\phi \cdot \psi ^{Op(\Gamma +x)}_{n_s} \tilde{\varphi })(w+x)\mathrm {d}x = \frac{1}{(2\pi )^{d+d_s}}\int _{\mathbb {R}^{d}} V^{n,-}_{n_s, \Gamma }(w,y) \tilde{\varphi }(y) \, \mathrm {d}y \end{aligned}$$

satisfy,Footnote 6

figure e

Replace lines 15–19 of p. 548 by: “To prove (60), recall (16) and note that

$$\begin{aligned} V^{n,-}_{n_s, \Gamma }(w,y)=&\int _{\eta \in \mathbb {R}^{d},\, x_-, \eta _s\in \mathbb {R}^{d_s}} e^{-ix(y,x_-)\eta } e^{i( \pi _{\Gamma +x(y,x_-)}(w+x(y,x_-))- z(y, x_-)) \eta _s}\\&\times \frac{ \phi (w+x(y,x_-))}{|\det D \mathcal {Y}_{\Gamma , x_-} (\mathcal {Y}_{\Gamma , x_-}^{-1}(y))|} \psi _{n_s}^{(d_s)}(\eta _s) \psi _{\Theta ', n,-}(\eta ) \mathrm {d}\eta \mathrm {d}\eta _s \mathrm {d}x_- , \end{aligned}$$

using for each \(x_-\in \mathbb {R}^{d_s}\) the \(C^r\) change of variable \(y=\mathcal {Y}_{\Gamma , x_-}(z,x_+):= \pi _{\Gamma +(x_-,x_+)}^{-1}(z)\) in \(\mathbb {R}^d\), with \(z\in \mathbb {R}^{d_s}\) and \(x_+\in \mathbb {R}^ {d_u}\), setting also

$$\begin{aligned} x(y,x_-)= (x_-, \Pi _+\big (\mathcal {Y}^{-1}_{\Gamma , x_-}(y))\big ) , \quad z(y,x_-)= \pi _{\Gamma +x(y,x_-)} (y) , \end{aligned}$$

where \(\Pi _+:\mathbb {R}^{d_s+d_u}\rightarrow \mathbb {R}^{d_u}\) is defined by \(\Pi _+(x_-, x_+)=x_+\). Next just like in [10, 11] (see also [Lemma 2.34, 2]), using that \(\pi _{\Gamma +x}(w+x)=\pi _\Gamma (w)+x_-\) if \(x=(x_-,x_+)\in \mathbb {R}^{d_s}\times \mathbb {R}^{d_u}\) and that \(\Gamma \in \mathcal {F}\), first integrate by parts (see Appendix 3) \((r-1)\) times with respect to \(x_-\in \mathbb {R}^{d_s}\) in the formula for \(V^{n,-}_{n_s, \Gamma }(w,y)\), and second, noticing that \(\Vert y-w\Vert > \epsilon \) implies that either \(\Vert \pi _{\Gamma }(w)-\pi _{\Gamma +x(y,x_-)}(y)\Vert > \epsilon /(2C_0)\) or \(\Vert \Pi _+(\mathcal {Y}^{-1}_{\Gamma , x_-}(y))\Vert >\epsilon /(2C_0)\), integrate by parts with respect the other variables as many times as necessary. It is an enlightening exercise to prove (60) for affine \(\Gamma \).”

There is a minor typo in the left-hand side of (64) on p. 549, which should read:

figure f

Finally, (64) follows from (60) and the leafwise Young inequality (38). \(\square \)

7 Typos

On p. 537, the condition \(\mathbb {R}^{d_s}\times \{0\} \subset \mathbf {C}_-\) must be replaced by “\(\mathbb {R}^{d_s}\times \{0\}\) is included in \((\mathbb {R}^{d}\setminus \mathbf {C}_+)\cup \{0\}\)” (thrice, including Defs 3.2–3.3). Also, the assumptions ensure that \(\Pi _\Gamma \) is surjective. Same page, 6 lines after (17), the norms are equivalent uniformly in \(\Gamma \), not equal, due to the Jacobian. In Lemma 4.2, one must assume that F can be extended by a bilipschitz regular cone hyperbolic diffeomorphism \(\tilde{F}\) of \(\mathbb {R}^d\), with \(\Vert \tilde{F}\Vert _+\), \(1/\Vert \tilde{F}\Vert _{-}\), \(1/\Vert \tilde{F}\Vert _{--}\) and \(1/|\det (D\tilde{F}|_{(\mathbf {C}'_+)^\perp })|\) controlled by twice the corresponding constants for F. In lines 2–3 of p. 555, the sum is over all \(\ell \ge 0\), and in line 2, one of the \((1+\Vert x_-\Vert )^{Q_1}\) must be replaced by \((1+\Vert x_+\Vert )^{Q_2}\), while \(\psi ^{Op}_\ell \upsilon (x)\) should be replaced by \(\Vert \psi ^{Op}_\ell \upsilon \Vert _{L_\infty }\).