数理解析研究所講究録 1244 巻 2002 年 24-31

24

AMATHEMATICAL APPROACH TO
INTERMITTENCY (1), (2)Michiko Yuri
Department of Business Administration, Sapporo University
平成 13 年 12 月 16 日

1Piecewise$C^{0}$

Convertible Systems

Let $(T,X, Q=\{X_{i}\}_{i\in I})$ be apiecewise
Convertible system i.e., $X$ is acompact metricspace with metric , : $Xarrow X$ is anoninvertible map which is not necessarily continuous,and $Q=\{X_{i}\}:\in I$ is acountable disjoint partition $Q=\{X\dot{.}\}:\in I$ of $X$ such that
intXi
is dense in $X$ and satisfy the following properties.
$C^{0}$$d$$T$$\bigcup_{\in I}\dot{.}$(01) For each i$(T|_{int\mathrm{x}_{:}})^{-1}$with intX:
extends to ahomeomorphism$\in I$$\dot{.}\neq\emptyset,T|_{intx_{:}}$(02)$T( \bigcup_{intX\dot{.}=\emptyset}X\dot{.})\subset\bigcup_{intX_{i}=\emptyset}X\dot{.}$(03)$\{X_{i}\}_{i\in I}$

Let$\underline{i}=$generates$\mathcal{F}$$(i_{1}\ldots i_{n})\in I^{n}$$intX_{i}arrow T(intX_{\dot{\iota}})$$v\dot{.}$on

is ahomeomorphism and.

$d(T(intX\dot{.}))$.

, the sigma algebra of Borel subsets of X.

satisfy$int(X_{1}.\cdot\cap T^{-1}X_{2}\dot{.}\cap\ldots T^{-(n-1)}X_{n}\dot{.})\neq\emptyset$. Then we define

which is called acylinder of rank and wfite
. By
(01),
:
is ahomeomorphism andextends to ahomeomorphism:.
We impose on $(T, X, Q)$ the next condition which gives anice countable states symbolic
dynamics similar to sofic shifts (cf. [11]):$X_{\underline{i}}:=X_{i_{1}}\cap T^{-1}X_{2}\dot{.}\cap\ldots T^{-(n-1)}X_{i_{n}}$$T^{n}|_{intx_{:_{1}\ldots:_{n}}}$$n$$arrow T^{n}(int(X_{i_{1\cdots n}}.\cdot))$$intX_{i_{1}\ldots i_{n}}$$(T^{n}|\dot{.}ntX_{1\cdots n}.\cdot:)^{-1}$$v_{i_{1}}\circ v_{i_{2}}\circ\ldots\circ v_{i_{n}}=v:_{1\ldots n}\dot{.}$(Finite Range Structure)

many open subsets$U_{1}$:$d(T^{n}(intX_{\underline{|}}.))arrow d(intX\underline{\dot{.}})$$\mathcal{U}=\{int(T^{n}X_{\dot{1}_{1}\cdots i_{n}})$ $\forall X_{i_{1}\ldots i_{n}},\forall n>0\}$\ldots$U_{N}$$|\underline{i}|=n$consists of finitelyof X.

In particular, we say that $(T, X, Q)$ satisfies Bernoulli property if $cl(T(intX\dot{.}))=X$ (Vi$I)$ so that
{intX} and that $(T, X, Q)$ satisfies Markov property if int(d(intX:)$cl(intTX_{j}))\neq\emptyset$ implies $cl(intTX_{j})\supset d(intX\dot{.})$ . $(T,X, Q)$ satisfying
Bernoulli (Markov)
property is called apiecewise
Convertible Bernoulli (Markov) system respectively. We saythat $X_{i}\in Q$ is afull cylinder if $cl(T(intX_{i}))=X$ . We assume further the next condition:$\in$$\mathcal{U}=$$\cap$$C^{0}$

25(Transitivity) intX$\mathrm{A}\ovalbox{\tt\small REJECT}$E {1,2, \ldots N},

T’$U_{k}$$(intX^{(l)}’(s_{\mathit{1}1}))\ovalbox{\tt\small REJECT}$

such that for eachand Vlc {1,2, \ldots N}, SO $<\$ )$X^{()}"(s_{l})$of rank such that
contains an interior of acylinder)$\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}$$|<\mathrm{O}\mathrm{O}$$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}_{\ovalbox{\tt\small REJECT}}^{j}.U_{k\mathit{1}}$$s_{l}$$U_{\mathit{1}}$.

2Topological pressure for potentials of weak boundedvariation
Definition We say that is apotential of weak bounded variation(WBV) if there exists
satisfying $\lim_{narrow\infty}(1/n)\log C_{n}=0$ andasequence of positive numbers,$\phi$$\forall n\geq$$\{C_{n}\}$$1,\forall X_{i_{1}\ldots i_{n}}\in \mathrm{V}_{j=0}^{n-1}T^{-j}Q$$\frac{\sup_{x\in X_{i_{1\cdots\dot{\cdot}n}}}\exp(\Sigma_{j=0}^{n-1}\phi(T^{j}x))}{\inf_{x\in X_{i_{1}.i_{n}}}\exp(\Sigma_{j=0}^{n-1}\phi(T^{j}x))}\leq C_{n}$.

(C.f.[11,13,15-19])
We define apartition function for each $n>0$ and for eachZn$(U_{k}, \phi)$$U_{k}\in \mathcal{U}$as follows :.

$:= \sum_{\underline{i}:|\underline{i}|=n,int(TX_{i_{n}})=U_{k}\supset intX_{1}}\dot{.}\sum_{v_{\underline{i}}x=x\in cl(intX_{\underline{t}})}\exp[\sum_{h=0}^{n-1}\phi T^{h}(x)]$We further define :$\overline{Z}_{n}(U_{k}, \phi)=$$\underline{i}:|\underline{i}|=n,int(TX_{t})=U_{k}\supset intX_{i_{1}}\sum_{n}\sup_{x,\in X_{\underline{i}}}\exp[\sum_{h=0}^{n-1}\phi T^{h}(x)]$and$Zn(Uk, \phi)=\sum_{\underline{i}:|\underline{i}|=n,int(TX_{i_{n}})=U_{k}\supset intX_{i_{1}}}\inf_{x\in X_{\underline{i}}}\exp[\sum_{h=0}^{n-1}\phi T^{h}(x)]$.

-invertible Markov system with finite
Lemma 2.1 $(fl 7f)$ Let $(T, X, Q)$ be a piecewise
range structure satisfying the transitivity. Let be a potential of $WBV$. For each,,exist and the limits does not depend on. Furthermore,$C^{0}$$U_{k}\in$$\phi$$\mathcal{U}$$\lim_{narrow\infty}\frac{1}{n}\log\overline{Z}_{n}(U_{k}, \phi)$$\lim_{narrow\infty}\frac{1}{n}\log\underline{Z}_{n}(U_{k}, \phi)$$k$$P_{top}(T, \phi):=\lim\underline{1}\log Z_{n}(X, \phi)$$narrow\infty n$$= \lim_{narrow\infty}\frac{1}{n}\log Z_{n}(U_{k}, \phi)=\lim_{narrow\infty}\frac{1}{n}\log\overline{Z}_{n}(U_{k}, \phi)=\lim_{narrow\infty}\frac{1}{n}\log\underline{Z}_{n}(U_{k}, \phi)$

,

where
$\log Z_{n}(X, \phi):=\sum_{\underline{i}:|\underline{i}|=n,int(TX_{i_{n}})\supset intX_{i_{1}}}\sum_{v_{\underline{i}}x=x\in d(intX_{\underline{i}})}\exp[\sum_{h=0}^{n-1}\phi T^{h}(x)]$.

We define$\mathcal{W}_{0}(T):=${$\phi$:$Xarrow \mathrm{R}|\phi$satisfies WBV and$P_{\mathrm{t}\mathrm{o}\mathrm{p}}(T,$Then we can easily see that the pressure function Ptop $(T$, . :nuity for bounded functions and convexity$)$$\phi)<\infty$}.

$\mathcal{W}_{0}(T)arrow \mathrm{R}$satisfies conti-

26
Weak Gibbs measures associated to potentials of
WBV

3

is called aweak Gibbs
Definition $([7],[11],[13],[15-19])$ ABorel probability measureofmeasure for afunction with aconstant P if there exists asequence
positive numbers with $\lim_{narrow\infty}(1/n)\log K_{n}=0$ such that -a.e.x,$\nu$$\{K_{n}\}_{n>0}$$\phi$$\nu$$K_{n}^{-1} \leq\frac{\nu(X_{i_{1}\ldots i_{n}}(x))}{\exp(\Sigma_{i=0}^{n-1}\phi T^{i}(x)+nP)}\leq K_{n}$where$X_{i_{1}\ldots i_{n}}(x)$,denotes the cylinder containing x.

Definition ABorel probability measure$\frac{d(\nu T)|\chi}{d\nu|_{X_{i}}}=f|_{X_{*}}.(\forall i\in I)$$\nu$

on $X$ is called a -conformal measure if$f$.

-invertible Markov system with $FRS$
Lemma 3.1 ([17]) Let $(T, X, Q)$ be a piecewise. Let
and be an $\exp[Ptop(T, \phi)-\phi]-$satisfying the transitivity andconformal measure. Then is a weak Gibbs measure for $\phi with-Ptop(T, \phi)$ .
$C^{0}$$intX\in \mathcal{U}$$\phi\in \mathcal{W}_{0}(T)$

$\nu$

$\nu$

For

$\phi:Xarrow \mathrm{R}$

we define the Ruelle-Perron-Probenius operator$\mathcal{L}_{\phi}$by$\mathcal{L}_{\phi}g(x)=\sum_{i\in I}\exp[\phi(v\dot{.}(x))]g(v:(x))(\forall g\in C(X),\forall x\in X)$.

Lemma 3.2 ([11],[13]) If there $\dot{m}tp>0$ and a Borel probability measure on $X$ satis, then is an $\exp[\log p-\phi]$ -conformal measure and $p=\exp[P_{top}(T. ’\phi)]$ .
fying
$e$

$\nu$

$\nu$

$\mathcal{L}_{\phi}^{*}\nu=p\nu$

4Indifferent periodic points associated to potentials
of WBV.

Lemma 4.1

$P_{top}(T, \phi)\geq\frac{1}{q}\Sigma_{h=0}^{q-1}\phi T^{h}(x_{0})(\forall x_{0}\in X,T^{q}x_{0}=x_{0})$is called an indifferent periodic point with period. If there exists ansure , then satisfies

Definition$x_{0}$$P_{\mathrm{t}\mathrm{o}\mathrm{p}}(T, \phi)=\frac{1}{q}\Sigma_{h=0}^{q-1}\phi T^{h}(x_{0})$

$\nu$

$q$

with respect to if-conformal mea$\phi$$\exp[P\mathrm{t}\mathrm{o}\mathrm{p}(T, \phi)-\phi]$$x_{0}$.

$\frac{d(\nu T^{q})}{d\nu}|_{\mathrm{x}_{:_{1}..:_{q}}(x_{\mathrm{O}})(x_{0})=\exp[qP_{\mathrm{t}\mathrm{o}\mathrm{p}}(T,\phi)-\sum_{h=0}^{q-1}\phi T^{h}(x_{0})]=1}$

If$x_{0}$

is not indifferent, then we call

$x_{0}$

arepelling per iodic point.

Proposition 4.1 ([16-17]) Let
be an indifferent periodic point with period. Let be an $\exp[Ptop(T, \phi)-\phi]$ -conformal measure. Thento$x_{0}$$\phi\in \mathcal{W}_{0}(T)$

(i)

(ii)

$\forall s\geq 1$$\nu$, $P_{top}(T, s\phi)=sP_{top}(T, \phi)$ and $\forall s<1$ , PtOp(T,$\nu(X_{i_{1}\ldots i_{n}}(x_{0}))$$q$

decays subexponentially fast$s\phi$) $\geq sP_{top}(T, \phi)$ .

with respect27
5Jump transformations. Define $B_{1}:=\{X_{i}\in Q:X_{i}\subset$
Let be asubset of the index set I and let
and for each $n>1B_{n}:= \{X_{i_{1}\ldots i_{n}}\in\bigvee_{i=0}^{n-1}T^{-i}Q$ : $X_{i_{k}}\subset B_{1}^{c}(k=1, \ldots, n-1)$ ,. Define afunction $R:Xarrow \mathrm{N}\cup\{\infty\}$ by $R(x)= \inf\{n\geq 0:T^{n}x\in B_{1}\}+1$ . Then we
see that $B_{n}:= \{x\in X|R(x)=n\}=\bigcup_{X_{i_{1}}}.i_{n}\in B_{n}Xi_{1}\ldots i_{n}$ and $D_{n}:=\{x\in X|R(x)>n\}=$by $T^{*}x=T^{R(x)}x$ .
:. Now we define the jump transformation.
andWe denote-invertible Markov
is apiecewise
Then it is easy to see that$:B_{n+1}=D_{n}\capT^{-n}B_{1}$is valid for $n\geq 1$ . Let
system with FRS and the property (1). We assume further the next:
be apotential of WBV withcondition :$J$$B_{1}= \bigcup_{i\in J}X_{i}$$X_{i_{n}}\subset$$B_{1}\}$$B_{1}\}$$T^{*}$$\bigcap_{i=0}^{n}T^{-i}B_{1}^{c}$$\bigcup_{n=1}^{\infty}B_{n}arrow X$$I^{*}:= \bigcup_{n\geq 1}\{(i_{1}\ldots i_{n})\in I^{n} :$X^{*}:=X \backslash (\bigcup_{i=0}^{\infty}T^{*-i}(\bigcap_{n\geq 0}D_{n}))$$C^{0}$$(T^{*}, X^{*}, Q^{*}=\{X_{\underline{i}}\}_{\underline{i}\in I}*)$$\phi$$Xarrow \mathrm{R}$

X_{i_{1}\ldots i_{n}}\subseteq B_{n}\}$$P_{\mathrm{t}\mathrm{o}\mathrm{p}}(T, \phi)<\infty$(Local Bounded Distortion)that$\exists\theta>0$

and$\forall X_{i_{1}\ldots i_{n}}\in B_{n}$, $\exists 0<L_{\phi}(i_{1}\ldots i_{n})<\infty$ such$|\phi v_{i_{1}\ldots i_{n}}(x)-\phi v_{i_{1}\ldots i_{n}}(y)|\leq L_{\phi}(i_{1}\ldots i_{n})d(x, y)^{\theta}$

and$\sup_{n\geq 1X_{i_{1}}}.\sup_{\in i_{n}B_{n}}\sum_{j=0}^{n-1}L_{\phi}(i_{j+1}\ldots i_{n})<\infty$.

by $\phi^{*}(x)=\sum_{i=0}^{R(x)-1}\phi T^{i}(x)$ and denote the local inverses tois afamily of equi-H\"older continuous functions and if
by . Thensatisfies summability of variation.
satisfies the next property thenDefine$\phi^{*}$:$\bigcup_{n=1}^{\infty}B_{n}arrow \mathrm{R}$$T^{*}$$\{\phi^{*}v_{\underline{i}}\}$$T^{*}|_{X_{\underline{i}}}(\underline{i}.\in I^{*})$$v_{\underline{i}}$$\phi^{*}$(Exponential Instability)$narrow\infty$$\sigma^{*}(n):=\sup_{\underline{i}\in I^{*}:|\underline{i}|=n}diamX_{\underline{i}}$decays exponentially fast as.

The summable variation allows one to show the existence of an unique equilibrium Gibbs-conformal measure on $X$
forunder the existence of anstate. The following formula gives a -invariant cr-finite
with $\nu(\bigcap_{n\geq 0}D_{n})=0$ and.
measure$\mu^{*}$$\exp[P\mathrm{t}\mathrm{o}\mathrm{p}(T, \phi)-\phi]$$\phi^{*}$

$\nu$

$T$

$\mu^{*}\sim\nu|_{X^{*}}$$\mu\sim\nu$(2) :$\mu(E)=\sum_{n=0}^{\infty}\mu^{*}(D_{n}\cap T^{-n}E)$.

, then is finite. In particular, $\mu(B_{1})=\mu^{*}(X^{*})>0$ , since $\nu(X^{*})=1$ .
If
If the reference measure is ergodic, then both , are ergodic, too.
$\Sigma_{n=0}^{\infty}\nu(D_{n})<\infty$

$\mu$

$\nu$

$\mu$

$\mu^{*}$

Theorem 5.1 (A construction of conformal measures) ([17]) Let $(T, X, Q)$ be a piecebe the jumpwise
-invertible Markov system with $FRS$ satisfying transitivity. Lettransfor mation associated to a union of full cylinders of rank 1which satisfies exponential
instability. Let :be a potential of $WBV$ satisfying (LBD), PtOp(T, ) $<\infty$ and$C^{0}$$T^{*}$$\phi$$Xarrow \mathrm{R}$$\phi$

28.

. Suppose either
0 orexists a Borel probability measure&on X supported on X’satisfying$||\mathrm{i}_{+}*1||<\mathrm{o}\mathrm{o}$$P_{top}(\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\rangle j^{\ovalbox{\tt\small REJECT}})\ovalbox{\tt\small REJECT}$$\mathrm{I}||<\mathrm{o}\mathrm{o}$$||\ovalbox{\tt\small REJECT} \mathrm{C}(\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} Ptop(r_{\ovalbox{\tt\small REJECT}}=,p=))$,then there$\frac{d\nu T}{d\nu}|_{X:}=\exp[P_{top}(T, \phi)-\phi](\forall i\in I)$

and$\nu(\bigcup_{:\in I}\partial X\dot{.})=0$.

We can associate the indifferent periodic points
sets.

$x_{0}$

with respect to$\phi$

to the Marginal$\bigcap_{n\geq 0}D_{n}$Proposition 5.1 ([17])(i) (Failure of bounded distortion)$C_{nq}(x_{0}):=.

monotonically as n

\cdot\sup_{x,y\in X_{1\cdots\cdot nq(x_{0})}}.\frac{\exp[\Sigma_{=0}^{nq-1}\phi\dot{T}(x)]}{\exp[\Sigma_{=0}^{nq-1}\phi T^{1}(y)]}..\cdot.\cdotarrow\infty$$arrow\infty$.

(ii) (Singularity of the invariant density)$x_{0} \in\bigcap_{n\geq 0}D_{n}$For a -invariant probability measure m on (X,tion of Q with respect to.
$T$$\mathcal{F})$,$I_{m}$

and $pdd\nu(x_{0})=\infty$ .

denotes the conditional informa-$T^{-1}\mathcal{F}$

Theorem 5.2 (Variational principle)
Let be the $\exp$ [Ptop(T,] -conformmeasure obtained under assumptions in Theorem 5.1. We assume further that
$Rd\nu<\infty$ and $H_{\nu}(Q^{*})<\infty$ ,consists of periodic points. Ifthen there eists
a -invariant ergodic probability measure equivalent to which satisfies the following
variational principle.
$([\mathit{1}7J)$$\bigcap_{n\geq 0}D_{n}$$\int_{X}$

$T$

$al$

$\phi)-\phi$$\nu$$\Gamma:=$

.

$\nu$

$\mu$

$P_{top}(T, \phi)=h_{\mu}(T)+\int_{X}\phi d\mu\geq h_{m}(T)+\int_{X}\phi dm$

for all-invariant ergodic probability measure $m$ on $X$ with $I_{m}+\phi\in L^{1}(m)$ satisfying$h_{m}(T)<\infty$ or $\int_{X}\phi dm>-\infty$ .
$T$

Corollary 5.1 (Phase transition) We assume all conditions in Theorem 5.2. If consists of indifferent periodic points with respect to , then the set of equilibrium states foris the convex hull of and the set of invariant Borel probability measures supported on .
$\Gamma$$\phi$$\mu$$\phi$$\Gamma$29
6Slow decay of correlations
We denoteand lettransfer operator with respect to , i.e.,$v_{i_{1}\ldots i_{n}}’(x)= \frac{d(\mu v_{i_{1}.i_{n}})}{d\mu}(x)$$P_{\mu}$: $L^{1}(\mu)arrow L^{1}(\mu)$ be the normalized

$\mu$

.

$P_{\mu}f(x)= \sum_{i\in I}v_{i}’(x)f(\psi_{i}(x))1_{TX_{i}}(x)(\forall f\in L^{1}(\mu))$In this section, we shall establish bounds on the -convergence of iterated transfer operatorsand bounds on the decay of correlations relative to bounded functions
satisfying aweak Lipschitz-type condition defined by :$L^{1}$$f$$\{P_{\mu}^{n}\}_{n\geq 1}$(6-1) $\exists 0<L_{f}<\infty$ such that$\dot{.}\sup_{X_{(m)}\subset D_{m}^{c}}\sup_{x,y\in X_{(m)}}.|f(x)-f(y)|\leq L_{f}\sigma(m)(\forall m>0)$

under the following conditions.
(6-2)$\Delta_{1}(k):=\sup_{n\geq 1}\sup_{i(n)\in A_{n}}\sup_{X_{j(k)}\subset D_{k}^{c}}\sup_{x,y\in X_{j(k)}}|1-\dot{.}\frac{\psi_{(n)}’(x)}{\psi_{i(n)}’(y)}|arrow 0$(6-3)$\Delta_{2}(k):=\sup_{X_{j(k)}\subset D_{k}^{\mathrm{C}}}\sup_{x,y\in X_{j(k)}}|1-\frac{(d\mu/d\nu)(x)}{(d\mu/d\nu)(y)}|arrow 0$

asHere $\sigma(m):=\sup_{i(m)\in A_{m}}diamX_{i(m)}$ , $i(m)$ denotes asequencedenotes.

$karrow\infty$

as k$arrow\infty$.

.

$i_{1}\ldots$ $i_{m}$of length $m$ and$D_{m}^{c}$$X\backslash D_{m}$Remark (1) If
is Holder continuous (with exponent 6),
can be boundedfrom above by $O(\Delta_{1}(m))+O(\sigma(m)^{\theta})$ . For all examples, we can easily estimate bothand.
$d\mu^{*}/d\nu$$\Delta_{2}(m)$$\sigma(m)$$\Delta_{1}(m)$Remark (2) The condition (6-1) is milder than the usual Lipschitz condition. For example, forfor any $0<\delta<\beta$ is anon-Lipschitz
unbounded function satisfying (6-1).
$S_{\beta}(x)=x+x^{1+\beta}(\mathrm{m}\mathrm{o}\mathrm{d}1)f(x)=x^{-\delta}$

We denote $\Delta(k):=\max_{i=1,2}\Delta_{i}(k)$ .
Theorem 6.1 (Polynomial bounds) Let $(T, X, Q)$ be a piecervise -invertible Bernoulli
system and let and be the probability measures obtained in Theorems 5.1 and 5.2 respectively. Suppose that (6-2) and (6-3) are satisfied. Assume further that all,
anddecay polynomially fast. Thensatisfying (6-1) we have the following
results.
$C^{0}$

$\nu$

$\mu$

$\mu(D_{n})$$\sigma(n)$l.(Rates of$\forall f\in L^{\infty}(\mu)$$L^{1}$-convergence of$\{P_{\mu}^{n}f\}_{n\geq 1}$)$\forall n\geq 1$

and$\forall 0<\epsilon<1$$||P_{\mu}^{n}f- \int_{X}fd\mu||_{1}\leq\max\{O(\mu(D_{[n^{\epsilon}]})), O(\Delta([n^{\epsilon}])), O(\sigma(2[n^{\epsilon}]))\}$.

$\Delta(n)$302.(Decay of correlations)$\forall g\in L^{\infty}(\mu)$and$\forall 0<\epsilon<1$.

$| \int_{X}f(gT^{n})d\mu-\int_{X}fd\mu\int_{X}gd\mu|\leq\max\{O(\mu(D[n^{\epsilon}])), O(\Delta([n^{\epsilon}])), O(\sigma(2[n^{\epsilon}]))\}$The next result gives sufficient conditions for (6-2).
Lemma 6.1 Suppose that
respectively. Then,and Vx,$\theta_{1}$$\{\phi\psi_{i}\}_{i\in I}$,$\forall X_{\dot{\iota}_{1}\ldots i_{m}}\subset D_{m}^{c}$$\theta_{2}$$X_{j_{k+1}\ldots j_{n}}\subset.D_{n-k}$$\mathrm{y}\in X$

are equi-H\"older continuous with exponents
andsuch thatand$\{\phi^{*}\psi_{\underline{i}}^{*}\}_{\underline{i}\in I}*$$\forall(j_{1}\ldots j_{n})\in A_{n}$$X_{j_{k}}\subset B_{1}$we have$|1- \frac{\psi_{j_{1}\ldots j_{\mathfrak{n}}}’(\psi_{\dot{1}1\cdots\dot{\cdot}m}x)}{\psi_{j_{1}\ldots j_{n}}’(\psi_{\dot{\iota}_{1}\cdots\dot{l}_{m}}y)}|$$\leq\max${ 0 (a$(m+n-k)^{\theta_{2}}$),$o( \sum_{:=[m/2]}^{\infty}\sup_{:}\dot{.}\{diamX_{\iota_{1}\ldots\iota_{:}}\}^{\theta_{1}}),$$O( \sigma([\frac{m}{2}])^{\theta_{2}})$}$X_{l_{1}\ldots l}\subset B^{\cdot}$

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