%\input /home/dario/formati.tex \def\mela{\relax} \magnification1200 \hsize 14cm %\input /home/bambusi/macrodario.tex %file macrodario.tex \catcode`@=11 % %------------------------- comandi riservati --------------------------- % \def\b@lank{ } \newif\if@simboli \newif\if@riferimenti \newif\if@bozze \newif\if@data \def\bozze{\@bozzetrue \immediate\write16{!!! INSERISCE NOME EQUAZIONI !!!}} \newwrite\file@simboli \def\simboli{ \immediate\write16{ !!! Genera il file \jobname.SMB } \@simbolitrue\immediate\openout\file@simboli=\jobname.smb \immediate\write\file@simboli{Simboli di \jobname}} \newwrite\file@ausiliario \def\riferimentifuturi{ \immediate\write16{ !!! Genera il file \jobname.aux } \@riferimentitrue\openin1 \jobname.aux \ifeof1\relax\else\closein1\relax\input\jobname.aux\fi \immediate\openout\file@ausiliario=\jobname.aux} \newcount\eq@num\global\eq@num=0 \newcount\sect@num\global\sect@num=0 \newcount\para@num\global\para@num=0 \newcount\const@num\global\const@num=0 \newcount\lemm@num\global\lemm@num=0 \newif\if@ndoppia \def\numerazionedoppia{\@ndoppiatrue\gdef\la@sezionecorrente{\the\sect@num}} \def\se@indefinito#1{\expandafter\ifx\csname#1\endcsname\relax} \def\spo@glia#1>{} % si applica a \meaning\xxxxx; butta via tutto quello % che produce \meaning fino al carattere > % (v. manuale TeX, pag. 382, \strip#1>{}). \newif\if@primasezione \@primasezionetrue \def\s@ection#1\par{\immediate \write16{#1}\if@primasezione\global\@primasezionefalse\else\goodbreak \vskip\spaziosoprasez\fi\noindent {\bf#1}\nobreak\vskip\spaziosottosez\nobreak\noindent} %--------------------------- Indice ------------------------------- \newif\if@indice \newif\if@ceindice \newwrite\file@indice \def\indice{ \immediate\write16{Genera il file \jobname.ind} \@indicetrue \immediate\openin2 \jobname.ind \ifeof2\relax\else \closein2\relax \@ceindicetrue\fi \if@ceindice\relax\else \immediate\openout\file@indice=\jobname.ind \immediate\write \file@indice{\string\vskip5pt \string{ \string\bf \string\centerline\string{ Indice \string}\string}\string\par} \fi } \def\quiindice{\if@ceindice\vfill\eject\input\jobname.ind\else\vfill\eject \immediate\write\file@indice{\string{\string\bf\string~ Indice\string}\string\hfill\folio} \null\vfill\eject\null\vfill\eject\relax\fi} % %------------------------------ a disp. dell'utente: sezioni ------------- \def\sezpreset#1{\global\sect@num=#1 \immediate\write16{ !!! sez-preset = #1 } } \def\spaziosoprasez{50pt plus 60pt} \def\spaziosottosez{15pt} \def\sref#1{\se@indefinito{@s@#1}\immediate\write16{ ??? \string\sref{#1} non definita !!!} \expandafter\xdef\csname @s@#1\endcsname{??}\fi\csname @s@#1\endcsname} \def\autosez#1#2\par{ \global\advance\sect@num by 1\if@ndoppia\global\eq@num=0\fi \global\lemm@num=0 \global\para@num=0 \xdef\la@sezionecorrente{\the\sect@num} \def\usa@getta{1}\se@indefinito{@s@#1}\def\usa@getta{2}\fi \expandafter\ifx\csname @s@#1\endcsname\la@sezionecorrente\def \usa@getta{2}\fi \ifodd\usa@getta\immediate\write16 { ??? possibili riferimenti errati a \string\sref{#1} !!!}\fi \expandafter\xdef\csname @s@#1\endcsname{\la@sezionecorrente} \immediate\write16{\la@sezionecorrente. #2} \if@simboli \immediate\write\file@simboli{ }\immediate\write\file@simboli{ } \immediate\write\file@simboli{ Sezione \la@sezionecorrente : sref. #1} \immediate\write\file@simboli{ } \fi \if@riferimenti \immediate\write\file@ausiliario{\string\expandafter\string\edef \string\csname\b@lank @s@#1\string\endcsname{\la@sezionecorrente}}\fi \goodbreak\vskip 48pt plus 60pt \noindent{\bf\the\sect@num.\quad #2} \if@bozze {\tt #1}\fi \par\nobreak\vskip 15pt \nobreak} \def\blankii{\blank\blank} \def\destra#1{{\hfill#1}} \font\titfnt=cmssbx10 scaled \magstep2 \font\capfnt=cmss17 scaled \magstep4 \def\blank{\vskip 12pt} \def\capitolo#1#2\par{ \global\advance\sect@num by 1\if@ndoppia\global\eq@num=0\fi \global\lemm@num=0 \global\para@num=0 \xdef\la@sezionecorrente{\the\sect@num} \def\usa@getta{1}\se@indefinito{@s@#1}\def\usa@getta{2}\fi \expandafter\ifx\csname @s@#1\endcsname\la@sezionecorrente\def \usa@getta{2}\fi \ifodd\usa@getta\immediate\write16 { ??? possibili riferimenti errati a \string\sref{#1} !!!}\fi \expandafter\xdef\csname @s@#1\endcsname{\la@sezionecorrente} \immediate\write16{\la@sezionecorrente. #2} \if@simboli \immediate\write\file@simboli{ }\immediate\write\file@simboli{ } \immediate\write\file@simboli{ Sezione \la@sezionecorrente : sref. #1} \immediate\write\file@simboli{ } \fi \if@riferimenti \immediate\write\file@ausiliario{\string\expandafter\string\edef \string\csname\b@lank @s@#1\string\endcsname{\la@sezionecorrente}}\fi \par\vfill\eject \destra{\capfnt {\la@sezionecorrente}\hbox to 10pt{\hfil}} \blankii\noindent{\titfnt\baselineskip=20pt \hfill\uppercase{#2}}\blankii \if@indice \if@ceindice\relax\else\immediate\write \file@indice{\string\vskip5pt\string{\string\bf \la@sezionecorrente.#2\string}\string\hfill\folio\string\par}\fi\fi \if@bozze {\tt #1}\par\fi\nobreak} \def\semiautosez#1#2\par{ \gdef\la@sezionecorrente{#1}\if@ndoppia\global\eq@num=0 \fi \global\lemm@num=0 \global\para@num=0 \if@simboli \immediate\write\file@simboli{ }\immediate\write\file@simboli{ } \immediate\write\file@simboli{ Sezione ** : sref. \expandafter\spo@glia\meaning\la@sezionecorrente} \immediate\write\file@simboli{ }\fi \s@ection#2\par} %------------------paragrafi---------------------------------------- \def\pararef#1{\se@indefinito{@ap@#1} \immediate\write16{??? \string\pararef{#1} non definito !!!} \expandafter\xdef\csname @ap@#1\endcsname {#1} \fi\csname @ap@#1\endcsname} \def\autopara#1#2\par{ \global\advance\para@num by 1 \xdef\il@paragrafo{\la@sezionecorrente.\the\para@num} \vskip10pt \noindent {\bf \il@paragrafo\ #2} \def\usa@getta{1}\se@indefinito{@ap@#1}\def\usa@getta{2}\fi \expandafter\ifx\csname @ap@#1\endcsname\il@paragrafo\def\usa@getta{2}\fi \ifodd\usa@getta\immediate\write16 {??? possibili riferimenti errati a \string\pararef{#1} !!!}\fi \expandafter\xdef\csname @ap@#1\endcsname{\il@paragrafo} \def\usa@getta{\expandafter\spo@glia\meaning \la@sezionecorrente.\the\para@num} \if@simboli \immediate\write\file@simboli{ }\immediate\write\file@simboli{ } \immediate\write\file@simboli{ paragrafo \il@paragrafo : pararef. #1} \immediate\write\file@simboli{ } \fi \if@riferimenti \immediate\write\file@ausiliario{\string\expandafter\string\edef \string\csname\b@lank @ap@#1\string\endcsname{\il@paragrafo}}\fi \if@indice \if@ceindice\relax\else\immediate\write \file@indice{\string\noindent\string\item\string{ \il@paragrafo.\string}#2\string\dotfill\folio\string\par}\fi\fi \if@bozze {\tt #1}\fi\par\nobreak\vskip .3 cm \nobreak} %------------------------------ a disp. dell'utente: equazioni ----------- \def\eqpreset#1{\global\eq@num=#1 \immediate\write16{ !!! eq-preset = #1 } } \def\eqlabel#1{\global\advance\eq@num by 1 \if@ndoppia\xdef\il@numero{\la@sezionecorrente.\the\eq@num} \else\xdef\il@numero{\the\eq@num}\fi \def\usa@getta{1}\se@indefinito{@eq@#1}\def\usa@getta{2}\fi \expandafter\ifx\csname @eq@#1\endcsname\il@numero\def\usa@getta{2}\fi \ifodd\usa@getta\immediate\write16 { ??? possibili riferimenti errati a \string\eqref{#1} !!!}\fi \expandafter\xdef\csname @eq@#1\endcsname{\il@numero} \if@ndoppia \def\usa@getta{\expandafter\spo@glia\meaning \il@numero} \else\def\usa@getta{\il@numero}\fi \if@simboli \immediate\write\file@simboli{ Equazione \usa@getta : eqref. #1}\fi \if@riferimenti \immediate\write\file@ausiliario{\string\expandafter\string\edef \string\csname\b@lank @eq@#1\string\endcsname{\usa@getta}}\fi} \def\eqsref#1{\se@indefinito{@eq@#1} \immediate\write16{ ??? \string\eqref{#1} non definita !!!} \if@riferimenti\relax \else\eqlabel{#1} ???\fi \fi\csname @eq@#1\endcsname } \def\autoeqno#1{\eqlabel{#1}\eqno(\csname @eq@#1\endcsname)\if@bozze {\tt #1}\else\relax\fi} \def\autoleqno#1{\eqlabel{#1}\leqno(\csname @eq@#1\endcsname)} \def\eqref#1{(\eqsref{#1})} %----------- Lemmi automatici: a disposizione dell'utente ---------------- \def\lemmalabel#1{\global\advance\lemm@num by 1 \xdef\il@lemma{\la@sezionecorrente.\the\lemm@num} \def\usa@getta{1}\se@indefinito{@lm@#1}\def\usa@getta{2}\fi \expandafter\ifx\csname @lm@#1\endcsname\il@lemma\def\usa@getta{2}\fi \ifodd\usa@getta\immediate\write16 { ??? possibili riferimenti errati a \string\lemmaref{#1} !!!}\fi \expandafter\xdef\csname @lm@#1\endcsname{\il@lemma} \def\usa@getta{\expandafter\spo@glia\meaning \la@sezionecorrente.\the\lemm@num} \if@simboli \immediate\write\file@simboli{ Lemma \usa@getta : lemmaref #1}\fi \if@riferimenti \immediate\write\file@ausiliario{\string\expandafter\string\edef \string\csname\b@lank @lm@#1\string\endcsname{\usa@getta}}\fi} \def\autolemma#1{\lemmalabel{#1}\csname @lm@#1\endcsname\if@bozze {\tt #1}\else\relax\fi} \def\lemmaref#1{\se@indefinito{@lm@#1} \immediate\write16{ ??? \string\lemmaref{#1} non definita !!!} \if@riferimenti\else \lemmalabel{#1}???\fi \fi\csname @lm@#1\endcsname} %--------------- bibliografia automatica: riservati ---------------------- \newcount\cit@num\global\cit@num=0 \newwrite\file@bibliografia \newif\if@bibliografia \@bibliografiafalse \newif\if@corsivo \@corsivofalse \def\title#1{{\it #1}} \def\rivista#1{#1} \def\lp@cite{[} \def\rp@cite{]} \def\trap@cite#1{\lp@cite #1\rp@cite} \def\lp@bibl{[} \def\rp@bibl{]} \def\trap@bibl#1{\lp@bibl #1\rp@bibl} \def\refe@renza#1{\if@bibliografia\immediate % scrive su .BIB \write\file@bibliografia{ \string\item{\trap@bibl{\cref{#1}}}\string \bibl@ref{#1}\string\bibl@skip}\fi} \def\ref@ridefinita#1{\if@bibliografia\immediate\write\file@bibliografia{ \string\item{?? \trap@bibl{\cref{#1}}} ??? tentativo di ridefinire la citazione #1 !!! \string\bibl@skip}\fi} \def\bibl@ref#1{\se@indefinito{@ref@#1}\immediate \write16{ ??? biblitem #1 indefinito !!!}\expandafter\xdef \csname @ref@#1\endcsname{ ??}\fi\csname @ref@#1\endcsname} \def\c@label#1{\global\advance\cit@num by 1\xdef % assegna il numero \la@citazione{\the\cit@num}\expandafter \xdef\csname @c@#1\endcsname{\la@citazione}} \def\bibl@skip{\vskip 5truept} %------------------------ bibl. automatica: a disp. dell'utente ------------ \def\stileincite#1#2{\global\def\lp@cite{#1}\global \def\rp@cite{#2}} \def\stileinbibl#1#2{\global\def\lp@bibl{#1}\global \def\rp@bibl{#2}} \def\corsivo{\global\@corsivotrue} \def\citpreset#1{\global\cit@num=#1 \immediate\write16{ !!! cit-preset = #1 } } \def\autobibliografia{\global\@bibliografiatrue\immediate \write16{ !!! Genera il file \jobname.BIB}\immediate \openout\file@bibliografia=\jobname.bib} \def\cref#1{\se@indefinito % se indefinito definisce {@c@#1}\c@label{#1}\refe@renza{#1}\fi\csname @c@#1\endcsname} \def\upcref#1{\null$^{\,\cref{#1}}$} \def\cite#1{\trap@cite{\cref{#1}}} % [5] \def\ccite#1#2{\trap@cite{\cref{#1},\cref{#2}}} % [5,6] \def\ncite#1#2{\trap@cite{\cref{#1}--\cref{#2}}} % [5-8] senza definire \def\upcite#1{$^{\,\trap@cite{\cref{#1}}}$} % ^[5] \def\upccite#1#2{$^{\,\trap@cite{\cref{#1},\cref{#2}}}$} % ^[5,6] \def\upncite#1#2{$^{\,\trap@cite{\cref{#1}-\cref{#2}}}$} % ^[5-8] senza def. \def\clabel#1{\se@indefinito{@c@#1}\c@label % sola definizione {#1}\refe@renza{#1}\else\c@label{#1}\ref@ridefinita{#1}\fi} \def\cclabel#1#2{\clabel{#1}\clabel{#2}} % def. doppia \def\ccclabel#1#2#3{\clabel{#1}\clabel{#2}\clabel{#3}} % def. tripla \def\biblskip#1{\def\bibl@skip{\vskip #1}} % spaziatura nella bibl. \def\insertbibliografia{\if@bibliografia % scrive la bibliografia \immediate\write\file@bibliografia{ } \immediate\closeout\file@bibliografia \if@indice \if@ceindice\relax\else\immediate\write \file@indice{\string\vskip5pt\string{\string\bf\string~ Bibliografia\string}\string\hfill\folio\string\par}\fi\fi \catcode`@=11\input\jobname.bib\catcode`@=12\fi } %--------- per comporre il file con la bibliografia -------------- \def\commento#1{\relax} \def\biblitem#1#2\par{\expandafter\xdef\csname @ref@#1\endcsname{#2}} % ricordare: una lista in chiaro della bibliografia si % ottiene eseguendo $ TEX BIBLIST %---------------- titolo in cima alla pagina, data.----------------- \def\data{\number\day.\number\month.\number\year} \def\datasotto{\@datatrue \footline={\hfil{\rm \data}\hfil}} \def\titoli#1{\if@data\relax\else\footline={\hfil}\fi \xdef\prima@riga{#1}\voffset+20pt \headline={\ifnum\pageno=1 {\hfil}\else\hfil{\sl \prima@riga}\hfil\folio\fi}} \def\duetitoli#1#2{\if@data\relax\else\footline={\hfil}\fi \voffset=+20pt \headline={\ifnum\pageno=1 {\hfil}\else{\ifodd\pageno\hfil{\sl #2}\hfil\folio \else\folio\hfil{\sl #1}\hfil\fi} \fi} } \def\la@sezionecorrente{0} % ------------------COSTANTI --------------------------------- \def\const@label#1{\global\advance\const@num by 1\xdef \la@costante{\the\const@num}\expandafter \xdef\csname @const@#1\endcsname{\la@costante}} \def\cconlabel#1{\se@indefinito{@const@#1} \const@label{#1}\fi} \def\constnum#1{\se@indefinito{@const@#1} \const@label{#1}\fi\csname @const@#1\endcsname} \def\ccon#1{C_{\constnum{#1}}} \catcode`@=12 %------------------ FORMATI TEOREMI E GENERALI -------------------- \def\abstract{ \vskip48pt plus 60pt \noindent {\bf Abstract.}\quad} \def\summary{\vskip48pt plus 60pt \noindent {\bf Summary.}\quad} \def\firma{\noindent \centerline{Dario BAMBUSI}\par\noindent \centerline{Dipartimento di Matematica dell'Universit\`a,}\par\noindent \centerline{Via Saldini 50, 20133 Milano, Italy.}\par} \def\teorema#1#2{\par\vskip4pt \noindent {\bf Teorema \autolemma{#1}.}{\sl \ #2} \par\vskip10pt} \def\esempio#1#2{\par\vskip4pt \noindent {\bf Esempio \autolemma{#1}.}{\sl \ #2} \par\vskip10pt} \def\definizione#1#2{\par\vskip4pt \noindent {\bf Definizione \autolemma{#1}.}{\sl \ #2} \par\vskip10pt} \def\dimostrazione{\par\noindent{\bf Dimostrazione.}\ } \def\theorem#1#2{\par\vskip4pt \noindent {\bf Theorem \autolemma{#1}.}{\sl \ #2} \par\vskip10pt} \def\semitheorem#1{\par\vskip4pt \noindent {\bf Theorem.}{\sl \ #1} \par\vskip10pt} \def\lemma#1#2{\par\vskip4pt \noindent {\bf Lemma \autolemma{#1}.}{\sl \ #2} \par\vskip4pt} \def\proof{\par\noindent{\bf Proof.}\ } \def\proposizione#1#2{\par\vskip4pt \noindent {\bf Proposizione \autolemma{#1}.}{\sl \ #2} \par\vskip10pt} \def\proposition#1#2{\par\vskip4pt \noindent {\bf Proposition \autolemma{#1}.}{\sl \ #2} \par\vskip10pt} \def\corollario#1#2{\par\vskip4pt \noindent {\bf Corollario \autolemma{#1}.}{\sl \ #2} \par\vskip10pt} \def\osservazione#1#2{\par\vskip4pt \noindent {\bf Osservazione \autolemma{#1}.}{\sl \ #2} \par\vskip10pt} \def\corollary#1#2{\par\vskip4pt \noindent {\bf Corollary \autolemma{#1}.}{\sl \ #2} \par\vskip10pt} \def\remark#1#2{\par\vskip4pt \noindent {\bf Remark \autolemma{#1}.}{\sl \ #2} \par\vskip4pt} \def\definition#1{\par\vskip2pt \noindent {\bf Definition.}{\sl \ #1} \par\vskip2pt} \def\definitionn#1#2{\par\vskip4pt \noindent {\bf Definition \autolemma{#1}.}{\sl \ #2} \par\vskip4pt} %------------------- ROUTINE DI USO GENERALE ----------------------- \def\norma#1{\left\Vert#1\right\Vert} \def\perogni{\forall\hskip1pt} \def\meno{\hskip1pt\backslash} \def\frac#1#2{{#1\over #2}} \def\fraz#1#2{{#1\over #2}} \def\interno{\vbox{\hbox{\vbox to .3 truecm{\vfill\hbox to .2 truecm {\hfill\hfill}\vfill}\vrule}\hrule}\hskip 2pt} \def\quadratino{ \hfill\vbox{\hrule\hbox{\vrule\vbox to 7 pt {\vfill\hbox to 7 pt {\hfill\hfill}\vfill}\vrule}\hrule}\par} \font\strana=cmti10 \def\lie{\hbox{\strana \char'44}} \def\ponesotto#1\su#2{\mathrel{\mathop{\kern0pt #1}\limits_{#2}}} \def\Sup{\mathop{{\rm Sup}}} %\def\Sup#1{\hskip2pt\ponesotto{{\rm Sup}}\su{#1}} \def\tdot#1{\hskip2pt\ddot{\null}\hskip2.5pt \dot{\null}\kern -5pt {#1}} \def\diff#1#2{\frac{\partial #1}{\partial #2}} \def\base#1#2{\frac{\partial}{\partial#1^{#2}}} \def\charslash#1{\setbox2=\hbox{$#1$} \dimen2=\wd2 \setbox1=\hbox{/}\dimen1=\wd1 \ifdim\dimen2>\dimen1 \rlap{\hbox to \dimen2{\hfil /\hfil}} #1 \else \rlap{\hbox to \dimen1{\hfil$#1$\hfil}} / \hfil\fi} \def\Re{{\rm \kern 0.4ex I \kern -0.4 ex R}} \def\Sh{{\rm Sh}\hskip1pt} \def\Ch{{\rm Ch}\hskip1pt} \def\poisson#1#2{\left\{#1 ,#2\right\} } \def\toro{{\bf T}} \def\Na{{\bf N}} \def\ra{{\bf Z}} \def\Ra{{\bf Z}} \def\id{{\bf 1}} \def\Cm{{\bf C}} \def\reale{{\rm Re}\hskip2pt} \def\imma{{\rm Im}\hskip2pt} \def\rin{{\bf Z}} \def\pmb#1{\setbox0=\hbox{#1}\ignorespaces \hbox{\kern-.02em\copy0\kern-\wd0\ignorespaces \kern.05em\copy0\kern-\wd0\ignorespaces \kern-.02em\raise.02em\box0 }} \def\vett#1{\pmb{$#1$}} \def\A{{\cal A}} \def\B{{\cal B}} \def\C{{\cal C}} \def\D{{\cal D}} \def\E{{\cal E}} \def\F{{\cal F}} \def\G{{\cal G}} \def\H{{\cal H}} \def\I{{\cal I}} \def\L{{\cal L}} \def\M{{\cal M}} \def\N{{\cal N}} \def\O{{\cal O}} \def\P{{\cal P}} \def\Q{{\cal Q}} \def\R{{\cal R}} \def\S{{\cal S}} \def\T{{\cal T}} \def\U{{\cal U}} \def\V{{\cal V}} \def\W{{\cal W}} \def\Z{{\cal Z}} \def\J{{\cal J}} \def\sym{\nabla^\Omega} \def\uno{{\kern+.3em {\rm 1} \kern -.22em {\rm l}}} \def\unpo{\vskip3pt} \def\unp{\vskip6pt} \def\a{\`a\ } \def\o{\`o\ } \def\e{\`e\ } %------------------------------ F I N E --------------------------------- %\input /home/bambusi/dariobiblio.tex \commento{cmf\cmf} \commento{abbreviazioni} \def\cmp{Commun. Math. Phys.} \def\cmf{Commun. Math. Phys.} \def\gio{A. Giorgilli} \def\gal{L. Galgani} \def\bgg{G. Benettin, \gal, \gio} \def\zamp{J. Appl. Math. Phys. (ZAMP)} \def\catene{bam93a} \def\giro{bam91b} \def\breather{bam96} \def\adiabat{bam95b} \def\dipolouno{bam94??} \def\lia{bam91a} \def\dipolo{bam94?} \def\bamnek{bam96?} \riferimentifuturi \autobibliografia \datasotto \def\data{17.3.2003} \numerazionedoppia %\bozze \autobibliografia \def\im{{\rm i}} \def\es{{\rm e}} \def\Na{{\bf N}} \def\bna{\bar\Na\null^n} \def\bi{{\bf k}}\def\bk{{\bf k}}\def\bl{{\bf l}} \def\bt{{\bf t}} \def\bx{{x}} \def\bl{{\bf l}} \def\bji#1{j^{(#1)}} \def\bki#1{k^{(#1)}} \def\bii#1{i^{(#1)}} \def\bli#1{l^{(#1)}} \def\ells{\ell^2_{s_*}} \def\balpha{{\bf k}} \def\hi#1{\norma{\bii{#1}}} \def\hj#1{\norma{\bji {#1}}} \def\hjj#1{({\bji {#1}})\null} \def\lapla{\Delta} \def\bbi{\underline\bk} \def\der#1#2{\frac{d^{#1}\omega_{#2}}{dm^{#1}}} \def\bj{{\bf j}} \def\bbj{\underline\bj} \def\rass{\bar\ra}\def\rassn{\bar\ra^n} \def\sumras{\sum_{\bj\in\rassn}} \def\rs{R^{(s)}} \def\homega{H_0} \def\nbi#1{\norma{\bii{#1}}} \def\nbj#1{\norma{\bji {#1}}} \def\hatl#1{\hat{#1}\null^l} \def\birkhof{bam03} \def\nlw{bam98} \def\geoschro{bam97??} \biblitem{bou03}J. 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Kuksin: \title{An Averaging Theorem for Distributed Conservative Systems and its Application to Von Karman's Equation,} \rivista{PMM U.S.S.R.} {\bf 53}, 150-157 (1989). \biblitem{bam96?}D.~Bambusi, N.N. Nekhoroshev: \title{A property of exponential stability in nonlinear wave equation near the fundamental linear mode}. \rivista{Physica D} {\bf 122}, 73-104 (1998). \biblitem{bam97??}D.~Bambusi: \title{Nekhoroshev theorem for small amplitude solutions in nonlinear Schr\"odin\-ger equations.} \rivista{Math. Z.} {\bf 130}, 345--387, (1999). \biblitem{ben85} \bgg: \title{A Proof of Nekhoroshev's Theorem for the Stability Times in Nearly Integrable Hamiltonian Systems} \rivista{Cel. Mech.} {\bf 37}, 1-25, (1985). \biblitem{bam98}D.~Bambusi: \title{On long time stability in Hamiltonian perturbations of nonresonant linear PDE's,} \rivista{Nonlinearity} {\bf 12}, 823-850, (1999). \biblitem{you97a} J. Xu, J. You, Q. Qiu:\title{ Invariant tori for nearly integrable Hamiltonian systems with degeneracy} \rivista{Math. Z.} {\bf 226}, 375--387 (1997). \duetitoli{D.~Bambusi}{Averaging theory for Hamiltonian PDE's.} {\bf \centerline{AN AVERAGING THEOREM FOR QUASILINEAR HAMILTONIAN PDEs} } \vskip20pt \centerline{Dario BAMBUSI} \vskip20pt \centerline{Dipartimento di Matematica,} \centerline{Universit\`a degli studi di Milano} \centerline{ Via Saldini 50, 20133 MILANO, Italy.} \unp \abstract We study the dynamics of Hamiltonian quasilinear PDEs close to elliptic equilibria. Under a suitable nonresonance condition we prove an averaging theorem according to which any solution corresponding to small amplitude smooth initial data remain very close to a torus up to long times. An application to quasilinear wave equations in an $n$ dimensional paralleliped is given. \autosez{1}{Introduction} In this paper we study the dynamics of Hamiltonian partial differential equation of quasilinear type (in the sense of Kato [\cref{kat75},\cref{kat85}]). In particular we concentrate on small amplitude solutions. Assuming (a) that the frequencies of small oscillation fulfill a new nonresonance condition that is satisfied in very general cases and (b) a natural estimate on the Lyapunov exponents of the system, we will construct infinitely many approximate integrals of motion. We will deduce that solutions corresponding to ``smooth'' initial data remain $\O(\epsilon^{M})$ close to a finite dimensional torus up to times $\O(\epsilon^{-1})$, $\epsilon$ being the norm of the initial datum and $M$ an arbitrary integer. The abstract theorem is then applied to a quasilinear wave equation in an $n$-dimensional parallelepiped (see eq. \eqref{nlw} below). In particular we show that the abstract nonresonance condition is fulfilled for almost all values of the parameters and we use Kato's theory with the addition of quantitative estimates of the constants in order to verify the assumption on the Lyapunov exponent. \unp The proof of the abstract theorem is obtained in some steps. The first one consists in making a Galerkin cutoff, namely in approximating the infinite dimensional Hamiltonian system by a finite dimensional one. The second consists in applying to such a finite dimensional system a classical recursive algorithm [\cref{whi16},\cref{che24a},\cref{che24b},\cref{gioihp}] of construction of integrals of motion. It is well known that such an algorithm gives rise to divergent series, but truncating the construction at a given order one obtaines objects (the ``integrals'') having a small Poisson Brackets with the (finite dimensional) Hamiltonian. Thirdly we show that the Poisson Brackets of the ``integrals'' with the {complete infinite dimensional Hamiltonian} is very small in a small ball of a phase space of ``smooth'' functions. This is not enough to ensure that the integrals remain approximatively constant along the solutions; indeed one has still to ensure that the norm of the solution remains small up to the times we are interested in. This will be obtained by adding an assumption on the Lyapunov exponents of the system, an assumption which is automatic in semilinear models and that can be verified using Kato's theory in quasilinear models. The construction of the approximate integrals is quite delicate since one has to keep into account the dependence on the dimension of the truncated system of all the constants entering in the estimates. This is obtained by using a precise computation of the constants due to Giorgilli \cite{gioihp}. \unpo As far as we know, the main result of the present paper constitutes the first averaging type theorem applicable to {\it quasilinear} equations, namely equations in which the nonlinearity contains as many derivatives as the linear part. We also emphasize that this is important in view of physical applications like Magnetohidrodynamics or Elastodynamics, where the equations are quasilinear. The main limitation of the result of the present paper rests in its time of validity. Indeed known results for finite dimensional systems and for some semilinear partial differential equations (see [\cref{bou00}, \cref{\birkhof}, \cref{bou96a}, \cref{BG03}]) describe the dynamics over a time scale much longer than $O(\epsilon^{-1})$, precisely over a time scale of order $\epsilon^{-M}$ with arbitrary $M$. We point out that such results apply only to equations in which the frequencies of small oscillation fulfill a nonresonant condition stronger than that of the present paper (such a stronger condition is only exceptionally satisfied in more than one space dimensions). A further result applicable to quite general {\it semilinear} models in arbitrary space dimension was obtained by Kuksin\upcite{kuk90}, but the time scale covered by that paper is of order $O(\epsilon^{-1})$. Moreover Kuksin's result allows to describe only the dynamics corresponding to initial data in which the energy is initially essentially concentrated on a finite (and fixed) number of linear modes. Finally we recall the papers [\cref{ver87},\cref{pal96},\cref{kro89},\cref{mat02}] where Galerkin truncations have been used together with averaging techniques in order to study the dynamics of nonlinear PDEs. In particular the paper \cite{mat02} had a great influence on the present paper. Related results are also those of [\cref{sha85},\cref{cra96},\cref{bam96?},\cref{\geoschro}]. \unp Plan of the paper. In sect.~\sref{abs} we state the main results of the paper. In Sect.~\sref{wave} we give the application to the nonlinear wave equation. In sect.~\sref{for} we recall the algorithm of construction of the integrals of motion in finite dimensional Hamiltonian systems. In sect.~\sref{pa} we prove the main result of the paper. In sect.~\sref{mea} we show that the diophantine condition we use is fulfilled with probability one in the wave equation. In sect. \sref{app} we use Kato's theory to verify the assumptions on Lyapunov exponents in the case of nonlinear wave equation. Finally in the appendix we prove a result on the geometry of level surfaces of the approximate integrals. \noindent {\it Acknowledgement.} I thank Antonio Giorgilli for some interesting discussions and for explaining me the key ideas leading to the proof of proposition \lemmaref{nonsca}. I also thank Alberto Arosio for some discussions on Kato's theory. This work was developed under the partial support of Gruppo Nazionale di Fisica Matematica (INdAM). \autosez{abs}Abstract results \noindent{\sl \sref{abs}.1 The Approximate Integrals} \vskip\spaziosottosez Consider a Hamiltonian system of the form $$ H(p,q)=\sum_{j\geq 1}\omega_j\frac{p_j^2+q_j^2}{2}+P(p,q)\autoeqno{ham} $$ where $\{\omega_j\}$ is a sequence of strictly positive quantities and $(p_j,q_j)$ are conjugated variables. We will denote by $z\equiv\{(p_j,q_j)\}_{j\geq 1}$ a phase point and by $X_H(z)\equiv \left(-\diff{H}{q_j}(z),\diff{H}{p_j}(z)\right)$ the (formal) Hamiltonian vector field of a function $H$. To define precisely the phase space consider the Hilbert space $\ell^2_s$ of the sequences $\{x_j\}_{j\geq 1}$ such that $$ \norma{x}_s^2:=\sum_{j\geq1} j^{2s}x_j^2<\infty\ , $$ and denote $\P_{s}:=\ell_s^2\times\ell_s^2$. If $z\equiv (p,q)\in\P_s$ we will denote by $$ \norma{z}_s:=\sqrt{\norma{p}_s^2+\norma{q}_s^2}\ , $$ its norm and by $B_s(R)$ the open ball of radius $R$ in $\P_s$. For any positive (large) $N$ denote by $\omega^{(N)}:=(\omega_1,...,\omega_N)$ the truncation of length $N$ of the frequency vector. Having fixed a positive (large) $r_*$, we assume: \item{H1)} (Nonresonance) There exist $\alpha$ and $\gamma>0$ such that for any $N$ one has $$ |\omega^{(N)}\cdot k|\geq\frac{\gamma}{N^\alpha}\ ,\ \perogni k\in\ra^N\ ,\ 0<|k|\leq r_*+2 $$ where $|k|:=|k_1|+...+|k_N|$. \quadratino \noindent \remark{nonr}{Consider the case where $$ \omega_j=j^{d_1}+\frac{\sigma_j}{j^{d_2}}\ ,\quad \sigma_j\in [0,1]\ ,\quad d_1, d_2\geq 0 $$ one can show that H1 is fulfilled if the parameters $\sigma_j$ are chosen in a set of probability one with respect to the product measure.} To ensure that $$ H_0(p,q):=\sum_{j\geq 1}\omega_j\frac{p_j^2+q_j^2}{2}\autoeqno{H0} $$ describes the linearization of the system at the origin we assume \item{H2)} There exist $s_*$ and a neighbourhood of the origin $\U\subset \P_{s_*}$, a positive C and $\nu\geq 1$ such that $$ |P(z)|\leq C\norma{z}_{s_*}^{\nu+2}\ ,\quad\perogni z\in \U . $$ \quadratino \noindent Then we need some smoothness of the Hamiltonian and of its vector field, so we assume that \item{H3)} One has $H\in C^{r_*+3}(\U)$.\quadratino \item{H4)} There exists a real $d$ with the following property: for any positive $s\geq s_*$ there exists a neighbourhood of the origin $\U_{s+d}\subset\P_{s+d}$ such that $X_{H}\in C^{r_*+2}(\U_{s+d},\P_s)$. \quadratino \remark{ss}{Actually it is enough that the above assumptions are fulfilled for an infinite unbounded set of values of $s$. } \noindent In order to ensure that the finite dimensional algorithm of construction of the integrals of motion is applicable we assume that \item{H5)} The nonlinearity is reversible, namely one has $P(-p,q)=P(p,q)$. \quadratino \unp \noindent Denote the linear actions by $I_j:=(p_j^2+q_j^2)/2$. In the following we will always denote by $C$ a positive (usually large) constant whose value can change from line to line. \theorem{const}{Fix $M\geq 4$, and assume that H1-H5 hold with some $r_*>\frac{4}{3}M-3$ then there exist positive constants $s'$, $\epsilon_*$, and $\beta$ with the following properties: for any $0<\epsilon<\epsilon_*$ define $$ N:=[\epsilon^{-\frac1{4\beta}}]\ , $$ then \item{1)} there exist $N$ approximate integrals of motion $\{\Phi^{(j)}\}_{j=1}^N$, which are analytic in $B_{s_*}(\epsilon_*)$, and fulfill $$ \eqalign{ \sup_{z\in B_{s_*+s'}(\epsilon)}\left|\poisson{H}{\Phi^{(j)}}(z)\right|\leq C\epsilon^M \cr \sup_{z\in B_{s_*}(\epsilon)}\left|\Phi^{(j)}(z)-I_j(z) \right| \leq C\epsilon^2\epsilon^{1/4} }\autoeqno{sti.1} $$ \item{2)}For any $s\geq s_*$ there exits $\epsilon_s$ and $C_s$ such that one has $$ \sup_{z\in B_{s+s'}(\epsilon)}\left|\poisson{H}{I^{(s,N)}}(z)\right| \leq C_s\epsilon^M\ ,\quad\perogni \epsilon<\epsilon_s $$ where $I^{(s,N)}:=\sum_{j>N} j^{2s}I_j $. } \remark{rinfty}{In the $C^{\infty}$ completely nonresonant case $M$ can be chosen arbitrarily large; then the quantity $s'$ turns out to be $M$ dependent.} If we assume also local well posedness of the dynamics then we can add some dynamical properties of the solutions. So, consider the Hamilton equations of motions $$ \dot z=X_H(z)\ ,\autoeqno{ham4}\ $$ and assume: \item{H6)}(Local well posedness) $\perogni s\geq s_*$ there exits $\epsilon_{s+d}>0$ with the following property: $\perogni z_0\in B_{s+d}(\epsilon_{s+d})$ there exist $T>0$ and a unique solution $$ z\in C^0\left([0,T];\P_{s+d}\right)\cap C^1([0,T],\P_s) $$ of \eqref{ham4} with $z_0=z(0)$.\quadratino \theorem{evo1}{Under the same assumptions of theorem \lemmaref{const}, assume also H6, then there exists $\epsilon_s^*$ with the following properties: Let $z_0\in\B_{s+s'}(\epsilon)$ with $\epsilon<\epsilon_s^*$, and denote by $z(t)$ the solution of \eqref{ham4} with initial datum $z_0$. Denote also $$ \toro_{z_0}:=\left\{ w\in\P_s\ :\ \Phi^{(j)}(w)=\Phi^{(j)}(z_0)\ ,\quad j=1,...,N\ , I_j(w)=0\ \perogni j>N\right\} $$ and let $\tau_\epsilon(z_0)$ be the escape time of the solution from $B_{s+s'}(\epsilon)$, namely the infimum of the times for which $z(t)\not\in B_{s+s'}(\epsilon) $, then one has $$ \eqalign{ |\Phi^{(j)}(t)-\Phi^{(j)}(0)|& \leq {C\epsilon^{M_1}}\ ,\ j=1,...,N \cr \left|I^{(s,N)}(t)-I^{(s,N)}(0)\right|&\leq {C\epsilon^{M_1}} \cr d_{s}(z(t),\toro_{z_0})&\leq C\epsilon^{M_2} }\autoeqno{diff2} $$ for all the times $t$ such that $$ |t|\leq\min\left\{\frac{1}{C\epsilon^{M_3}},\tau_\epsilon(z_0)\right\} $$ where $M_1,M_2,M_3$ are such that $M_1+M_2\leq M$, $2M_2+M_3+2\frac{s+1}{\beta}\leq M$, and $d_s(\cdot;\cdot)$ denotes the distance in $\P_s$. } \unp\unp \vskip\spaziosoprasez \noindent{\sl \sref{abs}.2 Quasilinear Equations} \vskip\spaziosottosez We are now going to bound from below the escape time in the case of quasilinear systems. We begin with some notations and definitions. Let $X$ and $Y$ be Hilbert spaces, with $Y$ densely embedded in $X$. The space of bounded linear operators from $Y$ to $X$ will be denoted by $\B(Y,X)$. We assume the Hamilton equations of \eqref{ham} to be quasilinear, i.e. that \item{K1)}(Quasilinearity) there exists $s_*$, and, for any $s\geq s_{*}$ a positive $R_{s+d}$, a map $$ B_{s+d}(R_{s+d})\ni z\mapsto A(z)\in\B(\P_{s+d},\P_s) $$ and a map $g:B_{s+d}(R_{s+d})\to \P_{s+d}$ such that the Hamilton equations of $H$ take the form $$ \dot z=A(z)z+g(z)\equiv X_{H}(z)\ .\autoeqno{qlin} $$ \quadratino \noindent The key assumption concerns the properties of the linearized flow. For any function $\zeta\in C^0([0,T],\P_{s+d})\cap C^1([0,T],\P_s)$ we will consider the family of linear operators $A^\zeta(t):=A(\zeta(t))$ and the corresponding linear time dependent differential equation $$ \dot z=A^{\zeta}(t)z\ .\autoeqno{lin1} $$ We recall that, under suitable assumptions there exists a corresponding evolution operator $U(t,s)$, strongly continuous as a map in $\B(X,X)$ and also as a map in $\B(Y,Y)$. The operator $U(t,s)$ is defined by the property that the solution of \eqref{lin1} fulfilling $z(s)=z_0$ is given by $z(t)=U(t,s)z_0$. We assume: \item{K2)} (Linear estimate) There exists $\nu\geq 1$ such that, for any $s\geq s_{*}$, any $\epsilon>0$, any $T>0$ and any function $\zeta\in C^0([0,T],\P_{s+d})\cap C^1([0,T],\P_s)$ fulfilling $$ \sup_{t\in[0,T]}\norma{\zeta(t)}_{s+d}+\sup_{t\in[0,T]}\norma{\dot \zeta(t)}_{s}\leq \epsilon \autoeqno{sti.2} $$ the evolution operator $U(t,s)$ associated to equation \eqref{lin1} exists and fulfills the estimate $$ \sup_{0\leq t\leq \tau\leq T}\norma{U(t,\tau)}_{\ell^2_{s+d}\to \ell^2_{s+d} }\leq M \es^{\beta \epsilon^{\nu} T}\ , \autoeqno{stalin} $$ with some constants $M,\beta$ independent of $\zeta,T,\epsilon$. \quadratino \item{K3)} $g$ is of class $C^1(\U_{s+d},\P_{s+d})$ and has a zero of order $\nu+1$ at the origin, namely $$ \norma{g(z)}_s\leq C_s\norma z_s^{\nu+1}\ . $$ \quadratino \remark{ex.1}{In the semilinear case where the linear operator $A$ is independent of $z$, one has $A=X_{H_0}$. It follows that assumption K2 is automatically true.} \remark{ex.2}{In the true quasilinear case one can use Kato's theory\upccite{kat75}{kat85} to verify assumptions K2. This is what we will do in the application to the nonlinear wave equation. } \theorem{evo}{Under the same assumption of theorem \lemmaref{const}, assume also K1-K3, then there exists $s'$ and, for any $s\geq s_*$ there exists a constant $\epsilon_s$ such that, if the initial datum $z_0$ fulfills $$ \epsilon:=\norma{z_0}_{s+s'}\leq \epsilon_s\ ,\autoeqno{ini} $$ then along the corresponding solution one has $$ \eqalign{ |\Phi^{(j)}(t)-\Phi^{(j)}(0)|& \leq {C\epsilon^{M_1}}\ ,\ j=1,...,N \cr \left|I^{(s,N)}(t)-I^{(s,N)}(0)\right|&\leq {C\epsilon^{M_1}} \cr d_s(z(t),\toro_{z_0})&\leq C\epsilon^{M_2} }\autoeqno{diff1} $$ for all times $t$ fulfilling $$ |t|\leq \frac{1}{C\epsilon^{\nu}}\autoeqno{time} $$ where $M_1=M-\nu$, $M_2=\frac{M-\nu}{2}-\frac{s+1}{\beta}$. } \remark{rtime}{The strongest limitation of the present result rests in its time of validity. Indeed, in the finite dimensional case and also in the case of semilinear equations in one space dimension\upcite{\birkhof}, a similar description of the dynamics has been proved to hold up to times of order $\epsilon^{-M}$ for any $M$. Due to results of the kind of \cite{klamaj} we expect the time \eqref{time} to be optimal for quasilinear models. However the nonresonance condition H1 is not satisfied by the model considered in \cite{klamaj}, so it is not a counterexample to the result of the present paper. We think it would be very interesting to find an example satisfying the assumptions of theorem \lemmaref{evo} but developing singularities over a the time scale $\epsilon^{-\nu}$. } \remark{ltime}{The theory of \cite{\birkhof} can be extended to some semilinear equations in higher space dimension (see \cite{BG03}) like the Nonlinear Schr\"odinger equation thus extending the validity time of the estimates \eqref{diff1}. The main property needed to obtain such an extension is a nonresonance condition stronger than H1. However such a condition is not fulfilled in general high dimensional examples like the nonlinear wave equation.} \autosez{wave}Application to quasilinear wave equations Having fixed a diophantine vector $a=(a_1,...,a_n)\in\Re^n$ with $a_i>0$\footnote{\null$^{1}$}{We recall that it means that there exist a positive $\bar \gamma$ and a real $\bar\tau$ such that $$ \left|a_1k_1+...+a_nk_n\right|\equiv \left|a\cdot k\right|\geq\frac{\gamma_1}{|k|^{\bar\tau}} \ ,\quad\perogni k\in\ra^n\meno\left\{0\right\}\ $$ and that diophantine vectors form a set of full measure in $\Re^n$. }, consider the $n$ dimensional parallelepiped $\R$ with sides of length $\pi/\sqrt{a_i}$, namely $$ \R:=\left\{\bx\equiv (x_1,...,x_n)\in\Re^n\ :\ 00$, then there exists a subset $\J\subset [0,b]$ of measure $b$ such that, if $m\in \J$, then the frequencies $\omega_j$ fulfill the assumption $H1$.} This theorem will be proved in section \sref{mea}. Assumptions H2 and H3 are a trivial consequence of Sobolev embedding theorems, and also H6 is automatically true. To fulfill the smoothness assumptions of the vector field we assume that the potential $W$ is even in each of the arguments, namely $$ W((-)^{c_0}u,(-)^{c_1}s_1,...,(-)^{c_n}s_n)=W(u,s_1,...,s_n)\ ,\quad \perogni (c_0,c_1,...,c_n)\in (Z_2)^{n+1}\ ,\autoeqno{sym0} $$ from which in particular it follows $\nu\geq 2$. To verify assumptions H4,H6 and K1-K3 it is convenient to proceed as follows: First rescale the domain in order to transform it into a standard cube with sides of length $\pi$; then identify the space $\F_s$ with the space $\A$ of the functions of class $H^{\tilde s+1}(\toro^n)\times H^{\tilde s}(\toro^n))$ (where $\toro=\Re/2\pi \ra$) which are skewsymmetric in each variable, namely that fulfill $$ u((-)^{c_1}x_1,...,(-)^{c_n}x_n)=(-)^{c_1+...+c_n}u(x_1,...,x_n)\ . $$ Clearly one has $\F_s\simeq\A\cap ( H^{\tilde s+1}(\toro^n)\times H^{\tilde s}(\toro^n))$. Due to the symmetry properties that \eqref{sym0} induces on the nonlinearity, the subspace $\A$ is invariant under the dynamics of \eqref{nlw} in $\toro^n$. So the validity of H4 is a simple consequence of Sobolev inequalities. The same is true for K1 and K3. Local existence and uniqueness (i.e. H6) was proven by Kato (see \cite{kat85} theorem 14.3). \theorem{exi}{Assume $W\in C^\infty$ and \eqref{sym0}, then property K2 holds.} \noindent For the proof see sect.\sref{app}. So the general theory applies and in particular one has \theorem{main}{For any positive large $M$ there exists a set $\J\subset [0,b]$ of measure $b$ such that, for any $m\in\J$ there exists a positive constant $s'$, and for any $s\geq s_*$ a constant $\epsilon_s$, such that if the initial datum $z_0\equiv(u_0,v_0)$ fulfills $$ \epsilon:=\norma{z_0}_{s+s'}\leq \epsilon_s\autoeqno{ini.1} $$ then along the corresponding solution of \eqref{nlw} one has $$ d_{s}(z(t),\toro_{z_0})\leq C\epsilon^M\quad {\rm for} \quad |t|\leq\frac{1}{C\epsilon^\nu} \autoeqno{diff3} $$ where $\toro_{z_0}$ is a finite dimensional smooth torus. } We conclude by some remarks on the possibility of extending the result to more general domains; to this and there are two kinds of difficulties. The first one is related to the verification of the nonresonance condition. Following the proof of theorem \lemmaref{mes} one can show that if the Dirichlet eigenvalues $\mu_j$ of the Laplacian in the considered domain fulfill the gap estimate $$ \mu_{j+1}-\mu_j>C/j^{\delta} $$ with a given $\delta$, then the nonresonance condition H1 is fulfilled for most values of the parameter $m$. However we do not know whether there exist domains different form the parallelepiped in which the gap condition is fulfilled. The second point concerns smoothness of the solutions. Indeed it is well known that there are some compatibility conditions which are necessary and sufficient for the smoothness of solutions of the wave equation (which in turn is essential for our approach). We have no clear ideas on how to deal with such conditions in the case of general domains. \autosez{for}The classical algorithm Here we recall the classical algorithm of construction of the approximate integrals of motion and a result by Giorgilli\upcite{gioihp} that we need. Consider a Hamiltonian system with $N$ degrees of freedom, of the form $$ H=\sum_{k\geq 0}H_k\autoeqno{Hk} $$ where $H_k$ is a homogeneous polynomial of degree $k+2$ in the phase variables and $H_0$ is of the form \eqref{H0}. Let $I_j=(p_j^2+q_j^2)/2$ be an action of the linear system, we aim to construct an integral of motion of the form $\Phi^{(j)}:=I_j+\sum_{l\geq1}\Phi_l$ with $\Phi_l$ a homogeneous polynomial of degree $l+2$. Clearly $\Phi^{(j)}$ has to fulfill the equation $\poisson{H}{\Phi^{(j)}}=0$. Inserting the Taylor expansions of $H$ and of $\Phi$ and equating terms of the same degree we get the recursive system $$ L_{H_0}{\Phi_l}=\Psi_l \ ,\ l\geq 1\ ,\quad L_{H_0}:=\poisson{H_0}{\, .\, } \autoeqno{hom} $$ where $\Psi_l$ is determined by $$ \eqalign{ \Psi_1&=\poisson{I}{H_1} \cr \Psi_l&=\sum_{j=1}^{l-1}\poisson{\Phi_j}{H_{l-j}}+\poisson{I}{H_l} }\autoeqno{psi} $$ So, provided one is able to solve the homological equation \eqref{hom}, the formal integral of motion can be computed. In order to solve the homological equation we remark that at the $l$-th step \eqref{hom} appears as a linear equation in the finite dimensional space of the polynomials of degree $l+2$. So, if one is able to diagonalize the linear operator $L_{H_0}$, then equation \eqref{hom} is easily solved. To diagonalize $L_{H_0}$ introduce the variables $$ \xi_j:=\frac1{\sqrt2} \left(q_j -\im p_j\right)\ ,\quad \eta_j:=\frac1{\sqrt2} \left( q_j +\im p_j\right)\ , \autoeqno{four} $$ which give the unperturbed Hamiltonian the form $H_0=\sum \omega_l\xi_l\eta_l$ (and the symplectic form becomes $\im d\xi_l\wedge d\eta_l$), so that one has $$ L_{H_0}{\xi^j\eta^k}=\left[\im(k-j)\cdot \omega\right]\ \xi^j\eta^k\ ,\quad \xi^j\eta^k\equiv \xi_1^{j_1}...\xi_N^{j_N}\eta_1^{k_1}... \eta_N^{k_N} $$ i.e. the monomial $\xi^j\eta^k$ form the basis of the space of polynomials on which $L_{H_0}$ is diagonal. So, \eqref{hom} is easily solvable provided $\Psi_l$ has no component on the kernel of $L_{H_0}$, a fact that can be easily verified recursively\footnote{$\null^{*}$}{Indeed, a component on the Kernel of $L_{H_0}$ is always even in the momenta $p$, while from the construction of $H_k$ such a function always turns out to be either independent of $p$ or skewsymmetric in it (see \cite{gioihp}).} in the case where $H_k(p,q)$ is even in $p$. As is well known the series defining the integral of motion are in general divergent, so, following \cite{gioihp}, one has to stop the construction at a given order and to estimate the Poisson bracket of the so obtained function with the Hamiltonian. To this end remark that, defining $$ \Phi^{(j,r_*)}:=I_j+\sum_{l=1}^{r_*}\Phi_l\ , $$ one has $$ \dot\Phi^{(j,r_*)}=\poisson{H}{\Phi^{(j,r_*)}}=O(\left|(p,q)\right| ^{r_*+3}) \autoeqno{resto} $$ where we denoted for simplicity $\Phi_0:= I_j$. Finally, in the finite dimensional case, one can obtain a long time estimate of the solution repeating the construction for all $j$'s and then using the approximate constant of motion as Lyapunov functions in order to control that the domain of construction of such functions is not left by the solution up to the times one is interested in. For the application to infinite dimensional systems we have to keep into account the dependence of all the constants on the number $N$ of degrees of freedom. To this end we recall a quantitative result by Giorgilli \upcite{gioihp}. We first have to introduce some notations. Fix $R\equiv(R_1,...,R_N)$ with $R_j>0$ and consider a polynomial $f(p,q)=\sum_{j,l} f_{jl}p^jq^l$, then the size of $f$ is measured by the norm $$ \norma{f}_R:=\sum_{i,l}|f_{jl}|R^{j+l}\ . $$ To control the small denominators assume that there exists a non-increasing sequence $\{\alpha_s\}_{s=1}^{r_*}$ such that $$ |\omega\cdot k|\geq\alpha_s \ {\rm for} \ k\in\ra^N\ ,\ 0<|k|\leq s+2 \ .\autoeqno{nonre} $$ Finally consider the complex neighbourhood of the origin $$ \Delta_{\rho R}:=\left\{(p,q)\in\Cm^{2N}\ :\ (|p_l|^2+|q_l|^2)^{1/2}\leq \rho R_l \right\} $$ and denote $$ \Lambda:=(\min_{l}R_l)^{-1}\ . $$ \theorem{gio}{[Giorgilli] Consider a Hamiltonian system of the form \eqref{Hk}, even in the momenta $p$, and assume that, for a given $R\in \Re^{N}_+$ there exist constants $h> 0$ and $E>0$ such that $\norma{H_k}_R\leq h^{k-1}E$ for $k\geq1$; assume that the linear frequencies fulfill \eqref{nonre}. Then, for any integer $01\ , \cr C_r&=8E\left(12\Lambda^2E+\frac{8}{9}h\alpha_1\right)^r \frac{(r+2)!}{\prod_{l=1}^{r}\alpha_l } \ ,\ r\geq 1\ . }\autoeqno{3.9} $$ } \autosez{pa}Proof of the abstract results. Expand the perturbation $P$ in Taylor series up to order $r_*+2$, $$ P=\sum_{l=1}^{r_*} P_l+\R_* $$ where $P_l$ are homogeneous polynomial of degree $l+2$ and $\R_*$ is the remainder. We will denote $$ H_*:=H_0+\sum_{l=1}^{r_*} P_l $$ \remark{analit}{The polynomials $P_l$ are analytic functions, and therefore, introducing the complexification $\P_{s_*}^{\Cm}$ of $\P_{s_*}$, one has that there exist constants $R_*$, $C$ such that $$ \left|{P_l(z)}\right|\leq C\norma{z}_{s_*}^{l+2}\ ,\ \perogni z\in \P_{s_*}^{\Cm}\ \quad \norma{z}_{s_*}\leq R_*\ . $$ } To make the Galerkin cutoff fix a large $N$ that will eventually be related to $\epsilon$, introduce the projector $\Pi_N$ defined by $$ \Pi_N(p,q)=(p_1,...,p_N,q_1,...,q_N)\ ,\autoeqno{piN} $$ and put $H^{(N)}:=H_0^{(N)}+\sum_{l=1}^{r_*}H_l$, with $$ H_0^{(N)}:=H_0\circ \Pi_N\ ,\quad H_l:=P_l\circ \Pi_N\ . $$ One has $$ H=H^{(N)}+(H_*-H^{(N)})+\R_*\ . $$ \lemma{gal}{For any $s\geq s_*$ there exists a constant $C$ and a domain $\U_{s}$ such that, for any $r\geq 0$, and any $N\geq 0$, one has $$ \eqalign{ \norma{X_{\R_*}(z)}_{s}\leq C \norma{z}_{s+d}^{r_*+2}\ ,\quad \perogni z\in\U_{s+d} \cr \norma{X_{H_*-H^{(N)}}(z)}_s\leq \frac{C}{N^{r}}\norma{z}_{s+r+d}\ , \quad \perogni z\in\U_{s+d+r}\ . }\autoeqno{r*} $$ } \proof First remark that $$ \norma{\uno-\Pi_N}_{s+r\to s}=\frac{1}{N^{r}}\ .\autoeqno{npi} $$ and that $$ X_{H_*-H^{(N)}}(z)= \left[ X_{H_*}(z)-X_{H_*}(\Pi_Nz) \right]+ (\uno-\Pi_N) X_{H_*}(\Pi_Nz)\ , $$ By H4, using the norm of the differential to estimate the Lipschitz constant of $X_{H_*}$, the first square bracket is estimated by $$ \sup_{z\in\U_{s+d}}\norma{dX_{H_*}(z)}_{s+d\to s}\norma{\uno-\Pi_N}_{s+d+r\to s+d}\norma{z}_{s+d+r} \leq \frac C{N^r}\norma{z}_{s+r+d} $$ The second term is trivially estimated using \eqref{npi}. The estimate of $X_{R_*}$ is obtained by applying Lagrange estimate of the remainder of the Taylor expansion.\quadratino We apply now theorem \lemmaref{gio} to $H^{(N)}$. \lemma{gale}{There exist $N$ truncated analytic integrals $\Phi^{(l,r_*)}=I_l+\sum_{s=1}^{r_*}\Phi^{(l)}_s$ for the Hamiltonian system $H^{(N)}$. Moreover for any $z\in\P_{s_*}$ such that $$ \norma{z}_{\infty}:=\sup_{k=1,...,N}\{\sqrt{p_k^2+q_k^2}\} \leq \frac1{CN^{\beta+\alpha}}\autoeqno{aper} $$ the following estimates hold $$ \eqalign{ \left|\left( \Phi^{(l,r_*)}-I_l \right)(z)\right|&From now on we will always denote simply by $\Phi^{(l)}$ the approximate integrals $\Phi^{(l,r_*)}$. To prove that the level surfaces of the approximate integrals are tori and that the distance from such tori is controlled by the difference between the values of the functions one has to pay some attention to the fact that the action $I_l$ is singular on the hypersurface $p_l=q_l=0$. We have the following \proposition{nonsca}{Assume $\norma{z}_\infty<\epsilon$ and $\norma{\bar z}_\infty<\epsilon$ with some $$ \epsilon<\frac{1}{CN^{3s_*+6+\alpha}} $$ then there exits $\eta_*$ such that $$ \left|\Phi^{(j)}(z)-\Phi^{(j)}(\bar z)\right|<\epsilon^2\eta\ ,\quad \perogni j=1,...,N\ ,\quad \Pi_N(z)=\Pi_N(\bar z)=0 $$ with $\eta<\eta_*$ implies $$ d_{\infty}(\bar z,\toro_z)1/C\epsilon^{\nu}$ with a suitable $C$.\quadratino \noindent{\bf Proof of theorem \lemmaref{evo}}. Define $\tilde \epsilon:=C\norma{z}_{s+s'}$ with the constant of \eqref{esti.0}, then by lemma \lemmaref{lya} one has $\norma{z(t)}_{s+s'}\leq \tilde \epsilon $. Thus, from theorem \lemmaref{const} (item 3)it follows $$ \left|\Phi^{(j)}(t)-\Phi^{(j)}(0)\right|\leq C\tilde \epsilon^{M_1}\ ,\quad |t|\leq\frac{1}{\tilde \epsilon^\nu}\ . $$ defining $\epsilon:=\norma z_{s+s'}$ and redefining the constants one has the thesis.\quadratino \autosez{mea}Proof of theorem \lemmaref{mes} For $k\in\ra^N\meno\left\{0\right\}$ with $0<|k|\leq r_*+2$ define $$ \R_{k}(\gamma,\alpha):=\left\{m\in \I\ :\ \left|\omega^{(N)}\cdot k \right| <\frac{\gamma}{N^{\alpha}}\right\} $$ We will estimate the measure of the union of such sets. First one has \lemma{fre2}{The eigenvalues $\mu_j$ fulfill the gap condition $$ \mu_{j+1}-\mu_j\geq \frac{C}{j^{\delta}}\ ,\quad \perogni j\geq 1\ .\autoeqno{gap} $$ with $\delta=4\bar \tau/n$, $\bar \tau$ being the exponent in the diophantine condition for $a$. } \proof Since $a$ is a diophantine vector one has $$ \left|\mu_{\bi}-\mu_{\bj} \right|=\left|\sum_{l}a_l(k_l^2-j_l^2)\right| \geq\frac{\gamma_1}{\left( \sum_{l}|k_l^2-j_l^2| \right)^{\bar\tau}} \ , $$ but $$ \sum_{l}|k_l^2-j_l^2|\frac{C}{i^{\delta+2/n}j^{\delta+2/n}} \autoeqno{m.2} $$ Denoting $y_i:={i}^{-(\delta+2/n)}$, \eqref{van} is estimated by $$ \prod_{l=2}^{K}\prod_{N=1}^{l-1}Cy_{\bii l}y_{\bii N}= C^{\sum_{l=2}^{K}(l-1)} \prod_{l=2}^{K}\left(y_{\bii l}^{l-1}\prod_{N=1}^{l-1}y_{\bii N}\right)= C\prod_{l=1}^{K} y_{\bii l}^K\ , $$ from which the thesis immediately follows. \quadratino The rest of the proof follows very closely the proof of theorem 6.16 of \cite{\birkhof}. We repeat the main steps for completeness. \lemma{m1.1}{(Proposition of Appendix B in \cite{ben85}) Let $u^{(1)},...,u^{(K)}$ be $K$ independent vectors with $\norma{u^{(i)}}_{\ell^1}\leq1$. Let $w\in\Re^K$ be an arbitrary vector, then there exist $i\in[1,...,K]$, such that $$ |u^{(i)}\cdot w|\geq\frac{\norma w_{\ell^1}\det(u^{(i)})}{K^{3/2}}\ , $$ where $\det(u^{(i)})$ is the determinant of the matrix formed by the components of the vectors $u^{(i)}$. } For the proof see \cite{ben85}. \corollary{m1.2}{Let $w\in\Re^{\Na}$ be a vector with only $K$ components different from zero, namely those with index $\bii1,...,\bii{K}$; assume $K\leq N$, $K\leq r_*+2$. Then, for any $m\in \I$ there exists, an index $i\in[0,...,K-1]$ such that $$ \left|w\cdot\frac{d^i\omega}{dm^i}(m)\right|\geq C\frac{\norma{w}_{\ell^1}}{N^\rho} $$ where $\omega$ is the frequency vector, and $\rho=K^2(\delta+\frac{2}{n})-\frac{2K}{ n}+n-1/n$} \proof The proof is obtained by considering the vectors $$ u^{(i)}:=\left\{ \matrix {\frac1{\norma{\frac{d^i\omega}{dm^i}}} \frac{d^i\omega}{dm^i} &{\rm if} & \norma{\frac{d^i\omega}{dm^i}}>1 \cr \frac{d^i\omega}{dm^i} &{\rm if} & \norma{\frac{d^i\omega}{dm^i}}\leq1 } \right. $$ and applying lemma \lemmaref{m1.1}. For more details see \cite{\birkhof} proof of corollary 6.10.\quadratino \lemma{m.4}{(Lemma 8.4 of \cite{\nlw}) Let $g:\I\to\Re$ be $N$ times differentiable, and assume that \item{(1)}$\perogni m\in \I$ there exists $i\leq K-1$ such that $\frac{d^ig}{dm^i}(m)>d$ \item{(2)}there exists $A$ such that $|g^{(i)}(m)|\leq A$ $\perogni m\in \I$, and $\perogni i$ with $1\leq i\leq K$. \goodbreak Define $$ \I_h:=\left\{m\in\I\ :\ |g(m)|\leq h\right\}\ , $$ then $$ \frac{|\I_h|}{|\I|}\leq \frac{A}d2(2+3+...+r+2d^{-1})h^{1/K} $$ } For the proof see \cite{\nlw} and \cite{you97a}. By combining lemma \lemmaref{m.4} and lemma \lemmaref{m1.2} we get the following \lemma{m.2}{For any choice of $k$ one has $$ \left|\R_{k}(\gamma,\alpha )\right|\leq |\I|C \frac{\gamma^{1/K}}{N^{\varsigma}}\autoeqno{m.21} $$ with $\varsigma=-2\rho+\alpha /K$. } \noindent {\bf Proof of theorem \lemmaref{mes}}. By \eqref{m.21} one has $$ \left|\bigcup_{k}\R_k(\gamma,\alpha)\right|\leq \sum_{k}\left|\R_k(\gamma,\alpha) \right|<\frac{(2N)^{r_*+3}|\I|C\gamma^{1/K}}{N^{\varsigma}} \ , $$ which, provided $\alpha$ is chosen large enough, is bounded uniformly with respect to $N$. Taking the union over $\gamma$ one obtains the thesis. \quadratino \autosez{app}Proof of theorem \lemmaref{exi}. \vskip\spaziosottosez First of all we introduce scaled variables $\tilde x_i=\sqrt{a_i}x_i, \tilde u$ by $u=\epsilon \tilde u$, so that equation \eqref{nlw} is converted into $$ u_{tt}-a_{ij}^\epsilon\partial_i\partial_j u+mu=\epsilon^\nu g_\epsilon(u,\nabla u) \autoeqno{nlw1} $$ where we omitted tildes, and used the notations $$ a_{ij}^\epsilon =a_i\delta_{ij}+\epsilon^\nu b^\epsilon_{ij}\ ; $$ in what follows we will omit the index $\epsilon$ from $g$, $a_{ij}$ and $b_{ij}$. We consider equation \eqref{nlw1} in $\toro^n$; moreover we will assume that the initial datum has zero average and that the vector field lives invariant the space of functions with zero average. Since we are interested in the case of Dirichlet boundary condition this is not restrictive. We will prove the theorem by considering directly equation \eqref{nlw1} using the spaces $Y=H^{s+1}\times H^{s}$ and $X:=H^{s}\times H^{s-1}$. Thus the unbounded part of \eqref{nlw1} is $\partial_t \zeta =A(\zeta)\zeta $ with $\zeta\equiv (u,v)$, and $$ A(\zeta):=\left( \matrix{ 0& \uno \cr \A(u) & 0 } \right)\ ,\quad \A(u):=a_{ij}(u,\nabla u)\partial_{i}\partial_j-m $$ Having fixed $w\in C^0(I;H^{s+1})\cap C^1(I;H^{s})$ with $I:=[0,T]$, we have to show that $A^w(t):=A(w(t))$ fulfills assumption K2. We begin by estimating the stability constants of the operator family $A^w(t)$ in $X$. To this end it is useful to make reference to the the second order equation $$ \ddot u=\A^w(t)u\autoeqno{lint} $$ where $\A^w(t):=a_{ij}(w(t), \nabla w(t))\partial_i\partial_j-m$. We remark that since $w(t)$ has the property \eqref{sti.2}, one has $b_{ij}\in C^{0}(I;H^s)\cap C^1(I;H^{s-1})$ with bounded norms. The proof will be based on proposition 9.3 of \cite{kat85}, so first of all we recall its statement. \proposition{kat2}{[Proposition 9.3 of \cite{kat85}]. Assume that: \goodbreak \noindent (i'a) $H$ and $H'$ have structures of real Hilbert spaces, with inner products $\langle\ .\ \rangle_t$ and $\langle\ .\ \rangle_t'$ depending on $t\in I$, such that the norms induced by them are equivalent to the standard norms $\norma{\ }_H$ and $\norma{\ }_{H'}$, respectively. These inner products are Lipschitz continuous in $t$, in the sense that $|\langle u,v\rangle_t-\langle u,v\rangle_s|\leq c|t-s|\norma u_H\norma v_H $, and similarly for $\langle\ .\ \rangle'_t$. \goodbreak \noindent (i'b) $\{ \A(t) \}$ is a family of (unbounded) linear operators in $H$, such that $\A(t)=\A_S(t)+\A_R(t)$, where $\A_S(t)$ is selfadjoint and uniformly positive definite with respect to inner product $\langle\ .\ \rangle_t$, while $\A_R(t)$ is uniformly bounded (with bound $K$, say) from $H'$ to $H$. Moreover $D(\A_S(t))\subset H'$ and $$ \langle\A_S(t)u,v\rangle_t=\langle u,v\rangle'_t\qquad (u\in D(\A_S(t)), v\in H' )\ .\autoeqno{sckat} $$ Then the family $$ A(t):= \left( \matrix{ 0& \uno \cr \A(t) & 0 } \right) $$ is a stable family in $H'\oplus H$ with stability constants $M$, $\beta:=2c+K$, where $M$ depends on the relation between the equivalent norms $\norma\ _H$ and $\langle\ ,\ \rangle_0^{1/2}$. } \lemma{linX}{The evolution operator of the equation \eqref{lin1} fulfills the estimate $$ \sup_{0\leq s\leq t\leq T}\norma{U(s,t)}_{X,X}\leq M \es^{C\epsilon^\nu T} $$ with constants $M,C$ independent of $T$ and $\epsilon$. } \proof We define the spaces $H:=H^{s-1}$, and $H':=H^s$. The standard scalar product in $H^{s-1}$ is defined by $$ \left\langle u,v\right\rangle_{H}\equiv \left\langle u,v\right\rangle_{H^{s-1}} \equiv\int (D^{s-1}u)(D^{s-1}v)\ ,\quad D:=\left(-\Delta\right)^{1/2} $$ and the integral is taken over the whole torus. The standard scalar product in $H'$ is defined in the same way with $s$ in place of $s-1$. Define the scalar product $\langle\ ,\ \rangle_t:= \left\langle\ ,\ \right\rangle_{H}$ (independent of $t$), and the operator $\A_S$ $$ \A_S u:= -D^{-(s-1)}\left(\partial_j\left(a_{jk}\partial_k(D^{s-1}u) \right)\right)+mu\ . $$ Then $\langle\ ,\ \rangle'_t$ is defined by \eqref{sckat}. It is easy to see that the operator $\A_S$ is uniformly elliptic (just use positivity of $a_{ij}$). We will study the relation of this operator with $\tilde \A:=\partial_j(a_{jk}\partial_k)+m$, who in turn is related to $\A^w(t)$ by $$ \tilde \A u=\A u+(\partial_ja_{jk})(\partial_ku)\ . $$ Remark that the second term defines an operator whose norm, as an operator from $H'$ to $H$, is bounded by a constant times $\epsilon^\nu$. We now compare the operators $\tilde \A$ and $\A_S$. For simplicity we restrict to the case where $s-1=2r$, i.e. $s$ is odd\footnote{$\null^{*}$}{The general case can be dealt with using the operators $S_i$ defined in the proof of lemma \lemmaref{linY}}. One has $$ \A_S=\tilde\A+D^{-(s-1)}\left[\tilde \A,D^{s-1}\right]=\tilde\A+ D^{-(s-1)} \partial_j[a_{jk}\partial_k,D^{s-1}]\ . $$ So, it is enough to compute $[a_{jk}\partial_k,\left(\partial_l\partial_l\right)^r]u$ (summation convention!). To this end consider $$ \left(\partial_l\partial_l\right)^r(a_{jk}\partial_k u)= \partial_{l_1}\partial_{l_1} ...\partial_{l_r}\partial_{l_r}\left(a_{jk}\partial_ku \right) $$ which is the sum of terms of the form $$ c_{\alpha}\left(\frac{\partial^{|\alpha|}} {\partial x^{\alpha}}a_{jk} \right) \left(\frac{\partial^{|\beta |}}{\partial x^{\beta}}u\right) $$ where $\alpha$ and $\beta$ are multindices fulfilling $|\alpha|+|\beta|=2r$. The only term with $|\beta|=2r$ is the one where the Laplacians act entirely on $u$ so it disappears when computing the commutator. It is easy to see that all the other terms define operators which are bounded as operators from $H'$ to $H$, provided $s>[n/2]$. And moreover the norms of such operators are bounded by a constant times $\epsilon^\nu$. Applying the remaining two operators (namely $D^{-(s-1)}$ and $\partial_j$) one has that $$ \A=\A_S+\A_R\ , $$ with $\A_R:H'\to H$ having norm bounded by a constant times $\epsilon^\nu$. Finally, it is very easy to see that $$ |\left\langle u,v\right\rangle'_t-\left\langle u,v\right\rangle'_{s}|\leq C\epsilon^\nu\norma{u}_{H^s}\norma{v}_{H^s}|t-s|\ . $$ So, proposition \lemmaref{kat2} gives the result.\quadratino To estimate the linear flow in $Y$ we construct an isomorphism $S:Y\to X$ such that $SA^wS^{-1}=A^w+\epsilon^\nu B$ with an operator $B:X\to X$ uniformly bounded. We define $$ S:=\left(\matrix{D&0\cr0&D}\right)\autoeqno{S}\ . $$ \lemma{linY}{One has $$ SA^wS^{-1}=A^w+\epsilon^\nu B $$ with $B$ uniformly bounded as an operator from $X $ to $X$ } \proof One has $$ SA^wS^{-1}=\left( \matrix{ 0 & \uno \cr D\A^wD^{-1} &0 } \right)\ , $$ so it is enough to prove that $$ D\A^wD^{-1}=\A^w+\epsilon^\nu\B $$ with $\B:H^s\to H^{s-1}$ uniformly bounded. To prove this fact remark that $$ \epsilon^\nu\B=[D,\A^w]D^{-1}= [D,\epsilon^\nu b_{ij}\partial_i\partial_j]D^{-1} $$ so the result is equivalent $[D,b_{ij}\partial_i\partial_j]:H^{s+1}\to H^{s-1}$ continuously. But $ [D,b_{ij}\partial_i\partial_j]u=[D,b_{ij}]\partial_i\partial_ju $, so it is enough the prove that $\perogni b\in H^s$ one has $[D,b]:H^{s-1}\to H^{s-1}$ continuously. To prove this fact we compute the commutator in the Fourier space. Denoting by a ``hat'' the (space) Fourier coefficients of a function one has $$ \eqalign{ \left\{[D,b]u\right\}^{\wedge}(\bk)= \norma{\bk}\sum_{\bl}\hat b_{\bk-\bl} \hat u_{\bl}-\sum_{\bl}\hat b_{\bk-\bl} \norma{\bl}\hat u_{\bl} \cr =\sum_{\bl}\hat b_{\bk-\bl} \hat u_{\bl}\left(\norma{\bk}-\norma{\bl}\right) } $$ The last bracket can be rewritten in the form $$ \frac{k_1^2+...+k_n^2-l_1^2-...-l_n^2}{\norma{\bk}+\norma{\bl}} = (k_1-l_1)\frac{k_1+l_1}{\norma{\bk}+\norma{\bl}}+...+(k_n-l_n) \frac{k_n+l_n}{\norma{\bk}+\norma{\bl}} \ . $$ Define now the bilinear operators $S_i$ by $$ \left\{S_i(b,u)\right\}^{\wedge}(\bk):=\sum_{\bl}\hat b_{\bk+\bl} \hat u_{\bl}\frac{k_i+l_i}{\norma{\bk}+\norma{\bl}}\ , $$ then one has $$ [D,b]u=\sum_iS_i(\partial_i b, u)\ . $$ We prove now that $S_i:H^{s-1}\times H^{s-1}\to H^{s-1}$ continuously, and from this the thesis will immediately follow. Since $S_i$ is bilinear it is enough to prove that $$ \norma{S_i(v,u)}_{H^{s-1}}\leq C\norma{v}_{H^{s-1}}\norma{u}_{H^{s-1}}\ . $$ One has $$ \eqalign{ \norma{S_i(v,u)}^2_{H^{s-1}}&=\sum_{\bk}\norma{\bk}^{2(s-1)} \left( \sum_{\bl}\hat v_{\bk-\bl} \hat u_{\bl}\frac{k_i+l_i}{\norma{\bk}+\norma{\bl}} \right)^2 \cr &\leq \sum_{\bk}\norma{\bk}^{2(s-1)} \left( \sum_{\bl}|\hat v_{\bk-\bl} \hat u_{\bl}| \right)^2\left[ \sup_{\bk,\bl}\left|\frac{k_i+l_i}{\norma{\bk}+\norma{\bl}} \right| \right]^2 \cr &=C \sum_{\bk}\norma{\bk}^{2(s-1)} \left( \sum_{\bl}|\hat v_{\bk-\bl}|\, | \hat u_{\bl}| \right)^2\ . }\autoeqno{hat} $$ Define now $u^+(\bx):=\sum_{\bk}|\hat u_{\bk}|\es^{\im\bk\cdot \bx}$, and similarly for $v^+$. Then one has $\norma{u^+}_{H^{s-1}}=\norma{u}_{H^{s-1}}$, and therefore the square root of the last line of \eqref{hat} is given by $$ C\norma{u^+v^+}_{H^{s-1}}\leq C \norma{u^+}_{H^{s-1}}\norma{v^+}_{H^{s-1}} =C \norma{u}_{H^{s-1}}\norma{v}_{H^{s-1}} $$ (with a different $C$), which is the thesis. \quadratino Then the estimate (3.3) of \cite{kat75} gives \corollary{cor1}{The evolution operator of equation \eqref{lin1} fulfills the estimate $$ \sup_{0\leq s\leq t\leq T}\norma{U(s,t)}_{Y,Y}\leq M \es^{C\epsilon^\nu T} $$ with constants $M,C$ independent of $T$ and $\epsilon$. } \vskip\spaziosoprasez \semiautosez{A}Appendix. In this appendix we will prove proposition \lemmaref{nonsca}. We will work $\Re^{2N}$. In order to keep into account the dependence of all we will do on $N$ it is worth to scale the coordinates, namely to define coordinates $(\tilde p, \tilde q)$ defined by $$ \epsilon\tilde p=p\ ,\quad \epsilon\tilde q=q $$ and to consider the integrals $$ \tilde\Phi^{(l)}(\tilde p,\tilde q,\epsilon):= \frac{1}{\epsilon^2}\Phi^{(l)} (\epsilon \tilde p,\epsilon\tilde q)\ . $$ >From now to the end of the appendix we will use only these new variables and integrals, so will omit tildes. The role of small parameter will be played by $$ \mu:=N^{3a+\alpha}\epsilon\ . $$ \lemma{coor}{There exist analytic coordinates $(\xi_l,\eta_l)$ defined in the ball $$ \norma{(\xi_l,\eta_l)}<1/C $$ In which one has $$ \Phi^{(l)}=\frac{\xi_l^2+ \eta_l^2 }{2}+O(\left|\xi_l,\eta_l\right|^3) \ ,\quad \norma{(\xi,\eta)-(p,q)}_{\infty} \leq C\mu\ . $$ The big $O$ is uniform with respect to $\mu$.} \proof Write $$ \Phi^{(l)}(p,q)=\frac{p_l^2+q_l^2}{2}+\mu g_l(p,q,\mu)\ , $$ by the first of the estimates \eqref{gale1} the functions $g_l$ are smooth and bounded provided $\mu$ is small enough. Denote by $\hatl p$ the set of variables $$ \hatl p:=(p_1,...,p_{l-1},p_{l+1},...,p_N) $$ and similarly for $\hatl q$. First remark that, provided $\mu$ is small enough, there exists a surface $\S_l$ where the function $\Phi^{(l)}$, considered as a function of $p_l,q_l$ has a minimum (just use implicit function theorem applied to the differential of $\Phi^{(l)}$). Moreover such a surface is analytic and has equation $$ p_l=\mu G^l_p(\hatl p, \hatl q,\mu)\ ,\quad q_l=\mu G^l_q(\hatl p, \hatl q,\mu)\ . $$ Introduce new coordinates by $$ x_l:=p_l-\mu G^l_p(\hatl p, \hatl q,\mu)\ ,\quad y_l:=q_l-\mu G^l_q(\hatl p, \hatl q,\mu)\ . $$ Using the implicit function theorem one can see that, provided $\mu<\mu_*$ with a suitable $\mu_*$, this coordinate system is well defined in a neighbourhood (in the $\norma ._{\infty}$ norm) of the origin (independent of $\mu$). In these coordinates one has $$ \Phi^{(l)}(x,y)=\left[B_{l}(\hatl x,\hatl y) \right](x_l,y_l)+O(\left|x_l,y_l\right|^3) $$ where the quadratic form $\left[B_{l}(\hatl x,\hatl y) \right]$ is positive definite. It follows that there exists a rotation $R_l(\hatl x,\hatl y,\mu)$ in the plane $x_l, y_l$ which diagonalize such a quadratic form. Moreover, for $\mu=0$ the transformations $R_l$ reduce to the identity. Introduce the coordinates $$ (\xi_l,\eta_l)=\left[R_l(\hatl x,\hatl y,\mu) \right](x_l,y_l) \ . $$ By implicit function theorem they are well defined in a neighbourhood of the origin and reduce the integrals to the form $$ \Phi^{(l)}=\frac{\lambda_l^1(\hatl\xi,\hatl\eta)\xi_l^2+ \lambda_l^2(\hatl\xi,\hatl\eta)\eta_l^2 }{2}+O(\left|\xi_l,\eta_l\right|^3) \ . $$ scaling the variables one has the thesis.\quadratino \lemma{coo2}{There exist $C_1,C_2$ and $\eta_*>0$ with the following properties: let $z\in \Re^{2N}$ and $\bar z\in\Re^{2N}$ be in a ball of radius smaller than $1/C_1$ and fulfill $$ \left|\Phi_j(z)-\Phi_j(\bar z)\right|< \eta\ ,\quad {\rm with} \ \eta<\eta_* $$ then $$ d_{\infty}(\toro_z,\bar z)\leq C_2\eta^{1/2} $$ (recall that we are using rescaled coordinates). } \proof Introduce the coordinates of lemma \lemmaref{coor}. Since they are analytic they are also Lipschitz and therefore, in such coordinates distances are estimated by a constant times the original distance. Introduce now polar coordinate $\rho_j,\phi_j$ in the planes $\xi_j,\eta_j$. Then one has $$ \Phi^{(j)}=\rho_j^2+\rho_j^3 f_j(\rho,\phi)\ , $$ with bounded and smooth $f_j$'s. So it is possible to use the contraction mapping principle to solve the above system with respect to the $\rho_j^2$ obtaining $$ \rho_j^2:=\Phi^{(j)}+(\Phi^{(j)})^{3/2}\tilde f_j(\Phi,\phi)\ . $$ One has $$ d_{\infty}(\bar z,\toro_z)=\sup_j\left|\rho_j-\bar \rho_j\right| $$ with an obvious notation. Fixing $j$ and omitting it, we have $$ \eqalign{ \left|\rho-\bar \rho\right|=\left| \sqrt{ \Phi+\Phi^{3/2}\tilde f(\Phi,\phi) }- \sqrt{ \bar \Phi+\bar\Phi^{3/2}\tilde f_j(\bar \Phi,\bar \phi) } \right| \cr = \frac{\left|\Phi+\Phi^{3/2}\tilde f_j(\Phi,\phi) - \bar \Phi+\bar\Phi^{3/2}\tilde f_j(\bar \Phi,\bar \phi) \right|}{\sqrt{ \Phi+\Phi^{3/2}\tilde f(\Phi,\phi) }+\sqrt{ \bar \Phi+\bar\Phi^{3/2}\tilde f_j(\bar \Phi,\bar \phi) }}\ . } $$ Now, the function $\Phi+\Phi^{3/2}\tilde f_j(\Phi,\phi)$ is Lipschitz in $\Phi$, and therefore the numerator is estimated by a constant times $\Phi-\bar \Phi$. Write $\bar\Phi=\Phi+\delta$, then the denominator is monotonically increasing function of $\Phi$, and therefore it has its minimum for $\Phi_j=0$. Evaluating at such a point one gets that it is larger than a constant times $\sqrt{\delta}$. From this the thesis easily follows. \quadratino Going back to the nonrescaled variables one has the thesis of proposition \lemmaref{nonsca}. \eject \vskip\spaziosoprasez \centerline{{\bf References}} \insertbibliografia \bye