% Ms. no. 64 - Math. Zeit. % Plain TeX file. In the file I made the following changes: % - New attribution notice in footnote 1 % - Updated reference 2 and 7 % - Clarifying assumption added in 4^9 % - Linear form changed to fraction on 6^8 % % File uses nonstandard macros (included below): % \t makes a theorem and defines first parameter as the % number corresponding to the theorem % \rf appends one position to bibliography and defines the % first argument as the reference number % Bibliography is set as one box (\box2), each \rf appends to % to this box. The box in unpacked at the end of the paper % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \input amssym.def \input amssym.tex \long\def\ppp #1. #2\par{\medbreak \noindent{ #1.\enspace}{\sl#2}\par \ifdim\lastskip<\medskipamount \removelastskip\penalty55\medskip\fi} \long\def\nic#1{} \baselineskip=18pt \def \rank {\hbox {rank} } \def \supp {\hbox {supp} } \def \nl {\par} \newcount\pno \pno=0 \newcount\sno \sno=1 %\font\Bbb=msbm10 %\input /za/hebisch/my.mor \def \Cal { \Bbb } \long\def \p #1 #2 {\advpnr {\edef \pnrp {\hbox{ \pnr } } \global \let #1=\pnrp } \ppp {\pnr. {\bf Lemma}}. {#2} \par} \def \t #1 #2 {\advpnr {\edef \pnrp {\hbox{ \pnr } } \global \let #1=\pnrp } \ppp {\pnr . {\bf Theorem}}. {#2} \par} \def \nr #1 {\advpnr { \edef \pnrp {\hbox{ \pnr } } \global \let #1=\pnrp } \pnr} \def\P{{\it Proof: }} \def \advpnr {\global\advance\pno by 1} \def \pnr {(\the \sno.\the \pno)} %***************************** \newcount\refno \refno=1 \def \rf #1 #2 {\setbox3=\box2 \global\edef #1{\the \refno} \setbox2=\vbox{\unvbox3 \vskip 6pt plus2pt \hbox { \hbox to 2em{[\the \refno \global\advance\refno by1]\hfil} \vtop {\advance\hsize by-2em \noindent \strut #2 \hfil \strut } } } } \global \def \diam {\mathop{\hbox{ diam } }} \def \supp {\mathop{\hbox{ supp }} } \edef \int {\int\limits} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % End of macros % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \magnification=1200 \baselineskip=16pt \def\lin{\mathop{\rm lin}} \def\tr{\mathop{\rm tr}} {\bf Spectral multipliers on exponential growth solvable Lie groups} \vskip0.4cm {by Waldemar Hebisch\footnote{${}^1$}{ Partially supported by KBN grant {\tt 2 P301 052 07}. Partially supported by ECC under program "Fourier Analysis", contract no: CIPDCT 940001.}} \vskip0.2cm {Institute of Mathematics, Wroc\l{}aw University, pl. Grunwaldzki 2/4, 50-384 Wroc\l{}aw, Poland. {\it e-mail}: {\tt hebisch@math.uni.wroc.pl} {\it www}: {\tt http://www.math.uni.wroc.pl/{\char126}hebisch/Title.html} } \vskip0.4cm \rf {\Ch} {M. Christ, $L^{p}$ bound for spectral multiplier on nilpotent groups, {\it TAMS} 328 (1991), 73-81.} \rf {\CM} {M. Christ, D. M\"uller, On $L^{p}$ spectral multipliers for a solvable Lie group Geom. Funct. Anal. 6 (1196), 860--876.} \rf {\CSt} {J. L. Clerc, E. M. Stein, $L^{p}$ -multipliers for non-compact symmetric spaces, Proc. Nat. Acad. Sci. USA (1974), 3911--3912.} \rf {\Cm} {M. Cowling, Harmonic analysis on semigroups, Ann. of Math. 117 (1983), 267--283. } \rf {\CGHM} {M. Cowling, S. Giulini, A. Hulanicki, G. Mauceri, Spectral multipliers for a distinguished Laplacian on certain groups of exponential growth, {\it Studia Math.} 111 (1994), 103--121.} \rf {\Hs} {W. Hebisch, %A note on the paper of M.Cowling et. al. The subalgebra of \hbox{$L^{1}(AN)$} generated by the laplacean, Proc. AMS 117 (1993), 547--549.} \rf {\Hbo} {W. Hebisch, Boundedness of $L^{1}$ spectral multipliers for an exponential solvable Lie group, {\it Coll. Math. } 73 (1997), 155--164.} \rf {\Hom} {L. H\"ormander, Estimates for translation invariant operators in $L^{p}$ spaces, {\it Acta Math. 104} (1960), 93--140.} \rf {\Hus} {A. Hulanicki, Subalgebra of $L _{1 } (G) $ associated with Laplacian on a Lie group, {\it Colloq. Math. 31} (1974), 259-287.} \rf {\MM} {G. Mauceri, S. Meda, Vector-Valued Multipliers on Stratified Groups, {\it Revista Math. Iberoamericana} (6) 141-154.} \rf {\MS} {D. M\"uller, E. M. Stein, On spectral multipliers for Heisenberg and related groups, J. de Math. Pure et Appl. 73(1994) 413-440. } \rf {\St} {E. M. Stein, Topics in harmonic analysis related to the Littlewood-Paley theory, Ann. of Math. Stud. 63, Princeton Univ. Press, Princeton 1970.} \rf {\Te} {M. Taylor, $L^{p}$ -Estimate on functions of the Laplace operator, Duke Math. J. 58 (1989), 773-793. } {\bf Introduction} Let $M$ be a measure space and let $L$ be a positive definite operator on $L^2(M)$. By the spectral theorem, for any bounded Borel measurable function $F:[0,\infty)\mapsto {\Bbb C}$ the operator $F(L)f = \int_0^{\infty}F(\lambda) dE(\lambda)f$ is bounded on $L^2(M)$. We are interested in sufficient conditions on $F$ for $F(L)$ to be bounded on $L^p(M)$, $p\ne2$. We direct the reader to [\Ch], [\CSt], [\Cm], [\Hom], [\Hus], [\MM], [\St] and [\Te] for more background on various multiplier theorems. In this paper we assume $F$ is compactly supported and have some smoothness (finite number of derivatives) and we consider only the case $p=1$. Our measure space $G$ is semidirect product of stratified nilpotent Lie group $N$ and the real line. The operator $L$ is (minus) sublaplacian on $G$. Our group has exponential volume growth. The earlier theory suggested that one needs holomorphic $F$ for $F(L)$ to be bounded on $L^{1}$, however the recent results [\CGHM], [\Hs], [\Hbo] showed that estimates on only a finite number of derivatives of $F$ imply boundedness of $F(L)$ on $L^{1}$ on some solvable $G$ of exponential growth. In this case we say that $G$ (more precisely $L$) has $C^{k}$-functional calculus. On the other hand, Christ and M\"uller give an example of a solvable Lie group on which $F$ must be holomorphic. The problem is to find the condition on $G$ (and possibly $L$) which decides whether $G$ has a $C^{k}$-functional calculus or not. %There must be some connection with the symmetry (or non-symmetry) %of $L^{1}(G)$. %However, the structure of subalgebra of $L^1$ generated by $L$ seem to %be somewhat special so our may be different. Here, our condition is in terms of roots of adjoint representation of the Lie algebra of $G$. Our groups are of ``rank one'', but unlike [\Hbo], we allow groups with commutant of arbitrarily large step of nilpotency. %Let $N$ be a nilpotent Lie group with dilations $e^{sA}$. %We assume that $N$ is stratified. %\vskip 0.1cm {\bf Preliminaries}\nobreak \vskip 0.2cm \nobreak Let $N$ be a stratified nilpotent Lie algebra of step $q$, that is, \goodbreak $$ N=\bigoplus_{j=1}^qV_j \ , $$ and $[V_j,V_i]\subset V_{i+j}$ for every $1\leq j,i\leq q$. We assume that $V_1$ generates $N$. A {\it dilation structure} on a stratified Lie algebra $N$ is a one parameter group $\{e^{sD}\}$ of automorphisms of $N$ determined by $$ DX=jX \hbox{\qquad for }\ X\in V_j.$$ If we consider $N$ as a nilpotent Lie group with the multiplication given by the Campbell-Hausdorff formula $$ xy=x+y+ {1\over 2} [x,y]+...,$$ then $\{e^{sD}\}$ forms a group of automorphisms on the group $N$, and the nilpotent Lie group $N$ equipped with the dilations $\{e^{sD}\}$ is said to be a {\it stratified homogeneous group}. One easily checks that the Lebesgue measure on $N$ is also biinvariant Haar measure. There exists a number $Q>0$ such that for all bounded measurable $F\subset N$ $$|e^{sD}F|=e^{sQ}|F|,$$ this $Q$ is called the {\it homogeneous dimension} of $N$. It is evident that $$Q=\tr(D)=\sum_{j=1}^{q} j\cdot \hbox{dim}V_j.$$ We choose and fix a {\it homogeneous norm } on $N$, that is, a continuous, positive, symmetric, and smooth away from $0$ function $x\mapsto |x|$ which vanishes only for $x=0$, and satisfies $|e^{sD}x|=e^{s}|x|$. Let $G={\Bbb R}\times N$, with the multiplication given by the formula $$ (u_1,n_1)(u_2,n_2)=(u_1+u_2,e^{-u_2D}n_1n_2).$$ Then $$ (u,n)^{-1}=(-u,e^{uD}n^{-1}).$$ Let a weight function $w$ be defined as $w(u,n)=|n|^Q$. Note, that the Lebesgue measure is left invariant, modular function $\delta$ is given by $$ \delta(u,n)=e^{uQ},$$ and we have $$ \int f(g)dg=\int \delta(g)f(g^{-1})dg,$$ $$\int f(g)w(g)dg=\int\delta^2(g)f(g^{-1})w(g)dg.$$ Assume $X_1,\dots,X_m$ generate $N$ and are of order $1$ (that is $\lin\{X_1,\dots,X_m\}=V_1$). %$AX_j=X_j$, $j=1,\dots,m$). We will identify $X_j$ with left invariant vector fields on $G$. We denote by $\tilde X_j$ the corresponding right invariant vector fields on $G$. We put $\tilde X_0=\partial_u$. Right invariant vector fields generate {\it left} translations so $$ \langle \tilde X_j f, f \rangle = - \langle f,\tilde X_j f \rangle \qquad \hbox{\rm for $j=0,\dots,m$}.$$ We have $\delta|_N=1$ so $$ \langle X_j f, f \rangle = - \langle f,X_j f \rangle \qquad \hbox{\rm for $j=1,\dots,m$}.$$ We write $$ L=\sum_{j=0}^{m}\tilde X_j^2.$$ %Let $p_t$ be the convolution kernel of The heat kernel $p_t$ is defined by the formula $e^{tL}f=p_t*f$. In the sequel we will identify convolution operators with kernels (functions on $G$). The real operators in the algebra generated by $L$ are self-adjoint, which in terms of kernel reads: $$ F(L)(g)=\delta(g)F(L)(g^{-1}).$$ Note that the formula is valid for complex $F$. % (consider Let $d$ be (right) invariant riemannian metric on $G$. There is a constant $C$ such that $$ B_r=\{ (u,n) : d((u,n),0) < r \} \subset $$ $$ \{ (u,n): |u|0$ we compute $$ \langle \tilde X^2_j f,wf\rangle= \int (\tilde X^2_j f)(g)\bar f(g)w(g)dg$$ $$=\int (\delta(g)(X^2_j f)(g^{-1})) (\delta(g)\bar f(g^{-1}))w(g)dg$$ $$=\int \delta^2(g)( X^2_j f)(g^{-1}) \bar f(g^{-1})w(g)dg$$ $$=\int X^2_j f(g) \bar f(g)w(g)dg= \langle X^2_j f,wf\rangle. $$ %Similarly %$$\|w^{1/2}\tilde X_j f\|^2=\langle \tilde X_j f,w\tilde X_j f\rangle= %\langle X_j f,wX_j f\rangle =\|w^{1/2}X_j f\|^2 $$ Next $$ \langle X^2_j f,wf\rangle=-\langle X_j f,wX_j f\rangle- \langle X_j f,(X_jw) f\rangle.$$ Because of the homogeneity $|X_jw|\leq C |w|^{(Q-1)/Q}$ so $$ \Re\langle (i+\alpha)\tilde X^2_j f,wf\rangle= \Re(-(i+\alpha)\langle X_j f,wX_j f\rangle-(i+\alpha) \langle X_j f,(X_jw) f\rangle$$ $$ \leq -\alpha\|w^{1/2} X_j f\|^2 +C\|w^{1/2} X_j f\|\|(X_jw)w^{-1/2}f\|$$ $$ \leq -\alpha\|w^{1/2} X_j f\|^2 + \alpha\|w^{1/2} X_j f\|^2 +{1\over\alpha}C'\|w^{Q-2\over 2Q}f\|^2= {1\over\alpha}C'\|w^{Q-2\over 2Q}f\|^2$$ $$ \leq {1\over\alpha}C'\|w^{1/2}f\|^{2Q-4\over Q}\|f\|^{4\over Q}.$$ For $j=0$, $$ \Re \langle (i+\alpha)\tilde X^2_0 f,wf\rangle= -\alpha\|w^{1/2}\tilde X_0 f\|^2$$ Adding over $j$, we get $$\partial_s\phi(s) \leq C''{1\over\alpha} \|w^{1/2}f\|^{2Q-4\over Q}\|f\|^{4\over Q}$$ $$\leq C'''{1\over\alpha}\phi(s)^{Q-2\over Q}.$$ Hence %putting $\alpha=1/2s$ $$ \phi(s)\leq (\phi(0)^{2\over Q}+C{s\over\alpha})^{Q\over2}.$$ In terms of the semigroup (and putting $\alpha={1\over2s}$): $$ \|w^{1/2}%\exp((1+(i+\alpha)s)L)\delta_0 p_{1+is}\|\leq C(1+|s|)^{Q\over2}.$$ {\bf Improvements and open problems} \vskip 0.2cm One can easily generalize \aa to a larger class of groups (and sublaplacians). In fact, what counts is that $G$ is a semidirect product of nilpotent group and real line and that the matrix $D$ has eigenvalues with positive real parts. Then one can build $w$ making it homogeneous with respect to dilations generated by $D$. Inspection of the proof of \aa shows that the crucial role is played by the inequality $|X_j w|\leq C|w|^{(Q-1)/Q}$. However, this inequality may be replaced by weaker one: $|X_j w|\leq C(|w|^q+1)$ with $q<1$, which is valid even when $X_j$ are not eigenvectors of $D$. Moreover, it is not necessary to assume that $L$ is sum of squares of vector fields with distinguished field $\tilde X_0$ and other fields from Lie algebra of $N$. It is enough to assume that $L=\sum b_{j,k}\tilde X_j\tilde X_k$ with $b_{j,k}$ positive definite. Such an $L$ may be written in our restricted form at the cost of changing the one dimensional subgroup complementing $N$. The counterexample of M. Christ and D. M\"uller shows that one can not simply drop the assumption about eigenvalues. Finding correct conditions is an open problem. Also, while it is quite likely that theorem holds for $AN$ groups with multidimensional $A$, new ideas are needed to handle those. In the spirit of H\"ormander multiplier theorem one would like the have $$\|F(-tL)\|_{L^1}0$ and $s$. Our estimates gives much larger result for big $l$ (and moderate $|s|/l$). It may be interesting to note that the exponent $M$ we get seems to be far too big (on the Iwasawa $AN$ groups with one dimensional $A$ one has $M=3/2$), and getting the correct one should also give the correct scaling (dependence on $l$). Finally, let us mention that currently is not clear which properties of $L$ are really needed. Significant part of our argument goes for larger classes of operators, however, we don't know whether \aa remains valid for Schr\"odinger operators or for some higher order operators (say sums of even powers of vector fields). {\bf Acknowledgments}. I would like to thank J. Dziuba\'nski for valuable suggestions. {\bf References} \vskip 0.3cm \unvbox2 \end