% Author: Waldemar Hebisch % Title: Spectral multiplier on metabelian groups % % % Plain TeX with macros (included below). Uses AMS symbols % (amssym.def and amssym.tex) % \magnification=1200 \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} \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 8pt plus2pt% \hbox{\hbox to2em{[\the \refno \global\advance\refno by1]\hfil}% \vtop{\advance\hsize by-5em\noindent% #2\hfil}}}} %\strut#2\hfil\strut}}}} \global \def \diam {\mathop{\hbox{ diam } }} \def \supp {\mathop{\hbox{ supp }} } \edef \int {\int\limits} \long\def\nic#1{} \baselineskip=20pt \tolerance=9000 \hsize=5.6 true in \rf {\Al} {G. Alexopoulos, Spectral multipliers on Lie groups of polynomial growth, Proc. AMS 120 (1994), 973-979.} \rf {\An} {J. Ph. Anker, $L^p$ Fourier multipliers on Riemannian symmetric spaces of the non-compact type, Ann. of Math. 132 (1990), 597-628.} \rf {\And} {J. Ph. Anker, Sharp estimates for some functions of the Laplacian on noncompact symmetric spaces, Duke Math. J. 65 (1992), 257--297.} \rf {\Ast} {F. Astengo, Multipliers for a distinguished Laplacean on solvable extensions of H-type groups, Monatsh. Math. 120 (1995), 179--188.} \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 (1996), 860--876.} \rf {\CS} {M. Christ, Ch. Sogge, The weak type $L^{1}$ convergence of eigenfunction expansions for pseudodifferential operators, Invent. Math. 94 (1988), 421--453.} \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 {\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 {\Hgh} {W. Hebisch, Multiplier theorem on generalized Heisenberg groups, {\it Coll. Math. } 65 (1993) 231--239. } \rf {\HZ} {W. Hebisch, J. Zienkiewicz, Multiplier theorem on generalized Heisenberg groups II, {\it Coll. Math.} 69 (1995) 29--36.} \rf {\Hbo} {W. Hebisch, Boundedness of $L^{1}$ spectral multipliers for an exponential solvable Lie group, {\it Coll. Math. } 73 (1997), 155--164.} \rf {\Hme} {W. Hebisch, Spectral multipliers on exponential growth solvable Lie groups, {\it Math. Z. } 229 (1998), 435--441.} \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 {\LM} {J. Ludwig, D. M\"uller, Sub-Laplacians of holomorphic $L^p$-type on rank one $AN$-groups and related solvable groups, preprint.} \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. {\it Pure et Appl.} 73 (1994), 413-440. } \rf {\Muh} {S. Mustapha, Multiplicateurs spectraux sur certains groupes non-unimodulaires, Harmonic Analysis and Number Theory, CMS Conf. Proceedings, Vol 21, 1997.} \rf {\Mu} {S. Mustapha, Multiplicateurs de Mikhlin pour une classe particuli\`ere de groupes non-unimodulaires, Ann. Inst. Fourier 48 (1998), 957--966.} \rf {\Si} {A. Sikora, Multiplicateurs associ\'es aux souslaplaciens sur les groupes homog\`enes, {\it C.R. Acad. Sci. Paris, S\'erie I,} 315 (1992), 417--419.} \rf {\Te} {M. Taylor, $L^{p}$ -Estimate on functions of the Laplace operator, {\it Duke Math. J.} 58 (1989), 773-793. } {\bf Spectral multipliers on metabelian groups} by Waldemar Hebisch\footnote{${}^1$}{Partially supported by KBN grant {\tt 2 P03A 058 14} and European Commision via TMR network ``Harmonic analysis''} \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/} } \vskip0.4cm {\bf Introduction} Let $G$ be a Lie group, $X_j$ right invariant vector fields on $G$, which generate (as a Lie algebra) the Lie algebra of $G$, $$L=-\sum X_j^2.$$ Then $L$ is called sublaplacian, and it well-known that $L$ is positive definite and essentially selfadjoint on $C^{\infty}_{c}(G)\subset L^2(G)$, where $L^2(G)$ is taken with respect to a left-invariant Haar measure $dg$. 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 $L^2(G)$. We are interested in the behavior of $F(L)$ on $L^{p}$. This question has a long history. Classical results for polynomial growth case are [\Hom], [\Hus], [\Ch], [\MM], [\Al], [\CS], [\Si] for exponential growth [\CSt], [\Te], [\An], [\And]. Newer results show that connection with growth is more complicated [\Hgh], [\MS], [\HZ], [\Hs], [\CGHM], [\Ast], [\CM], [\Hbo], [\Muh], [\Hme], [\Mu], [\LM]. In this paper we consider $L^1(G)$ boundedness of $F(L)$ for (some) metabelian $G$ and a distinguished $L$ on $G$. Of the main interest is that the group is of exponential growth, and possibly higher rank. Previously positive results about higher rank groups where only about Iwasawa type groups. Also, our groups may be unimodular, so it is the second positive result (after [\Hbo]) about unimodular groups, and the first giving a family of examples. {\bf Results} Let $G={\Bbb R}^n\ltimes{\Bbb R}^m$, adjoint action is semisimple, $L=L_0+L_1$, $L_0$ lives on ${\Bbb R}^n$, $L_1$ lives on ${\Bbb R}^m$ and is a sum of (squares of) eigenvectors for adjoint action. More precisely, assume that $\lambda_j$, $j=1,\dots,m$ are linear forms on ${\Bbb R}^n$, $e_j$, $j=1,\dots,m$ is the canonical basis of ${\Bbb R}^m$, linear operator $A(x):{\Bbb R}^m\mapsto{\Bbb R}^m$ is given by the formula $A(x)e_j=\lambda_j(x)e_j$ and $$(x_1,y_1)(x_2,y_2)=(x_1+x_2,\exp(A(-x_2))y_1+y_2).$$ The right-invariant vector fields are: $$X_j=\partial_{x_j}$$ and $$Y_j=\exp(-\lambda_j(x))\partial_{y_j}.$$ We assume that $$L=-\sum X_j^2 - \sum Y_j^2=L_0+L_1.$$ We can transform general $L_0$ to our form, but for $L_1$ the assumption is somewhat restrictive. In this paper we identify convolution operators with functions: $$\exp(-tL)f=\exp(-tL)*f.$$ \t{\gt}{If $G$ and $L$ are as above, then there exists $C$ such that $$\|\exp(-(1+is)L)\|_{L^1}\leq C(1+|s|^{3m+n}).$$ } \t{\gg}{For every compactly supported $F\in C^{3m+n+1}$ the operator $F(L)$ is bounded on $L^1(G)$.} Theorem \gg is a straightforward consequence of \gt. Before the proof of \gt{} we need a lemma about ``symbols''. We consider it as well-known, but the form given below is adjusted to our needs. \p{\pl} {There is $C$ such that if $E$ is a normed space, $f:{\Bbb R}\mapsto E$, $|f|$ is integrable, $b\geq 1$, $$\sup |\hat f| \leq a,$$ $$\sup |\omega \partial_\omega \hat f(\omega)| \leq ab,$$ $$\sup |\omega^2 \partial_\omega^2 \hat f(\omega)| \leq ab^2,$$ then $$|f(x)|\leq {Cab\over |x|}.$$ } {\bf Remark} The lemma remains valid as long as $\hat f$ is reasonably defined (like $f\in S({\Bbb R},E^{*})^{*}$, where $S({\Bbb R},E^{*})$ consists of $E^{*}$ valued Schwartz class functions). {\it Proof}: Let $\phi\in C^{\infty}({\Bbb R})$ be such that $\phi(x)=1$ for $|x|\leq 1$ and $\phi(x)=0$ for $|x|\geq 2$. Fix $x_0\ne 0 $ and let $r={b\over |x_0|}$. Put $\hat f_1(\omega)=\phi(\omega/r)\hat f(\omega)$ and $\hat f_2(\omega)=(1-\phi(\omega/r))\hat f(\omega)$. We have $$|f_1(x)|\leq \int |\hat f_1|d\omega \leq \int_{-2r}^{2r}a d\omega =4ar,$$ and $$|x^2f_2(x)|\leq \int |\partial_\omega^2 \hat f_2(\omega)|d\omega.$$ By the Leibnitz formula $$\partial_\omega^2 \hat f_2(\omega)= (1-\phi(\omega/r))\partial_\omega^2 \hat f(\omega) - 2 r^{-1}\phi'(\omega/r)\partial_\omega \hat f(\omega) + r^{-2}\phi''(\omega/r) \hat f(\omega)$$ so $$ \int |\partial_\omega^2 \hat f_2(\omega)|d\omega\leq \int_{|\omega|>r} {ab^2\over \omega^2}d\omega + \int_{2r>|\omega|>r} 2Cr^{-1} ab\omega^{-1} d\omega + \int_{2r>|\omega|>r} Cr^{-2} a d\omega$$ $$\leq ab^2r^{-1}+4Cabr^{-1}+2Car^{-1}\leq C'ab^2r^{-1}.$$ Now $$|f(x_0)\leq |f_1(x_0)|+|f_2(x_0)|\leq 4ar + C'ab^2r^{-1}|x_0|^{-2}$$ $$ =(4+C')ab|x_0|^{-1}=C''ab|x_0|^{-1}.$$ $\triangle$ {\it Proof}: of \gt. We decompose the regular representation of $G$ using Fourier transform in $y$ variable. In coordinates $$L=-\Delta_x -\sum \exp(-2\lambda_j(x))\partial_{y_j}^2$$ where $\Delta_x=\sum\partial_{x_j}^2$. If we denote by $H_z$ the Fourier transform (in $y$ variable) of $L$ at $z$, then $$ H_z=-\Delta_x +\sum z_j^2\exp(-2\lambda_j(x)).$$ $\Re H_z\geq 0$, provided that $\Re z_j>\Im z_j$, $j=1,\dots,m$, so $z\mapsto\exp(-tH_z)$ is bounded holomorphic in the area given by the inequalities. Considering $(t+is)H_z$ we see that $\exp(-(t+is)H_z)$ is bounded and holomorphic as long as $\Re (t+is)z_j^2\geq 0$, $j=1,\dots,m$. Moreover, we can estimate the integral kernels $$\|\exp(-(2t+is)H_z)\delta_0\|_{L^2} \leq \|\exp(-(t+is)H_z)\| \|\exp(-tH_z)\delta_0\|_{L^2}.$$ By the Feynmann-Kac formula $$\|\exp(-tH_z)\delta_0\|_{L^2}\leq\|q_t\|_{L^2}=ct^{-n/4}$$ where $q_t$ is ordinary euclidean heat kernel. Consequently, by the Cauchy integral formula (for real $z$) $$\|\partial_z^\alpha\exp(-(t+is)H_z)\delta_0\|_{L^2} \leq C_\alpha |z_1|^{-\alpha_1}\cdot\dots\cdot|z_m|^{-\alpha_m}(1+{|s|\over t})^{|\alpha|} t^{-n/4}$$ %\|\exp(-{t\over 2}\Delta_x)\|_{L^2}$$ Applying \pl $m$ times we get $$\|\exp(-(t+is)L)(\cdot,y)\|_{L^2}\leq C''(|y_1|\cdot\dots\cdot|y_m|)^{-1}(1+{|s|\over t})^{m}t^{-n/4}.$$ In [\Hme] (as the first step in proof of Theorem (1.1)) we proved that $$ \int | \exp(-(1+is)L)(g)| e^{d(g,0)}dg \leq C\exp (Cs^2)\leqno(1.4)$$ where $d(x,y)$ is the optimal control distance associated to $L$. One easily checks that $$\{g:d(g,0)r}|\exp(-(1+is)L)(g)|dg$$ $$+ \int_{|x|r}|\exp(-(1+is)L)(g)|dg \leq e^{-r}\int |\exp(-(1+is)L)(g)|\exp(d(g,0))dg$$ $$ \leq \exp(-Cs^2)C\exp(Cs^2)=C.$$ Next $$I_j\leq |A_j|^{1/2}\|\exp(-(1+is)L)\|_{L^2}\leq C|s|^n.$$ Finally $$I_0=\int_{\exp(-mcs^2)\leq|y_j|\leq\exp(cs^2)}\int_{|x|