% 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