\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{} \overfullrule=0pt \def\Gl{{\tilde \Gamma}} \def\FA{F} \def\FB{R} \def\FC{S} \def\FD{H} \def\id{\mathop{\rm id}\nolimits} \rf{\Br} {I. D. Brown, Dual topology of a nilpotent Lie group, Ann. Sci. \'Ecole Norm. Sup., 6 (1973), 407--411.} \rf {\CGGP} {M. Christ, D. Geller, P. G\l owacki, L. Pollin, Pseudodifferential operators on groups with dilations, Duke Math. J. 68 (1992), 31--65.} \rf {\CW} {R. Coifman, Weiss, Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes, Springer LN 242,Berlin 1971.} \rf {\CWt} {R. Coifman, G. Weiss, Transference methods in analysis, %. Lectures from the CBMS Regional Conference at the University % of Nebraska, May 31 -- June 4, 1976 AMS Regional Conference Series in Mathematics 31, Providence 1977.} \rf {\Dzm} {J. Dziuba\'nski, A remark on a Marcinkiewicz-H\"ormander multiplier theorem for some non-differential convolution operators, {\it Coll. Math.} 58 (1989), 77--83.} \rf {\FS} {G. B. Folland, E. M. Stein, Hardy spaces on Homogeneous Groups, Princeton University Press, 1982.} \rf {\GlInv} {P. G\l owacki, Stable semi-groups of measures as commutative approximate identities on non-graded homogeneous groups, Invent. Math. 83 (1986), 557--582.} \rf {\Glis} {P. G\l owacki, The inversion problem for singular integral operator on homogeneous groups, Studia Math. 87 (1987), 53--69.} \rf {\Glo} {P. G\l owacki, The Rockland condition for nondifferential convolution operators, Duke Math. J. 58 (1989), no. 2, 371--395. } \rf {\GP} {P. G\l owacki, The Rockland condition for nondifferential convolution operators II, Studia Math. 98 (1991), 99--114. } \rf {\HN} {B. Helffer, J. Nourrigat, Caract\'erisation des op\'erateurs hypoelliptiques homog\`enes invariants \`a gauche sur une grupe gradu\'e, Com. P. D. E. 3 (1978), 889--958. } % ?? H-N 172-176 1985 % % JN Com. in PDE 1991 - dowód dla operatorów pseudoróżniczkowych \rf {\HJ} {A. Hulanicki, J.W. Jenkins, Nilpotent Lie groups and summability of eigenfunction expansions of Schr\"odinger operators, {\it Studia Math.} 80 (1984), 235--244. } \rf {\K} {A.A. Kirillov, Unitary representations of nilpotent Lie groups (in Russian), Uspekhi Mat. Nauk 17 (1962), 57--110. } %\rf {\Noi} {J. Nourrigat, Systemes sous-elliptiques II, %Inv. Math 104(1991), 341--400.} \rf {\Noe} {J. Nourrigat, In\'egaliti\'es $L^2$ et repr\'esentations de groupes nilpotents, J. of Funct. Anal. 74 (1987), 300--327.} \rf {\Nol} {J. Nourrigat, $L^2$ inequalities and representations of nilpotent groups, C.I.M.P.A. School of Harmonic Analysis, Wuhan.} {\bf On operators satisfying Rockland condition\footnote{} {\rm 1991 {\it Mathematics Subject Classification}: Primary 22E30.}} \vskip 0.5cm {by Waldemar Hebisch\footnote{${}^1$} {Praca wykonana w ramach projektu badawczego {\tt 2 P301 052 07}.}} \vskip0.5cm {\bf Abstract} Let $G$ be a homogeneous Lie group. We prove that for every closed, homogeneous subset $\Gamma$ of $G^{*}$ which is invariant under the coadjoint action, there exists a regular kernel $P$ such that $P$ goes to $0$ in any representation from $\Gamma$ and $P$ satisfy Rockland condition outside $\Gamma$. We prove a subelliptic estimate as an application. {\bf Introduction} %Helfer-Nourriat Theorem showed that Rockland operators The purpose of this paper is to construct operators which satisfy Rockland condition in a given set of representations $\Gamma$, and are equal to $0$ outside $\Gamma$. Rockland operators satisfy remarkable subelliptic estimates ([\HN], [\GlInv], [\Glo], [\GP], [\Noe] see also [\Nol]) making them good substitute for elliptic operators on homogeneous groups. Christ et al. [\CGGP] gave a calculus for pseudodifferential operators on homogeneous groups: the formulas for products and adjoints and criteria for existence of left or right parametrices (generalizing results of [\Glis]). However, one should note that great flexibility of classical calculus of pseudodifferential operators is in large part due to the ease of constructing scalar functions (cutoffs and partitions of unity). In homogeneous group case we want to pre-specify operators in a set of representations and still have a regular kernels --- this is not straightforward --- in fact not always possible. Our kernels may serve as cutoffs on spectral side (for the spatial cutoffs one simply uses multiplications with smooth functions). The conditions we impose seem to be necessary. We present also a simple application in which we derive some $L^p$ estimates. {\bf Acknowledgements} I would like to thank Jean Nourrigat for valuable suggestions. %Such operators may be used %as cutoff or give a good partition of unity. {\bf Preliminaries} We consider a homogeneous group $G$, that is a nilpotent Lie group equipped with a family of automorphisms ({\it dilations\/}) $\{\delta_{t}\}_{t>0}$ such that $\delta_{t}\delta_{s}=\delta_{ts}$ and for all $x\in G$ we have $\delta_{t}x\rightarrow e$ if $t\rightarrow 0$. The reader may wish to consult [\FS] (our definition is a bit more general). We will identify $G$ with its Lie algebra via the exponential map, and write $0$ instead of $e$. With our identification all $\delta_{t}$ became linear maps. % and one easily checks that As $\det(\delta_{t})$ must be a power of $t$ there exists a number $Q>0$ such that for all bounded measurable $A\subset G$ $$|\delta_{t}A|=t^{Q}|A|,$$ this $Q$ is called the homogeneous dimension of $G$. More general, one can take $t$ to be discrete, that is consider dilation operator $D$ such that $D^{-k}x\rightarrow e$ if $k\rightarrow \infty$. %but in this paper this gives equivalent results. %present situation A distribution $T$ on $G$ is said to be a kernel of order $r\in {\Cal C}$ if $T$ coincides with a locally finite measure away from the origin, and is homogeneous of degree $-r-Q$, that is satisfies $$ (f\circ \delta_{t},T)=t^{r}(f,T) $$ for all $f\in C^{\infty}_{c}(G)$ and $t>0$. We extend action of dilation to distributions by the formula $$ (f,\delta_{t}T)=(f\circ\delta_{t},T).$$ Then $T$ is a kernel of order $r$ iff for all $t>0$ $$\delta_{t}T=t^r T.$$ A kernel is called regular if it coincides with a smooth function away from the origin. In the sequel we will identify right-invariant vector fields on $G$ with distributions supported in $\{0\}$. More precisely, there is one to one correspondence between right-invariant differential operators and distributions supported in $\{0\}$. To get the identification we write $$ (X,f)=Xf(0).$$ Then $Xf=X*f$, and for the left-invariant field $\tilde X$ corresponding to $X$ we have $\tilde Xf=f*X$. We also note that dilating a vector field as an element of Lie algebra and as a distribution gives the same result. %In the same way one %may identify right-invariant differential operators with %distributions (supported in $\{0\}$) and in this case the correspondence %is one to one. For a unitary representation $\pi$ of $G$ on a Hilbert space $H$ and a kernel $T$ of order $r$, $\Re(r)>0$, the operator $\pi(T)$ is defined on the space $C^{\infty}(\pi)$ of smooth vectors for $\pi$ by $$ (g,\pi(T)f)=(\phi_{f,g},T) $$ where $\phi_{f,g}(x)=(g,\pi(x)f)$. Equivalent definition is: $$ \pi(T)f=T*\psi_{f}(e) $$ where $\psi_{f}(x)=\pi(x)f$. This definition also makes sense for uniformly bounded representations on Banach spaces. If $\Re(r)\leq 0$ the situation is more tricky. %Put $s=\Re(r)$. For a regular kernel $T$ one may find $h\in C^{\infty}_c(G)$ such that $$ Tf=\sum_{k}2^{-rk}\delta_{2^k}(h)*f.$$ For $r=0$ one must have $\int h=0$ (otherwise $T$ would not be a distribution), and the Cotlar-Stein lemma shows that the sum defining $T$ is strongly convergent in any unitary representation of $G$ to a bounded operator. If $\Re(r)<0$ then $T$ defines unbounded operator on $L^{2}(G)$ (to see that $T$ is densely defined see [\CGGP]). The following lemma shows that the problem is caused by trivial representation. \nic { \p {\ba} { Let $G$ be a nilpotent Lie group. Let $\pi$ be nontrivial irreducible unitary representation of $G$. Then for every $f\in C^{\infty}(\pi)$ there are vector fields $X_{j}$ and $f_j\in C^{\infty}(\pi)$ such that $f=\sum \pi(X_j)f_j$.} {\it Proof.} Let $X_j$ generate $G$. Put $L=\sum X_j^2$. It is known that $L$ is invertible in $\pi$. So $$f=LL^{1}f=\sum X_j(X_jL^{-1}f).$$ } \p {\bb} { Let $G$ be a homogeneous nilpotent Lie group, $\pi$ be nontrivial irreducible unitary representation of $G$ %$f\in C^{\infty}(\pi)$ and $h$ be a Schwartz class function on $G$. Then for any $m$ there is a continuous seminorm $C_m(\cdot)$ such that $\|\pi(\delta_t(h))\|\leq C_m(h)(1+t)^{-m}$.} {\it Proof.} We fix a scalar product (so also a norm $|\cdot|$) on the Lie algebra of $G$. Note, that there exists $\alpha>0$ such that if $X$ is an element of Lie algebra of $G$ then $|\delta_{t^{-1}}(X)|\leq C(1+t)^{-\alpha}$ for $t>1$. Let $X_j$ span the Lie algebra of $G$. Put $L=\sum X_j^2$. It is known that $L$ is invertible in $\pi$. So $$\id_\pi=\sum \pi(X_j)(\pi(X_j)(\pi(L)^{-1}))=\sum \pi(X_j)E_j$$ where $E_j=\pi(X_j)\pi(L)^{-1}$ are bounded operators in $\pi$. Next, %using \ba, we write $$ \pi(\delta_t(h))=\pi(\delta_t(h))\sum \pi(X_j)E_j= \sum \pi(\delta_t(h*\delta_{t^{-1}}(X_j)))E_j.$$ Inductively, for any natural $l$ $$ \pi(\delta_t(h))=\sum \pi(\delta_t(h* \delta_{t^{-1}}(X_{j_1})*\dots*\delta_{t^{-1}}(X_{j_l}))) E_{j_l}\cdot\dots\cdot E_{j_1}$$ so $$ \|\pi(\delta_t(h))\|\leq C_l \max \|\pi(\delta_t(h* \delta_{t^{-1}}(X_{j_1})*\dots*\delta_{t^{-1}}(X_{j_l})))\|\leq$$ $$C_l \max \|h*\delta_{t^{-1}}(X_{j_1})*\dots*\delta_{t^{-1}}(X_{j_l})\|_{L^{1}}\leq$$ $$C_l \max C_{h,l}|\delta_{t^{-1}}(X_{j_1})|\cdot\dots\cdot |\delta_{t^{-1}}(X_{j_l})|\leq$$ $$ C'(h,l)(1+t)^{-\alpha l}$$ which gives the claim. If we put $$V_s=\{f\in L^1_{loc}(G)\cap C^{\infty}(G-\{0\}): \forall_{\phi\in C^{\infty}_c(G-\{0\})}\lim_{t\rightarrow 0} t^s\phi\delta_tf=0;$$ $$\hbox{\rm uniformly with all derivatives}\}$$ then $V_s$ is a locally convex metrizable vector space and Schwartz class functions are dense in $V_s$. For $0>\Re(r)>-s$ regular kernels of order $r$ are in $V_s$ and \bb shows that $\pi$ has (unique) extension form Schwartz class to $V_s$. \vskip0.4cm\relax% {\bf Main Results}\sno=2\nobreak% \vskip0.4cm\nobreak\leavevmode% Let $\pi_{l}$ for $l\in G^{*}$ be the representation associated with $l$ according to Kirilov theory (cf. [\K]). \t{\ra} { Let $G$ be a homogeneous Lie group with dilations $\{\delta_{t}\}_{t>0}$, and $\Gamma$ be a closed subset of $G^{*}$ such that $Ad^{*}(G)\Gamma \subset \Gamma$, $\forall_{t>0}\delta_{t}\Gamma \subset \Gamma$. For every $\alpha\geq 0$ there exists a regular kernel $P$ of order $\alpha$ such that for all ${l\in\Gamma}$ we have $\pi_{l}(P)=0$ and for all ${l\notin\Gamma}$ the operator $\overline{\pi_{l}(P)}$ is positive definite and injective on its domain. For every $0>\alpha>-Q$ there exists a kernel satisfying conditions above, except for $l=0$. Moreover, there is a Schwartz class function $H$ on $G$ such that for all ${l\in\Gamma}$ we have $\pi_{l}(H)=0$ and for all ${l\notin\Gamma}$ the operator $\pi_{l}(H)$ is positive definite and injective. } \nic { \t{\rv} { Let $G$ be a homogeneous Lie group with discrete dilation $D$, and $\Gamma$ be a closed subset of $G^{*}$ such that $Ad^{*}(G)\Gamma \subset \Gamma$, $D\Gamma=\Gamma$. For every $\alpha\geq 0$ there exists a regular kernel of order $\alpha$ such that for all ${l\in\Gamma}$ we have $\pi_{l}(P)=0$ and for all ${l\notin\Gamma}$ the operator $\overline{\pi_{l}(P)}$ is positive definite and injective on its domain.} } {\it Proof.} It is enough to prove the theorem only for $-Q<\alpha \leq 0$ and small %positive $\alpha > 0$. Indeed, taking sufficiently high power of $P$ we get $\alpha$ as large as we wish, without destroying other properties of $P$. Moreover, we only need to prove the last claim, that is to construct a Schwartz class function $H$ such that for all ${l\in\Gamma}$ we have $\pi_{l}(H)=0$ and for all ${l\notin\Gamma}$ the operator $\pi_{l}(H)$ is positive definite and injective. If we have such a function and $\alpha$ is small enough, then $$P=\int_{0}^{\infty}t^{-\alpha}\delta_{t}H{dt\over t }$$ give as a regular kernel of order $\alpha$ having the required properties. Indeed, $0$ is in $\Gamma$ so $\int H=0$. Moreover there is $\alpha_{0}>0$ such that for all $00$, it is easy to check that $$\pi(P)\subset\int_{0}^{\infty}t^{-\alpha}\pi(\delta_{t}H){dt\over t }$$ and that the right hand side defines closed injective operator ( we compute the integral applying function under the integral to a vector --- the domain is the set of all vectors for which the integral is convergent). For $\alpha\leq 0$ we get the same conclusion using \bb (condition $\alpha >-Q$ is used to prove that $P$ is a regular kernel). We are going to build $H$. Let us recall that by [\Br] the set $\Gl=\{\pi_{l}:l\in\Gamma\}$ is closed in the Fell topology of the space of representations. Fix $p\notin\Gamma$. $Ad^{*}(G)\Gamma \subset \Gamma$ implies $\pi_{p}\notin\Gl$. By the definition of Fell topology and density of $C^{\infty}_{c}(G)$ in $C^{*}(G)$, there exists a function $\FA\in C^{\infty}_{c}(G)$ such that $$\|\pi_{p}(F)\|=1$$ and for all $l\in\Gamma$ $$\|\pi_{l}(F)\|<1/10.$$ Replacing $\FA$ by $\FA^{*}*\FA$ we may assume that $\FA$ is positive definite. Choose $\phi\in C^{\infty}_{c}(\Bbb R)$ such that $\phi(1)=1$, $\phi\geq 0$, $\supp(\phi)\subset[1/10,2]$. By the spectral theorem the operator $$\phi(\pi_{p}(\FA))\ne0$$ while for all $l\in\Gamma$ $$\phi(\pi_{l}(\FA))=0.$$ Using functional calculus, as for example in [\HJ], we show that there exists a Schwartz class function $\FB$ on $G$ such that (as a convolution operator on $L^{2}(G)$) $\FB=\phi(\FA)$. Approximating $\phi$ by polynomials we see that for all $l$ we have $$\phi(\pi_{l}(\FA))=\pi_{l}(\FB).$$ We also note that the set of $\pi$ such that $\pi(\FB)\ne0$ is open (by definition). To summarize, we constructed $\FB$ such that for all $\pi\in\Gl$ $$\pi(\FB)=0,$$ the set $U_{\FB}=\{\pi(\FB)\ne0\}$ is open and $\pi_{p}\in U_{\FB}$. Since Fell topology has a countable basis, there exists a sequence $\{\FB_{i}\}_{i\in\Bbb N}$ such that the complement of $\Gl$ is the union of $U_{\FB_{i}}$. Therefore, putting $\FC=\sum a_{i}\FB_{i}$ where $a_{i}$ are positive and small enough for $S$ to be a Schwartz class function we see that for all $\pi$ in $\Gl$ $$\pi(\FC)=0$$ and $\pi(\FC)\ne0$ on the complement of $\Gl$. To finish the proof we need the following lemma: \p {\rb} { If $\pi$ is an irreducible unitary representation of $G$, the sequence $\{g_{j}\}_{j\in\Bbb N}$ is dense in $G$, $f\in L^{1}(G)$, $\pi(f)\ne0$, $\pi(f)\geq 0$, the sequence $\{c_{j}\}_{j\in\Bbb N}$ is positive and summable, then $$A=\pi(\sum c_{j} \delta_{g_{j}^{-1}}f\delta_{g_{j}})$$ (where $\delta_{g_{j}}$ means convolution operator with unit mass at $g_j$) is injective.} {\it Proof.} Suppose, on the contrary that $A$ is not injective. Then, there exists a nonzero $v$ such that $$(Av,v)=\sum c_{j} (\pi(\delta_{g_{j}^{-1}})\pi(f)\pi(\delta_{g_{j}})v,v)=0,$$ hence for each $j$ $$ (\pi(f)^{1/2}\pi(\delta_{g_{j}})v,\pi(f)^{1/2}\pi(\delta_{g_{j}})v)=0,$$ or simply $$\pi(f)^{1/2}\pi(\delta_{g_{j}})v=0.$$ Since $\pi$ is irreducible, closed linear span of $\pi(\delta_{g_{j}})v$ gives the whole space, so $\pi(f)^{1/2}=0$. As $\pi(f)$ is nonzero this gives a contradiction. Choosing $c_{j}=\exp(-j -|g_{j}|)$ and applying $\rb$ we conclude that $$ \FD=\sum c_{j} \delta_{g_{j}^{-1}}\FC\delta_{g_{j}}$$ is a Schwartz class function such that for $\pi$ in the complement of $\Gl$ the operator $\pi(\FD)$ is injective and for $\pi$ in $\Gl$ $$ \pi(\FD)=0$$ which ends the proof. {\bf Remark} If $P$ is a regular kernel of order $\alpha$, $\Gamma=\{l:\pi_l(P)=0\}$, then $Ad^{*}(G)\Gamma \subset \Gamma$, $\forall_{t>0}\delta_{t}\Gamma \subset \Gamma$ and $\Gamma$ is closed. More precisely, if $\Re\alpha<0$, then $\Gamma-\{0\}$ is closed in $G^{*}-\{0\}$. The first condition is clear. $\Gamma$ is invariant under dilations because $P$ is homogeneous. To see that $\Gamma$ is closed assume first that $\Re \alpha>0$. Let $\phi_n$ be an approximate unit in $L^1$ consisting of $C^{\infty}_{c}$ functions. $\phi_n*P\in L^1$ so $\Gamma_n=\{l:\pi_l(\phi_n*P)=0\}$ is closed. As $\Gamma=\bigcap_n \Gamma_n$ we see that $\Gamma$ is closed. If $\Re \alpha \leq 0$, then we compose $P$ with a kernel $R$ such that $\pi_l(R)$ is injective on smooth vectors for all $l\ne 0$ and $P*R$ have order with positive real part. \vskip0.3cm {\bf An application}\sno=3\relax% \nobreak\vskip0.4cm\nobreak\leavevmode% As an application of our construction we will give an extension to $L^p$ of a theorem by J. Nourrigat ([\Noe] Th\'eor\`eme 1.3). We need some setup to state the theorem. Let $\Omega$ be measure space with measure $\mu$. Assume $G$ act on $\Omega$ preserving the measure. Let $\phi:G\times\Omega\mapsto \Bbb C$ be a (measurable) cocycle for this action, that is $|\phi|=1$ and for all $g_1,g_2\in G$ and all $x\in\Omega$ $$ \phi(g_1g_2,x)=\phi(g_1,x)\phi(g_2,g_1^{-1}x).$$ Then the formula $$ \pi(g)f(x)=\phi(g,x)f(g^{-1}x)$$ gives continuous representation of $G$ which act trough isometries on $L^{p}(\Omega)$, $1\leq p < \infty$ (on $L^{\infty}$ we get isometries, but the action is only weak-$*$ continuous). We say that $\pi$ is a {\it cocycle representation}. The set of smooth vectors $C^{\infty}(\pi)$ is defined as usual (of course it may depend on $p$). Let us note that $\pi(C_{c}^{\infty}(G))(L^1\cap L^{\infty})$ is dense in $C^{\infty}(\pi)$ so we may do all the calculations on the common core. We also note that the usual construction of $\pi_l$ gives cocycle representation so we may consider $\pi_l$ as representations on $L^p$. Let us also sketch the proof of the following well-known lemma: \p {\kol} {If $W$ is a regular kernel of order $\alpha$, $\Re(\alpha)=0$ and $W$ gives bounded operator on $L^{2}(G)$ then $W$ gives bounded operator on $L^{p}(G)$, $10}$, and $\Gamma$ be a closed subset of $G^{*}$ such that $Ad^{*}(G)\Gamma \subset \Gamma$, $\forall_{t>0}\delta_{t}\Gamma \subset \Gamma$. Let $\pi$ be a cocycle representation of $G$ such that all irreducible components of $\pi$ are of the form $\pi_l$ with $l\in\Gamma$. Let $R$ be a regular kernel of order $\alpha$, $\Re(\alpha)>0$ such that for all $l\in\Gamma$, $l\ne 0$ the operator $\pi_{l}(R)$ is injective on $C^{\infty}$ vectors of $\pi_{l}$. Then for every $1-Q$ operator $P^{s}$ is given by the regular kernel of order $s$, for small $s>0$ $P^s$ generates a semigroup of symmetric probability measures (see [\CGGP] Theorem 6.1 and [\GlInv]). Choose $s_01$ we simply replace $R$ by $R^k$. \nic { the semigroup $e^{tB}$ %a Rockland operator generated by $B$ satisfy $$|X^{\gamma}e^{tB}\delta_{0}|\leq ...$$ We define kernel $K$ on $G\times R$ by the formula: $$ =\int_0^{\infty}e^{-t}dt.$$ From the spectral theorem one easily checks that $\|B^{1/2}KB^{1/2}\|_{L^2,L^2}\leq 1$. As $\|Rf\|\leq \|B^{1/2}f\|$ so $$\|RKR^{*}\|_{L^2,L^2}\leq 1.$$ Similarly $\|KR^{*}\|_{L^2,L^2}\leq 1$, $\|RK\|_{L^2,L^2}\leq 1$ and $\|K\|_{L^2,L^2}\leq 1$. Using ... we have $\|AKR^{*}\|_{L^2,L^2}\leq C$ and $\|AK\|_{L^2,L^2}\leq C$. One easily checks using the estimates for $e^{tB}\delta_{0}$ that both $AKR^{*}$ and $AK$ are $L^1$ kernels giving Calderon-Zygmund operators. Hence both are bounded on $L^p(G\times R)$. Note that one may extend representation $\pi_l$ to the representation $\tilde \pi_l$ of $G\times R$ by the formula $\tilde \pi_l((x,t))=\pi_l(x)$. By the transference principle we have $$ \|\pi_l(A)f\|_{L^p}=\|\tilde \pi_l(AKR^{*}R+AK)f\|_{L^p}\leq$$ $$ \|\tilde \pi_l(AKR^{*})\tilde \pi_l(R)f\|_{L^p}+ \|\tilde \pi_l(AK)f\|_{L^p}$$ $$\leq \|AKR^{*}\|_{L^p(G\times R),L^p(G\times R)}\|\pi_l(R)f\|_{L^p}+ \|AK\|_{L^p(G\times R),L^p(G\times R)}\|f\|_{L^p}\leq $$ $$ C(\|\pi_l(R)f\|_{L^p}+\|f\|_{L^p}.$$ } \vfil \vskip1cm {\bf References\par}\nobreak\vskip1cm\nobreak\unvbox2 {Institute of Mathematics, \par Wroc\l{}aw University, \par pl. Grunwaldzki 2/4, \par 50-384 Wroc\l{}aw, %\par \hskip1cm Poland. {\it e-mail}: {\tt hebisch@math.uni.wroc.pl} {\it www}: {\tt http://www.math.uni.wroc.pl/{\char126}hebisch/Title.html} } and {Institute of Mathematics, Polish Academy of Sciences, ul. \'Sniadeckich 8, 00-950 Warszawa, \hskip1cm Poland} \end