\magnification=1200
%\def \supp {\hbox {supp} }
%\def \dom{\hbox {dom} }
%\def \nl {\par}
\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{}
%\input header
\def \dom{\hbox {dom} }
\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 {\DI} {J. Dixmier, $C^{*}$-algebras, North-Holland, 1977.}
\rf {\FS} {G.B. Folland and E.M. Stein, Hardy Spaces on
Homogeneous Groups, Princeton Univ. Press, 1982.}
\rf {\GI} {P. G\l owacki,
Stable semigroups of measures as commutative approximate identities on nongraded homogeneous groups,
Invent. Math. 83 (1986), no. 3, 557--582.}
\rf {\GD} {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 {\GS} {P. G\l owacki,
An inversion problem for singular integral operators on homogeneous groups.
Studia Math. 87 (1987), no. 1, 53--69.}
\rf {\GH} {P. G\l owacki, W. Hebisch, Pointwise estimates for the
densities of stable
measures on homogeneous groups, Studia Math 104 (1993) 243--258.}
\rf {\GHu} {P. G\l owacki, A. Hulanicki,
A semigroup of probability measures with nonsmooth differentiable
densities on a Lie group,
Colloq Math. 51 (1987), 131--139.}
\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.}
\rf {\Hg} {W. Hebisch, On operators satisfying Rockland condition,
Studia Math. (to appear)}
\rf {\K} {A.A. Kirillov, Unitary representations of nilpotent Lie
groups (in Russian), Uspekhi Mat. Nauk 17 (1962), 57--110.}
\rf {\Nol} {J. Nourrigat, $L^2$ inequalities and representations
of nilpotent groups, C.I.M.P.A. School of Harmonic Analysis,
Wuhan.}
%**************************************
%
% OZNACZENIA
\def \ft{\tilde \phi}
\def \ir{Irr(N)}
\def \Dil{D}
% \def \alg {Alg}
{\bf New proof of subelliptic estimates for Rockland operators}
by Waldemar Hebisch
{\it Incomplete draft}
{\bf Introduction}
Our results are more general than that of G\l owacki: we drop the
symmetry assumption and allow non-smooth operators of arbitrarily high
order.
Moreover trivial but useful changes - we require only discrete family of
dilations which are not
necessarily diagonal, we allow order to be a complex number and
we allow operator valued kernels.
We think that more important is simplification of the proof.
The idea of the proof is as follows. We proceed by induction on the
dimension of the group. The abelian case is easy. Then (as in [\GI])
we consider representations induced from the characters of the
center. Estimate in representation induced from trivial character is
the inductive assumption. This estimate extends to ''nearby''
representations. Homogeneity will give as the full set of
representations, provided we can estimate $L^2$ norm by our operator.
Here is the novelty of our approach - we observe that the estimate
$\|v\|\leq C\|Pv\|$ is equivalent to $\|(1+P^*P)^{-1}\|<1$. Next,
using the estimates from previous steps we prove that
$(1+P^*P)^{-1}$ is ''locally'' in $C^{*}(N)$. Then, by the theory of
$C^{*}$ algebras, the norm is realized in some irreducible
representation. But in irreducible representation our operator has
discrete spectrum, so the Rockland condition implies the estimate.
Tricky part is that we must use kind of bootstrap argument --- we
are unable to give direct proof of various properties of
$(1+P^*P)^{-1}$ and we derive them via comparison with ''good''
operator $R$. In other words, before we prove the theorems we
need to know that there is at least one operator for which the
conclusions are valid. Such an operator do exists --- we take (powers of)
a generator of (stable) semigroup of probability measures, for which
most of the theory is the same as general case, but few crucial steps
are much easier.
\nic {
\p {\pa} {If $D$ is an automorphism of a real Lie algebra $N$, then
here exists a positive integer $l$ and one parameter group $\{T(t)\}_{t\in R}$
of automorphisms of
$N$ such that for any integer $k$
$$T(e^{lk})=D^{lk}.$$
%If $N$ is complex then $$T(e^{k})=D^{k}.$$
}
P. Let ${\bar N}=N\oplus iN$ be the complexification of $N$. We consider
$D$ as an automorphism of ${\bar N}$ commuting with complex conjugation.
For a complex
$\alpha$ put
$$V_{\alpha}=\{x\in {\bar N}: (D-\alpha)^{dim N}x=0 \}.$$
Of course $$V_{\alpha}$ are preserved by $D$.
One easily checks that $[V_{\alpha},V_{\beta}]\subset V_{\alpha\beta}$.
As $D$ is injective $V_{0}={0}$. We need a real (that is commuting with
complex conjugation) logarithm $A$ of $D^{2}$ which also on each $V_{\alpha}$
has all eigenvalues equal. If we want complex $A$ that is easy: logarithm has
analytic branch in a neighbourhood of the spectrum of $D$ so we may apply
it to $D$ and get even a logarithm of $D$. For $D$ restricted to the
sum of $V_{\alpha}$ over all $\alpha$ which are not real negative we may
choose a branch of logarithm commuting with complex conjugation, so on
this subspace we get real logarithm of $D$. On $V_{\alpha}$ with negative
$\alpha$ we take a logarithm of $D^{2}$ as $D^{2}$ has positive eigenvalues
on those $V_{\alpha}$. This way we get real $A$ such that $D^{2}=\exp(A)$
and $A$ has on each $V_{\alpha}$ only one eigenvalue. Let $B$ be the
semisimple part of $A$ that is on each $V_{\alpha}$ $B$ equals to
the multiplication by the eigenvalue of $A$. Of course $B$ commutes with $A$.
For any real $t$ $\exp(tB)$ is an automorphism of ${\bar N}$
(or we need a correction ??).
Then $A_{1}=A^{l}\exp(-lB)$ is a unipotent automorphism of ${\bar N}$.
$A_{1}$ has a nilpotent logarithm. $\exp(tX)$ is a polynomial which for
infinitely many $t$ gives us an automorphism of ${\bar N}$, hence for all
$t$.
}
{\bf Preliminaries}
Let $N$ be a nilpotent Lie group with a discrete family of dilations
$\{\Dil^k\}_{k\in\Bbb Z}$. We assume that
$$\Dil^k(x)\Dil^k(y)=\Dil^k(xy),$$
and
$$\lim_{k\rightarrow-\infty}D^kx=0$$
The homogeneous dimension $q$ of $N$ is defined by the formula
$$|D^k(F)|=2^{kq}|F|$$
for all bounded measurable $F\subset N$.
A distribution $T$ on $N$ with values in linear operator between
some finite dimensional
vector spaces is said to be a kernel
if $T$ coincides with a locally finite measure outside any neighbourhood of
origin (more general the convolver norm of $T$ times any
$C^{\infty}_c(N-\{0\})$ function is finite and \dots).
A kernel $T$ is said to be a homogeneous kernel
of order $r\in {\Bbb C}$ if $T$
is homogeneous distribution of degree $-r-q$, that is satisfies
$$
(f\circ \Dil^{k},T)=2^{rk}(f,T)
$$
for all $f\in C^{\infty}_{c}(N)$ and $t>0$.
{\bf Remark.} Ability to use vector valued kernel seem to be
quite useful. We consider operator valued kernels just to make
the theory more symmetric. On the other hand, one may consider vectors
of distributions with different homogeneity on each coordinate
---
but we see no use of them. Also, multiplying values by operator
instead of scalar can be reduced (under the technical conditions we need)
to coordinates of different homogeneity.
A kernel $T$ is said to be an almost homogeneous kernel
of order $r\in {\Bbb R}$ if for every $f\in C_c^{\infty}(N-\{0\})$
$2^{-rk}\|f\circ \Dil^{k}T\|_{M^{1}}$ is bounded when $k$ goes to
$\infty$.
\def \order{of positive order at most}
We say that $T$ is a kernel \order\ $r$ if either $T$ is a homogeneous kernel
of order $s$, $0<\Re(s)\leq r$ or $T$ is an almost homogeneous kernel
of order $s$, $0~~0$, then we may aproximate $T$ by truncated kernels ---
defining $T_n$ by the formula $(f,T_n)=2^{r(n-1)}(f\circ D^{-n+1},T_1)$ we
have
$$
\|(T_n-T)*f\|\leq C2^{-\Re(r)n}\|f\|.$$
If $T$ is regular we will assume that $T_1$ is obtained from $T$ by
multiplication with smooth function.
{\bf Example} Let $h$ be a positive Schwartz class function on $N$ such
that $\int h=1$. Let
$$
=\sum_{k=-\infty}^{\infty}2^{rk}(f(0)-\int h(x)f(D^{k}x)dx).$$
For small enough positive $r$, $T$ is a regular homogeneous kernel of
order $r$. Moreover, $T$ generates a semigroup of probability measures.
As the Levi measure equals $T$ on $N-\{0\}$, it has density and
is infinite, so (by Janssen) the semigroup has densities in $L^1$.
For a unitary representation $\pi$ of $N$ on a Hilbert space $H$ and
a kernel $T$ of order $r$, $\Re(r)>0$, the operator $\pi(T)$ is defined
on the space
$C^{\infty}(\pi)$ of smooth vectors for $\pi$ by
$$
(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)
$$
where $\psi_{f}(x)=\pi(x)f$.
As a special case of representations we get images in quotient of $N$.
If the divisor is a homogeneous subgroup (that is invariant under $D$), then
on quotient is well defind dilation, and the image of a homogeneous kernel is
a homogeneous kernel of the same order (and similarly for almost
homogeneous kernels).
\p {\pla} {If $B$ is compactly supported distribution, $A$ and $AB$ belong to
$C^{*}(N)$, then for any unitary representation $\pi$ of $N$
$$
\pi(A)\pi(B)=\pi(AB)$$
on $C^{\infty}(\pi)$.}
{\bf Main theorems}
\t {\gta} {Let $P$ be a homogeneous kernel of order $r$,
$\Re(r)>0$. The following conditions are equivalent:
\item{a)}
%Assume that $P$ satisfies the following condition:\nl
%\vbox {\qquad \vbox{ \advance\hsize by-7em \noindent
There
exists homogeneous kernel $S$ of order $s$, $\Re(s)=\Re(r)$ such that for every nontrivial
irreducible unitary representation
$\pi$ of $N$, the operator $\overline{\pi(P)}$ is injective on
the domain of $\overline{\pi(S)}$.%}}\nl
%Then, if
\item{b)}If $T$ is
%a homogeneous kernel of order $t$, $0<\Re(t)\leq \Re(r)$
%or $T$ is almost homogeneous kernel of order $t$, $00}e^{-tP^{*}P}\in C^{*}(N)$$
\item{d)}
$$
\forall_{t>0}(1+tP^{*}P)^{-1}\in C^{*}(N).$$
\nl
\noindent If $T$ additionaly is a regular kernel, then each of the above is
equivalent to following:
\item{e)} For every nontrivial irreducible unitary representation
$\pi$ of $N$, the operator $\overline{\pi(P)}$ is injective on the space
$C^{\infty}(\pi)$.
\item{f)} For every integer $m>0$, and every kernel $T$ \order\ $2m\Re(r)$
there exists a constant C such that
$$
\forall f\in C^{\infty}_{c}( N)\qquad
(\|Tf\|\leq C(\|(P*P)^{m}f\|+\|f\|)).
$$\nl
\noindent If $T$ is regular and takes values in square matrices, then a--f is
equvalent to g:
\item{g)}
For every integer $m>0$, and every kernel $T$ \order\ $m\Re(r)$ there exists a constant C
such that
$$
\forall f\in C^{\infty}_{c}( N)\qquad
(\|Tf\|\leq C(\|P^{m}f\|+\|f\|)).
$$
If one of the equivalent conditions above is satisfied and $P_1$ is truncated
kernel corresponding to $T$, then $P_1^{*}P_1$ is essentialy selfadjoint on
$C^{\infty}_{c}(N)$.
}
{\bf Remark} The conclusion of \gta means that two operators satisfying
assumption of \gta have equal domain, so the domain in assumption of
\gta can be chosen in canonical way.
\nic{
\t {\gtb} {Let $P$ be a regular homogeneous kernel of order $r$, $\Re(r)>0$. Assume
that $P$ satisfies the following condition:\nl
\qquad \vbox{ \advance\hsize by-7em \noindent For every
nontrivial irreducible unitary representation
$\pi$ of $N$, the operator $\overline{\pi(P)}$ is injective on the space
$C^{\infty}(\pi)$%of smooth vectors of the representation
.}\nl
Then for every integer $m>0$, and every kernel $T$ \order\ $2m\Re(r)$ there exists
a constant C such that
$$
\forall f\in C^{\infty}_{c}( N)\qquad
(\|Tf\|\leq C(\|(P*P)^{m}f\|+\|f\|)).
$$
If $T$ take values in square matrices, then
for every integer $m>0$, and every kernel $T$ \order\ $m\Re(r)$ there exists a constant C
such that
$$
\forall f\in C^{\infty}_{c}( N)\qquad
(\|Tf\|\leq C(\|P^{m}f\|+\|f\|)).
$$
}
\t {\gtc} { For the homogeneous kernel $P$ the Rockland condition from
Theorem \gta
(or \gtb if the kernel is regular) is equivalent with each of the following:
%$$
%e^{-tP^{*}P}f=h_{t}*f\qquad\hbox{\rm and $h_{t}\in L^{1}(N)\cap L^{\infty}(N)$}$$
$$
\forall_{t>0}e^{-tP^{*}P}\in C^{*}(N)$$
%$$
%(1+tP^{*}P)f=w_{t}*f
%\qquad\hbox{\rm and $w_{t}\in L^{1}(N)$}$$
$$
\forall_{t>0}(1+tP^{*}P)^{-1}\in C^{*}(N).$$
If one of the equivalent conditions above is satisfied and $P_1$ is truncated
kernel corresponding to $T$, then $P_1^{*}P_1$ is essentialy selfadjoint on
$C^{\infty}_{c}(N)$.}
}
\t {\gtd} { If $T$ is positive definite, regular homogeneous kernel of
order $r>0$ and satisfies one of the equvalent conditions of \gta, then:
\item{a)} For every unitary representation $\pi$ of $N$ the operator
$\pi(T)$ is essentialy selfadjoint on $C^{\infty}(\pi)$, moreover $T$ is
essentialy selfadjoint on $C^{\infty}_c(N)$
\item{b)} For any complex $s$, $\Re(s)>-q/r$ the operator $T^{s}$ corresponds
to regular kernel of order $sr$ and if $\Re(s)>0$,
$T^{s}$ satisfies conditions of \gta
\item{c)} The semigroup $e^{-tT}$ generated by $T$ consists of smooth
functions and satisfies
$$
|\partial^{\alpha}e^{-tT}(x)|\leq C_\alpha(1+|x|)^{-q-r}.$$}
\t {\gte} { If\/ $T$ is regular homogeneous kernel of order $r$, $\Re(r)>0$,
and $T$ satisfies one of the equvalent conditions of \gta, then the
polar decomposition of $T$ consists of regular kernels, that is, there
exists a positive definite, regular homogeneous kernel $A$ of order $\Re(r)$
(which satisfies equvalent conditions of \gta) and a regular homogeneous
kernel $U$ order $\Im(r)$ such that
$$
T=UA$$
and $U$ gives (injective) isometry (also $A$ is injective).}
In the abelian case, the proof is easy: Fourier transform $\hat P$ of $P$ is a
continuous function, Rockland condition states simply
that $|\hat P(z)|=\det(\hat P(z)^{*}\hat P(z))^{1/2}\ne 0$ for $z\ne 0$
so $|\hat P|$ (by homogeneity) majorises
every continuous homogeneous function of the same order ---
which is the conclusion of \gta. Condition $e^{-tP^{*}P}\in C^{*}(N)$
is equivalent to $e^{-t|\hat P(z)|^2}\rightarrow 0$ when $z\rightarrow\infty$,
which, thanks to homogeneity, is equivalent to $\hat P(z)\ne 0$ for $z\ne 0$.
Denote the center of $N$ by $V$, and choose a linear complement $\tilde N$ to
$V$ invariant under the action of dilations.
For a functional $z\in V^{*}$ we shall denote by $\pi_{z}$ the unitary
representation of $N$ induced by the character
$$
v\rightarrow e^{i(v,z)}
$$
from the center $V$ of $N$.
The next three lemmas can be proved following the proof of (3.2) and
(3.19) in [\GI]. One should note that regularity of kernels in [\GI]
is only used to prove that truncated kernels of order $0$ are bounded
on $L^2$ - however the inductive argument in the proof of (3.19)
works also for truncated kernels of order $0$ (it {\it does not} work
without truncation). Also the kernels $T_I=X^IT$ are no longer
homogeneous, but only almost homogeneous. Fact that we admit complex
orders and discrete dilations only requires that we change the notation.
\p {\gla} {Let $P$ be a homogeneous kernel of order $r$, $\Re(r)>0$ such that for every kernel $T$
\order\ $\Re(r)$ there exists a constant $C$ such that
$$
\forall f\in C^{\infty}_{c}(\tilde N)\qquad
(\|\pi_{0}(T)f\|\leq C(\|\pi_{0}(P)f\|+\|f\|)).
$$
For all $t>0$ and
for every kernel $T$ \order\ $\Re(r)$ %of order $0~~~~0$ which satisfy the conclusion of \gta,
that is for every kernel
$T$ \order\ $\Re(r)$ there exists constant $C$ such that
$$
\forall f\in C^{\infty}_{c}( N)\qquad
(\|Tf\|\leq C(\|Pf\|+\|f\|))
$$
and let $P_{1}$ be the corresponding
truncated kernel. Let $\eta$ be a positive function such
that there exists a neighbourhood $K$ of $e$ and a constant $C$ satisfying
$$
\sup_{z\in K}\max_{|\alpha|\leq r+q}|X^{\alpha}\eta(xz)|\leq C\eta(z).
$$
Then for every $\epsilon>0$ there exists a constant $C$ such that
$$
\forall f\in C^{\infty}( N)\qquad
(\|[P_{1},\eta]f\| \leq \epsilon\|\eta P_{1}f\|+C%\|\eta (I-\Delta_V)^lf\|).
\|\eta f\|)
$$
$$
\forall f\in C^{\infty}( N)\qquad
(\|\eta^{-1}[P_{1},\eta]\eta f\| \leq \epsilon\|\eta P_{1}f\|+
C\|\eta f\|). %C\|\eta (I-\Delta_V)^lf\|).
$$
Moreover, $P_1^{*}P_1$ is essentially selfadjoint on $C^{\infty}_c(N)$.}
Fix a compact set $F\subset V$ such that $0\notin F$ and
$V-{0}=\bigcup_{k}D^k(F)$.
\p {\bz} {
$$
\exists C \forall z\in F,f\in C_{c}^{\infty}(\tilde N)
(\|f\| \leq C\|\pi_{z}(P)f\|)
$$}
Let as see that \gla and \bz easily imply \gta.
% and \gtb.
Indeed
%\indent\hbox{\bz}
\bz and \gla imply
$$
\forall z\in F
\forall f\in C^{\infty}_{c}(\tilde N)\qquad
(\|\pi_{z}(T)f\|\leq C(\|\pi_{z}(P)f\|+\|f\|)\leq
C_{1}\|\pi_{z}(P)f\|)
$$
Using dilations, we have
$$
\forall{z\in \bigcup_{k\geq1}D^k(F)}
\forall f\in C^{\infty}_{c}(\tilde N) \qquad
(\|\pi_{z}(T)f\|\leq C_{1}\|\pi_{z}(P)f\|)
$$
which, together with \gla gives the estimate.
So it remains to prove \bz.
\indent\hbox{\bz} is an immediate consequence of
\p {\bb} {$$
\sup_{z\in F} \|(I+\pi_{z}(P)^{*}\pi_{z}(P))^{-1}\|<1.
$$}
\def \alg{\rm Alg}
For $f\in L^{1}(N)$ define
$$
\|f\|_{\alg}=\sup_{z \in F} \|\pi_{z}(f)\|
$$
and let $\alg$ be the completition of $L^{1}(N)$
with respect to $\|\cdot\|_{\alg}$. Of course $\alg$ is a homomorphic
image of $C^{*}(N)$ so
that (non-degenerate) representations of $\alg$ are naturally
identified with representations of $N$. This also gives us a way to
associate elements of $\alg$ with distributions on $N$.
\p {\bc} { $\exists f\in \alg \forall \delta$
$$
%\delta((1+P^{*}P)^{-1})
\delta(f)=(1+\delta(P)^{*}\delta(P))^{-1}$$
%and
%$$(1+P^{*}P)^{-1}\in \alg$$
}
Taking \bc as granted we easily prove \bb: by the theory of
$C^{*}$-algebras (see [\DI] Lemma 3.3.6) there exists an irreducible
representation $\delta$ of \alg (hence $N$) such that
$$
\|f\|_{\alg}=\|\delta(f)\|.
$$
By Kirillov theory $\delta(f)$ is compact, hence
being positive either has norm smaller then $1$ (which is what we want)
or has eigenvector with eigenvalue $1$. But since
$$
\delta(f)
=(1+\delta(P)^{*}\delta(P))^{-1}
$$
such an eigenvector must
lie in the kernel of $\delta(P)$. As $\delta$ is nontrivial this
contradicts the Rockland condition.
So it remains to prove \bc.
\p {\hest} {Let $R$, $S$ be closed densely defined operators on a
Hilbert space.
Put $A=R^{\ast}R$, $B=S^{\ast}S$. Then
$$
\|Re^{-tA}\|\leq t^{-1/2},$$
$$
\|e^{-A}-e^{-B}\|\leq 4\|R-S\|.
$$ }
By the spectral theorem $\|Ae^{-tA}\|\leq t^{-1}$. For $v$ in $\dom(A)$
$$
\|Rv\|^{2}=(Rv,Rv)=(R^{\ast}Rv,v)=(Av,v)\leq \|Av\|\|v\|
$$
and we have
$$
\|Re^{-tA}v\|^{2}\leq \|Ae^{-tA}\|\|e^{-tA}\|\|v\|^{2}\leq t^{-1}\|v\|^{2}$$
If $\|R-S\|$ is finite $\dom(R)=\dom(S)$. Thanks to this, our operators
give at least well defined bilinear forms on $\dom(R)\times\dom(R)$ which
provides dense common domain for calculations.
We have
$$
A-B=R^{\ast}R-S^{\ast}S=R^{\ast}(R-S)+(R^{\ast}-S^{\ast})S.
$$
so
$$\eqalign {
e^{-tA}(A-B)e^{-(1-t)B}&=e^{-tA}(R^{\ast}(R-S)+(R^{\ast}-S^{\ast})S)e^{-(1-t)B}
\cr &=(Re^{-tA})^{\ast}(R-S)e^{-(1-t)B}+
e^{-tA}(R-S)^{\ast}Se^{-(1-t)B} \cr}
$$
and
$$\eqalign {
\|e^{-tA}(A-B)e^{-(1-t)B}\|&\leq \|(Re^{-tA})^{\ast}(R-S)e^{-(1-t)B}\|+
\|e^{-tA}(R-S)^{\ast}Se^{-(1-t)B}\|\cr &\leq
\|Re^{-tA}\|\|R-S\|\|e^{-(1-t)B}\|+
\|e^{-tA}\|\|R-S\|\|Se^{-(1-t)B}\|\cr
&\leq (t^{-1/2}+(1-t)^{-1/2})\|R-S\|.\cr }
$$
Hence, using perturbation formula
$$
\eqalign {
\|e^{-A}-e^{-B}\|&=\|-\int_{0}^{1}e^{-tA}(A-B)e^{-(1-t)B}dt\|\cr &\leq
\int_{0}^{1}\|e^{-tA}(A-B)e^{-(1-t)B}\|dt\cr &\leq
\int_{0}^{1}(t^{-1/2}+(1-t)^{-1/2})\|R-S\|dt=4\|R-S\|.\cr }
$$
\nic
{
\p {\ll} {Let $A$ and $A_{n}$ for $n=1,\dots$ be closed densely defined
operators
on a Hilbert space $H$ and let $T(t)$, $t>0$ be bounded operators. If
$$
\forall x\in D(A) (A_{n}x\rightarrow Ax),
$$
$$
\forall x\in D(A^{*}) (A^{*}_{n}x\rightarrow A^{*}x)
$$
and
$$
\forall x\in H (e^{-tA^{*}_{n}A_{n}}x\rightarrow T(t)x),
$$
then
$$
T(t)=e^{-tA^{*}A}.
$$ }
By the spectral theorem, we have
$$
\|e^{-tA^{*}_{n}A_{n} }v-v\|\leq t^{1/2}\|(A^{*}_{n}A_{n})^{1/2}v\|=t^{1/2}
\|A_{n}v\|.
$$
If $v\in D(A)$, then by assumption, $\|A_{n}v\|\leq C_{v}$ where $C_{v}$ is
independent of $n$ so
$$
\|T(t)v-v\|\leq t^{1/2}C_{v}.
$$
Since $D(A)$ is dense in $H$ the operators $T(t)$ form continuous
semigroup of contractions on $H$. We write $T(t)=e^{-t\tilde A}$.
We are going to prove that $\tilde A=A^{*}A$.
By \hesta,
$$
\|A_{n}e^{-tA^{*}_{n}A_{n}}\|\leq t^{-1/2}
$$
and in particular is bounded by a constant independent of $n$.
Also
$$
(A_{n}e^{-tA^{*}_{n}A_{n}}x,y)=(e^{-tA^{*}_{n}A_{n}}x,A^{*}_{n}y)
\rightarrow (e^{-t\tilde A}x,A^{*}y)
$$
for all $x\in H$ and all $y\in D(A^{*})$. Hence
$$
A_{n}e^{-tA^{*}_{n}A_{n}}\rightarrow Ae^{-t\tilde A}
$$
weakly (and $Ae^{-t\tilde A}$ is bounded). Consequently,
$$
(e^{-tA^{*}_{n}A_{n}}A^{*}_{n}A_{n}x,y)=
(A_{n}x,A_{n}e^{-tA^{*}_{n}A_{n}}y)
\rightarrow (Ax,Ae^{-t\tilde A}y)=(e^{-t\tilde A}A^{*}Ax,y)
$$
for all $x\in D(A)$ and all $y\in H$.
Since each $\{e^{-tA^{*}_{n}A_{n}}\}$
is holomorphic semigroup of contractions we have (choosing a subsequence,
if necessary)
$$
e^{-tA^{*}_{n}A_{n}}A^{*}_{n}A_{n}=
\partial_{t}e^{-tA^{*}_{n}A_{n}}
\rightarrow
\partial_{t}e^{-t\tilde A}= e^{-t\tilde A}\tilde A.
$$
Therefore
$$
e^{-t\tilde A}\tilde A=e^{-t\tilde A}A^{*}A
$$
on $D(A^{*}A)$. Taking $t$ going to zero we have
$$
A^{*}A\subset \tilde A.
$$
Since both $A^{*}A$ and $\tilde A$ is selfadjoint, $A^{*}A=\tilde A$ which
ends the proof.
}
\p {\bd} {If $R$ is a regular kernel generating a probabilistic semigroup,
and $R_{1}$ is corresponding truncated kernel $t,N>0$, then
$$
\int e^{tR_{1}}(x)^{2}|x|^{N} < \infty$$
$$
\int |R_{1}e^{tR_{1}}(x)|^{2}|x|^{N} < \infty$$
with uniform bound when $N$ and $t$ stay in a bounded set and
$t$ is bounded away from $0$.}
\def\ft{}
\p {\ax} {$$\forall t>0(\ft e^{-tP_{n}^{*}P_{n}}\in C^{*}(N))$$
and
$$\ft(P_{n}^{*}P_{n}+1)^{-1}\in C^{*}(N).$$}
P. The first claim implies the second, so we will prove the first.
\nic {
$$
|( P_{n}e^{-tP_{n}^{*}P_{n}}\ft f,
P_{n}\eta^{2} e^{-tP_{n}^{*}P_{n}}\ft f)
-\|\eta P_{n}e^{-tP_{n}^{*}P_{n}}\ft f\|^{2}|$$
$$= |( P_{n}e^{-tP_{n}^{*}P_{n}}\ft f,
[P_{n},\eta]\eta e^{-tP_{n}^{*}P_{n}}\ft f+
\eta [P_{n},\eta] e^{-tP_{n}^{*}P_{n}}\ft f)|$$
%$$+( P_{n}e^{-tP_{n}^{*}P_{n}}\ft f,
%\eta^{2} P_{n}e^{-tP_{n}^{*}P_{n}}\ft f)$$
$$
\leq
+\|\eta P_{n}e^{-tP_{n}^{*}P_{n}}\ft f\|
(\|[P_{n},\eta] e^{-tP_{n}^{*}P_{n}}\ft f\|+
\|\eta^{-1}[P_{n},\eta] \eta e^{-tP_{n}^{*}P_{n}}\ft f\|)$$
}
%\hbox {**************************************}
We choose $\eta$ to be polynomially growing,
satisfy assumptions of \glb, and such that $L^{2}(\eta)\subset L^{1}$
-- for example $(1+|x|^{m})$ with $m$ large enough. We have
$$
|( P_{n}^{*}P_{n} \ft f,
\eta^{2} \ft f)
-\|\eta P_{n} \ft f\|^{2}|=
|( P_{n} \ft f,
P_{n}\eta^{2} \ft f)
-\|\eta P_{n} \ft f\|^{2}|$$
$$= |( P_{n} \ft f,
[P_{n},\eta]\eta \ft f+
\eta [P_{n},\eta] \ft f)|$$
$$
\leq
\|\eta P_{n} \ft f\|
(\|[P_{n},\eta] \ft f\|+
\|\eta^{-1}[P_{n},\eta] \eta \ft f\|)$$
Hence, by \glb
$$\Re (P_{n}^{*}P_{n} \ft f, \eta^{2} \ft f)
\geq 1/2\|\eta P_{n}e^{-tP_{n}^{*}P_{n}}\ft f\|^{2}
- C\|\eta e^{-tP_{n}^{*}P_{n}}\ft f\|^{2},$$
$$
|\Im ( P_{n}^{*}P_{n} \ft f,
\eta^{2} \ft f)|\leq
1/2\|\eta P_{n}e^{-tP_{n}^{*}P_{n}}\ft f\|^{2}
- C\|\eta e^{-tP_{n}^{*}P_{n}}\ft f\|^{2},$$
and
$$\Re (P_{n}^{*}P_{n} \ft f, \eta^{2} \ft f)
+2C\|\eta e^{-tP_{n}^{*}P_{n}}\ft f\|^{2}\geq
|\Im ( P_{n}^{*}P_{n} \ft f,
\eta^{2} \ft f)|.$$
Next, if $|\Im(z)|\leq \Re(z)$,
$$
\partial_{t}\|\eta e^{-tzP_{n}^{*}P_{n}}\ft f\|^{2}
=-2 \Re (z \eta P_{n}^{*}P_{n}e^{-tzP_{n}^{*}P_{n}}\ft f,
\eta e^{-tzP_{n}^{*}P_{n}}\ft f)$$
$$
\leq \sqrt{2}|z|(
|\Im(\eta P_{n}^{*}P_{n}e^{-tzP_{n}^{*}P_{n}}\ft f,
\eta e^{-tzP_{n}^{*}P_{n}}\ft f)|-
\Re(\eta P_{n}^{*}P_{n}e^{-tzP_{n}^{*}P_{n}}\ft f,
\eta e^{-tzP_{n}^{*}P_{n}}\ft f)$$
$$
\leq C'\|\eta e^{-tzP_{n}^{*}P_{n}}\ft f\|^{2}$$
Consequently,
$
e^{-zP_{n}^{*}P_{n}}\ft$ is a holomorphic family of
bounded operators on $L^{2}(\eta)$.
That means $\forall M>0\exists C_{M}\forall t0$
$$\|\int_{\epsilon}^{t-\epsilon} e^{-sP_{n}^{*}P_{n}}({P_{n}^{*}P_{n}}-R_{1})e^{(t-s)R_{1}}\ft ds \|_{L^{2}(\eta)} $$
$$\leq t\sup_{\epsilon\leq s \leq t-\epsilon}\left\|
e^{-sP_{n}^{*}P_{n}}({P_{n}^{*}P_{n}}-R_{1})e^{(t-s-\epsilon/2)R_{1}}\ft
\right\|_{L^{2}(\eta),L^{2}(\eta)}\|e^{(\epsilon/2)R_{1}}\|_{L^{2}(\eta)}<\infty
$$
so $I_{1,\epsilon}\in L^{1}$. Also $e^{tR_{1}}\in L^{1}$ so
$\ft e^{-tP_{n}^{*}P_{n}}$ is a limit in operator norm of $L^{1}$
functions so it belongs to $C^{*}(N)$.
\p {\at} {$$\forall t>0(\ft e^{-tP^{*}P}\in C^{*}(N))$$}
P. Since $\|P-P_{n}\|$ tends to $0$ when $n$ goes to infinity,
\hest implies that $\ft e^{-tP^{*}P}$ is the norm limit of
$\ft e^{-tP_{n}^{*}P_{n}}$. Hence, our claim follows from \ax.
\p {\ay} {$$\forall \rho\in\ir(\rho(e^{-P_{n}^{*}P_{n}})
=e^{-\rho(P_{n})^{*}\rho(P_{n})}$$}
\nic {
\p {\az} {$$\forall \rho\in\ir(\rho(e^{-P^{*}P})
=e^{-\rho(P)^{*}\rho(P)}$$}
\p {\hl} {$$\forall \rho\in\ir(\rho(e^{-tP_{n}^{*}P_{n}}P_{n}^{*}P_{n})
=\rho(e^{-tP_{n}^{*}P_{n}})\rho(P_{n})^{*}\rho(P_{n})$$}
\p {\as} {$$e^{-tP^{*}P}\in \alg$$}
P. It is enough to show that
$$
\forall \rho \in Ir(\alg),f\in D(\rho(P))
(\|f\| \leq C\|\rho(P)f\|)
$$}
Let $R$ be a symmetric regular ...
By \gla we have
$$C_{0}(\|Pf\|+\|f\|)\geq \|Rf\|$$
as $P-P_{n}$ is bounded
$$C_{1}(\|P_{n}f\|+\|f\|)\geq \|Rf\|$$
$$C_{1}^{2}(P_{n}^{*}P_{n}+1)\geq R^{2}$$
since $\ft$ is central, $0\leq \ft$ and ...
$$(1+C_{1}^{2}(1+P_{n}^{*}P_{n}))^{-1}\ft\leq (R^{2}+1)^{-1}\ft.$$
As both sides of the inequality belong to $C^{*}(N)$ we have
$$\rho((1+C_{1}^{2}(1+P_{n}^{*}P_{n}))^{-1}\ft)\leq \rho((1+R^{2})^{-1}).$$
Similarly one shows that there is $C_{2}$ so that
$$\rho((1+C_{2}^{2}(1+R^{2}))^{-1})\leq
\rho((1+C_{1}^{2}(1+P_{n}^{*}P_{n}))^{-1}\ft)$$
so there is a positive definite operator ${\tilde A}$ such that
$$\rho((1+P_{n}^{*}P_{n})^{-1}\ft)=(1+{\tilde A})^{-1}.$$
%$$(1+C_{1}^{2}(1+{\tilde A}_{n}))^{-1}\leq (\rho(R)^{2})^{-1}$$
%$$C_{1}^{2}({\tilde A}_{n}+1)\geq (\rho(R)^{2})$$
Note that $\dom({\tilde A}_{n}^{1/2})=\dom(\rho(R))$ and
$$C_{2}(\|\rho(R)f\|+\|f\|)\geq\|{\tilde A}_{n}^{1/2}\|.$$
On $C^{\infty}_{\rho}$ we have
$${\tilde A}e^{-t{\tilde A}}f=-\partial_{t}e^{-t{\tilde A}}f=
-\partial_{t}\rho(e^{-tP_{n}^{*}P_{n}}\ft)f=$$
$$\rho(e^{-tP_{n}^{*}P_{n}}P_{n}^{*}P_{n}\ft)f=
\rho(e^{-tP_{n}^{*}P_{n}}\ft)\rho(P_{n}^{*}P_{n})f=$$
$$e^{-t{\tilde A}}\rho(P_{n})^{*}\rho(P_{n})f.$$
In the last line we used the fact that on smooth vectors
$$(\rho(P_{n}^{*})f,g)=(f,\rho(P_{n})g)$$
so $\rho(P_{n}^{*})\subset\rho(P_{n})^{*}$.
Hence we got equality
$$e^{-t{\tilde A}}{\tilde A}f=e^{-t{\tilde A}}\rho(P_{n})^{*}\rho(P_{n})f.$$
If $t$ goes to $0$ we have (on smooth $f$)
$${\tilde A}f=\rho(P_{n})^{*}\rho(P_{n})f.$$
As smooth vectors are a core of $\rho(P_{n})$ we have $\dom(\rho(P_{n}))
\subset \dom({\tilde A}_{n}^{1/2})$.
Since $\rho(R)$ majorises ${\tilde A}_{n}^{1/2}$ (hence also $\rho(P_{n})$) and
smooth vectors are a core of $\rho(R)$
%the same order as $P$
we also have $\dom(\rho(R))\subset\dom(\rho(P_{n}))$ so
$\dom({\tilde A}_{n}^{1/2})\subset\dom(\rho(P_{n}))$. This together imply that
bilinear forms associated to ${\tilde A}_{n}$ and $\rho(P_{n})^{*}\rho(P_{n})$
are equal so ${\tilde A}_{n}=\rho(P_{n})^{*}\rho(P_{n})$.
\nic {
\p {\cb} {Let $A$ be a kernel of order $0$. Then
$$
\int_{1<|x|<2} A =0.$$}
P. This condition is sufficient for a (homogeneous on $N-{0}$)
measure to have homogeneous extension from $N-{0}$ to onto $N$.
The condition is satisfied on set of measures of codimension $1$,
so any other measure with homogeneous extension would give us
homogeneous extension for all measures.
However $|x|^{-Q}$ have no homogeneous extension, so the condition
is also necessary.
\p {\ca} {For a truncated kernel $A$ of order $0$ and any $0<\epsilon<1$
there exists truncated kernel $B$ of order $\epsilon$ and constant $C$
such that for every $f\in C^{\infty}(G)$
$$
\|Af\|\leq C(\|Bf\| + \|f\|).$$}
}
\vskip 0.5cm
{\bf References}
\vskip 1cm
\unvbox2
\end
~~