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\vskip1cm
{\bf Functional calculus for slowly decaying kernels}
\vskip1cm
{\qquad by Waldemar Hebisch (Wroc\l aw)\footnote{${}^1$}{
Supported by KBN grant {\tt 2 P301 052 07}.}}
\vskip1cm

\leftline{\bf 1. Introduction.}
\smallskip

\def \PP{{{\it Proof}\/:}}
\def \varOmega{\Omega}
\def \demo#1{\PP}
\def \enddemo{}

\def \hfilll{\hfill}
\def \text{\hbox}
\let \nl=\par
\def \\ {$$ $$}
\def \sm {\sum\limits}
\def \pro {\prod\limits}
\def \inT {\int\limits}
\def \supp{\mathop{\rm supp}}% {\operatorname{supp}}
\def \spect{\mathop{\rm sp}}
\def \Int {\hbox {Int } }
\def \align{\global \catcode`&=9}
\def \aligned{\global \catcode`&=9}
\def \endalign{}
\def \endaligned{}
\def \gather{}
\def \endgather{}
\def \frac %#1 #2 { {#1 \over #2 } }
\def \endgather{}
\def \endproclaim{}


\rf {\Al} {G. Alexpoulos, Spectral multipliers on Lie groups 
of polynomial growth, Proc. AMS 120 (1994), 973-979.}
%\rf {\Cy} {J. Cygan, Heat kernels for class 2 nilpotent groups, 
%           {\it Studia Math.} 64 (1979), 227-238.}

\rf {\Ch} {M. Christ, $L^{p}$ bound for spectral multiplier on 
             nilpotent groups, {\it TAMS} 328 (1991), 73-81.}
\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 {\CW} {R. Coifman, Weiss, Analyse Harmonique Non-Commutative 
	sur Certains Espaces Homogenes, Springer LN 242,Berlin 1971.}

%\rf {\CoT} {L. Colcani, G. Travaglini, Estimates for Riesz kernels 
%of eigenfunction expansions of elliptic differential operators on compact
%manifolds }

\rf {\Di} {J. Dixmier, Op\'erateurs de rang fini dans les 
repr\'esentations unitaires, Publ. Math. IHES 6 (1960), p. 13--25.}

\rf {\Dz} {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 {\Ga} {B. Gaveau, Principe de moindre action, propagation 
%           de la chaleur et estimates sous elliptiques sur certains groupes 
%           nilpotents, {\it Acta Math.} 139 (1977), 95-153.}

\rf {\Gl} {P. G\l owacki, The Rockland condition for nondifferential 
                convolution operators, Duke Math. J. 58 (1989), 371--395.}

\rf {\Hms} {W. Hebisch, A multiplier theorem for 
  Schr\"odinger operators, {\it Coll. Math.} 60/61 (1990), 659-664.  }

\rf {\Hae} {W. Hebisch, Almost everywhere summability of eigenfunction 
 expansions  associated to elliptic operators,  
 {\it{     Studia Math.     }}          96 (1990), 263-275.}  

\rf {\Hs} {W. Hebisch, %A note on the paper of M.Cowling et. al.(4 p.) 
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 {\HS} {W. Hebisch, A. Sikora,
 A smooth subadditive homogeneous norm on a homogeneous group,
 96 {\it Studia Math}. 96 (1990) 231--236. }

\rf {\HZ} {W. Hebisch, J. Zienkiewicz, Multiplier theorem on generalized 
Heisenberg groups II, {\it Coll. Math.} 69 (1995) 29--36.}

\rf {\Hom} {L. H\"ormander, Estimates for translation invariant 
operators in $L^{p}$ spaces, {\it Acta Math. 104} (1960), 93--140.}

\rf {\Hos} {L. H\"ormander, The spectral function of an elliptic operator, 
Acta Math. 121 (1968), 193--218.}

\rf {\Hus} {A. Hulanicki, Subalgebra  of  $L _{1 } (G) $ associated  with  
Laplacian  on  a  Lie  group, {\it Coll. Math. 31}  (1974), 
259-287.}


\rf {\HJ} {A. Hulanicki, J.W. Jenkins, Nilpotent  Lie  groups  
and  summability  
of  eigen\-function  expansions  of  Schr\"odinger  operators, 
{\it Studia  Math.} 80 (1984) 235-244.}



\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 {\Py} {T. Pytlik, Symbolic calculus on weighted group algebras, 
        Studia. Math. 73 (1982), 169--176.}


\rf {\Se} {A. Seeger, Estimates near $L^{1}$ for Fourier multipliers and 
maximal functions, Arch. Math. 53 (1989), 188--193.}

\rf {\Sc} {C. Sogge, On  the  convergence  of  Riesz  means  on  compact  
manifolds, Ann. of  Math. 126  (1987), 439--447. }

\rf {\Tl} {M. Taylor, Pseudodifferential operators, Princeton University 
Press, Princeton 1981.} 

%\rf {\So} {Ch. Sogge, }



%   Let $\varOmega$ be a metric space with the  metric $d$ and a Borel 
%measure $\mu$.

   Let $\varOmega$ be a metric space with the  metric $d$ and a Borel 
measure $\mu$. Let 
$$
B(x,r) = \{ y\in\varOmega : d(x,y) < r \}. 
$$
We assume that there are constants $C$, $q$ 
({\it volume growth constants} of\/ $\Omega$) such that 
$$
\mu (B(x,sr)) \leq  C\mu( B(x,r))s^q  
$$
for all $s>1$.
The existence of such constants is equivalent to the doubling condition, 
but here we are 
interested in getting $q$ as small as possible (at the cost of
enlarging $C$).


   The purpose of the paper is to present a reasonably general 
approach to functional calculus, multipliers and 
almost everywhere convergence theorems on $\varOmega$. We consider integral 
operators with kernels decaying polynomialy away from diagonal. Our 
methods are based on $L^{2}$ estimates obtained from spectral 
theorem and careful use of weight functions. 

The main question is to find sufficient conditions on positive 
definite operator $A$ on $\Omega$ and on function $F$ such that 
some of the following holds:
$$
F(A) \hbox{\rm \qquad is bounded on $L^{1}$}$$
$$
F(A) \hbox{\rm \qquad is bounded on $L^{p}$, $1<p<\infty$ and of weak type (1,1).}$$
$$
F^{*}(A)f = \sup _{t>0} | F(tA)f | 
\hbox{\rm  \qquad is bounded on $L^{p}$, $1<p<\infty$ and of weak type (1,1).}$$

The subject has long history. Let us only mention works [\Hom], [\Di], 
[\Hus], %[\Tl] Theorem , 
[\Py], [\HJ]. Our topic is closely related to works on Riesz means, 
for example [\Hos], [\Sc], [\CS]. 
\nic{
Dixmier [\Di] used functional calculus to 
prove that $L^{1}$ of a nilpotent group contains elements going onto 
finite rank operators in any irreducible unitary representation. 
Hulanicki [\Hus] considered functions of laplacean 
(more precisely of heat kernel) to prove that subalgebra of $L^{1}(G)$ 
generated by laplacean is symmetric if $G$ is of polynomial growth. 
Dixmier considered only compactly supported functions. Method of Hulanicki 
(very similar to that of Dixmier) was limited to exponentially decaying 
$f$. Next Hulanicki  [??] invented a way to handle $f$ 
decaying polynomialy but fast enough. } 
%In the previously mentioned resu
%Because we use Schwartz 
%inequality we are limited to finite dimensional spaces 
%(however in infinite dimensional setting it is hard to 
%expect positive results anyway). 

%We only most fundamental properties
In a few classical examples our  results are weaker then those 
previously known. However, it seems that proofs of, say, 
multiplier theorems go as follows. One splits the multiplier into 
diadically supported pieces. For each piece one gets
estimates for dominant term via some kind of Plancherel 
formula, which requires a lot of specific knowledge - unavailable 
in our setting. Then there is error term estimate - in many 
cases long and very technical. Final step is use of covering or 
decomposition arguments to get estimates for the whole multiplier. 
 Our methods seem well suited 
to handle error term estimates while for the main term we use 
essentially a trivial estimate. However, if better estimates for 
main term are known, we can use them. It is quite possible that 
in our general setting the trivial estimate is the best one. 
We also give an improved way of gluing estimates for pieces to 
get full multiplier theorem --- unlike the case of singular 
integrals we need no smoothness assumptions (cancelations are 
provided by the $L^{2}$ theory).

The main factor affecting our estimates is the volume growth rate. 
When the work on this paper begun this was believed to be the correct 
factor. Recent works [\Hs], [\MS], [\Hgh], [\HZ] shows that 
the picture is much more complicated. 




\leftline{\bf  2.Banach Algebras }
\pno=0
\sno=2

   Definition: A continuous function 
$\varphi : \varOmega \times \varOmega \rightarrow R$ is  called 
submultiplicative if for all $x$,$y$,$z$ 
                       $$\varphi(x,y) \geq 1 $$
and 
$$
 \varphi(x,y)\varphi(y,z) \geq \varphi(x,z) .
$$
   Of course if $a$,$b>0$, then $\omega_{a} =(1+d)^{a}$, 
$e^{bd}$, $\omega_{a} e^{bd}$ are submultiplicative. 
   Our functional calculus is based on Banach $*$-algebras 
whose elements are kernels $K$, $K(x,y)$ being a complex  number. 
 For a submultiplicative function $\varphi$ we write 
$$ 
\| K \|_{B(\varphi)} = %\max \{ \sup_x\inT  |K(x,y)|\varphi(x,y)
%d\mu (y), $$
%$$
\sup_y \inT |K(x,y)|\varphi(y,x)d\mu (x) 
%\}
 $$
$$K^{*}(x,y)={\bar K(y,x)}$$
\def \bnorm{|}
$$ \bnorm K \bnorm_{B(\varphi)}= \max  \{\| K \|_{B(\varphi)},\| K^{*}
\|_{B(\varphi)} \} $$
and we define the Banach $*$-algebra with unit element by 
$$
B(\varphi) = \{ K : \bnorm K\bnorm_{B(\varphi)} < +\infty 
\} + CI. 
$$
The multiplication is defined by 
$$
K_1K_2(x,y) = \int K_1(x,s)K_2(s,y)d\mu (s).
$$
%and 
%$$
%K^*(x,y) =\bar K(y,x). 
%$$
Obviously
$$ 
\int |K_1K_2(x,y)|\varphi(x,y)d\mu (x) $$
$$ \leq  \int \int |K_1(x,s)K_2(s,y)|d\mu (s)\varphi(x,y)d\mu (x)$$
$$ \leq  \int \int |K_1(x,s)|\varphi(x,s)|K_2(s,y)|\varphi(s,y)d\mu 
(s)d\mu(x)$$
$$ \leq  \int
 \|K_1\|_{B(\varphi)}    |K_2(s,y)|\varphi(s,y)d\mu (s)$$
$$ \leq  \|K_1\|_{B(\varphi)}    \|K_2\|_{B(\varphi)} $$
for every $x$.
%Similarly
%$$
%\sup_y \int |K_1K_2(x,y)|\varphi(y,x)d\mu (x) \leq 
% \|K_1\|_{B(\varphi)}\|K_2\|_{B(\varphi)}
%$$
%so
Hence
$$
\|K_1 K_2 \|_{B(\varphi)}     \leq  \|K_1\|_{B(\varphi)}
 \|K_2\|_{B(\varphi)} 
$$
and
$$
\bnorm K_1 K_2 \bnorm_{B(\varphi)}     \leq\bnorm
K_1\bnorm_{B(\varphi)}\bnorm K_2\bnorm_{B(\varphi)}.$$
   For a given kernel $K$ we are going to estimate the norm  of 
the element $Ke^{inK}$ in $B(\omega_{a})$. We will use the abbreviations
$$
\|K\|_a  = \|K\|_{B(\omega_a)}, 
$$
$$
\bnorm K\bnorm_{a} = \bnorm K\bnorm_{B(\omega_a)}.$$
%\proclaim{(3.1) Theorem}
\t {\ba }
 {Let $a > b \geq 0$ and $\varOmega$ be as 
above. For every $L$ there  exists 
constant $M$ depending only on $a$, $b$, $L$ and  
volume growth constants $C$,  $q$  but 
independent of\/ $\varOmega$, $d$, $\mu$, $K$ such that if
$$ 
K=K^* ,$$
$$\|K\|_a  \leq  L, $$
$$\sup_{x} \mu(B(x,1))\int |K(x,y)|^{2}d\mu (y) \leq  1, $$ 
then for every $n$
$$\bnorm e^{inK}\bnorm_{b}  \leq  M(1 + |n|)^{\kappa} $$
where $\kappa = 2^{[(b+q/2)/(a-b)]}((b+q/2)(1+1/(a-b))+1)$.}
%\endproclaim
\demo{Proof}
Put $A=\exp(iK)$, then $\| A\|_{a}=\bnorm A\bnorm_{a}\leq \exp(L) $. Let $p= q/2+b $, $ \delta = a-b $, $ r\geq (n\| 
A\|_{a})^{1/\delta}$ ($r$ will be chosen later).
Then $ \bnorm A\bnorm_{a}r^{-\delta}\leq 1/n $.

We put 
$$ 
A_0(x,y)=\cases{ A(x,y)\text{\quad for\ $d(x,y)<er$} \cr 
0 \qquad \text{otherwise},\cr }
$$
and 
$$ 
A_k(x,y)=\cases {A(x,y)\text{\quad for\ $re^k\leq 
d(x,y)<re^{k+1}$}\cr 0 \qquad\text{otherwise}.\cr } 
$$ 
\par
If $k\geq 1$, 
$$ 
\|A_k\|_{b}\leq 
(re^k)^{b-a}\|A\|_{a}=C_1e^{-k\delta} 
$$
where $C_1=r^{-\delta}\|A\|_{a}$. Also 
$$ 
\|A_0\|_{L^2\rightarrow L^2}\leq 1+\|A-A_0\|_{L^2\rightarrow L^2}
%\|A\|_{L^2\rightarrow L^2} 
%  + \|\sum_{k=1}^{\infty}A_{k}\|_{L^2\rightarrow L^2}
\leq 1+r^{-\delta}\bnorm A\bnorm_{a}=C_0\leq 
1+1/n.
$$
\par

%\proclaim {(3.3).Lemma}
\p {\bb }
{Let $A_i$ be as above, $\alpha$ be a 
multiindex, $E$ a kernel such that 
$$
 \sup_y \mu(B(y,1))\|E(\cdot,y)\|_{L^2}^{2}\leq M^{2}
$$
and $E(x,y)=0$ for $d(x,y)>r_0$. 
Then 
$$
\| \pro_{i=1}^{|\alpha|}A_{\alpha(i)}E\|_b\leq 
MC(r_0+r\sum e^{\alpha(i)})^p \exp(-\delta\sum 
\alpha(i))C_1^{|\{i:\alpha(i)>0\}|}C_0^{|\alpha|}
$$
and 
$$
\bnorm \pro_{i=1}^{|\alpha|}A_{\alpha(i)}\bnorm_b\leq 
C|\alpha|\left(r\sum e^{\alpha(i)}\right)^p \exp(-\delta\sum 
\alpha(i))C_1^{|\{i:\alpha(i)>0\}|}C_0^{|\alpha|}.
$$}
%\endproclaim
\demo{Proof}
$$\| \pro_{i=1}^{|\alpha|}A_{\alpha(i)}\|_{L^2\rightarrow L^2}
\leq \pro_{i=1}^{|\alpha|}\|A_{\alpha(i)}\|_{L^2\rightarrow L^2}
\leq 
\exp(-\delta\sum 
\alpha(i))C_1^{|\{i:\alpha(i)>0\}|}C_0^{|\alpha|}$$\par
Fix $y$. Put $U=\{x:d(x,y)\leq r_0+er\sum e^{\alpha(i)}\}$. 
We have 
$$ 
\| \pro_{i=1}^{|\alpha|}A_{\alpha(i)}E(\cdot,y)(1+d(\cdot,y))^b\|_{L^1}$$
$$=
\| \chi_U 
\pro_{i=1}^{|\alpha|}A_{\alpha(i)}E(\cdot,y)(1+d(\cdot,y))^b\|_{L^1}$$
$$\leq \sup_{x\in U}(1+d(x,y))^b\|\chi_U\|_{L^2}
\| \pro_{i=1}^{|\alpha|}A_{\alpha(i)}E(\cdot,y)\|_{L^2}$$
$$\leq (1+r_0+er\sum e^{\alpha(i)})^b|B(r_0+er\sum 
e^{\alpha(i)},y)|^{1/2} $$
$$ \qquad \times\|\pro_{i=1}^{|\alpha|}A_{\alpha(i)}\|_{L^2\rightarrow L^2}
\|E(\cdot,y)\|_{L^2}$$
$$\leq C(r_0+r\sum e^{\alpha(i)})^p
\exp(-\delta\sum 
\alpha(i))C_1^{|\{i:\alpha(i)>0\}|}C_0^{|\alpha|}M
$$
which gives the first assertion.
We prove the second assertion inductively. If $\alpha(|\alpha|)>0$,
$$
\| \pro_{i=1}^{|\alpha|}A_{\alpha(i)}\|_b\leq
\| \pro_{i=1}^{|\alpha|-1}A_{\alpha(i)}\|_b
\|A_{\alpha(|\alpha|)}\|_b
$$
and we estimate the first factor by the inductive assumption. 
 If $\alpha(|\alpha|)=0$, $$A_0=I+A_0'$$ where 
$$ 
\sup_y \mu(B(y,1))
\|A_0'(\cdot,y)\|_{L^2}\leq
 \sup_y \mu(B(y,1)) 
\|(A-I)(\cdot,y)\|_{L^2}\leq C
$$ 
so
$$
\| \pro_{i=1}^{|\alpha|-1}A_{\alpha(i)}A_0\|_b
\leq
\| \pro_{i=1}^{|\alpha|-1}A_{\alpha(i)}\|_b+
\| \pro_{i=1}^{|\alpha|-1}A_{\alpha(i)}A_0'\|_b
$$
and we estimate the first term by the inductive assumption and the 
second by the first assertion of the lemma. Applying ${}^{*}$ to 
all the operators in the
proof we get not only estimate for $\|\cdot\|_{b}$ but also for $\bnorm\cdot\bnorm_{b}$. 
\enddemo

Let us note the following obvious conseqence of \bb.

%\proclaim{(3.4).Lemma}
\p {\bc }
{If $|\alpha|\leq n$, then
$$
\bnorm \pro_{i=1}^{|\alpha|}A_{\alpha(i)}\bnorm_b\leq 
C|\alpha|^{p+1}\left(re^{\max \alpha(i)}\right)^p \exp(-\delta\sum 
\alpha(i))C_1^{|\{i:\alpha(i)>0\}|}.
$$}
%\endproclaim


%\proclaim{(3.5).Lemma}
\p {\bd }
{For every $\varepsilon>0$ there exists constant $C$ such
that for every $b$, $\varOmega$, $f:N\mapsto R$, $n\in N$  
and kernel $A$ on $\varOmega$
if  
$$
\| \pro_{i=1}^{n}A_{\alpha(i)}\|_b\leq f(n)\exp(-\varepsilon\sum\alpha(i))
(1/ {n})^{|\{i:\alpha(i)>0\}|}
$$
then
$$
\bnorm A^{n}\bnorm_{b}\leq Cf(n).
$$}
%\endproclaim
\demo{Proof}
$$
\bnorm A^{n}\bnorm_{b}=\|A^n\|_b\leq \sm_{|\alpha|=n}\| \pro_{i=1}^{n}A_{\alpha(i)}\|_b$$
$$\leq f(n)\sm_{|\alpha|=n}\exp(-\varepsilon\sum\alpha(i))(1/n)^
{|\{i:\alpha(i)>0\}|}$$
$$\leq 
f(n)(1+1/n\sm_{k=1}^{\infty}e^{-\varepsilon k})^n$$
$$\leq 
f(n)\exp(\sm_{k=1}^{\infty}e^{-\varepsilon 
k})=C f(n).
$$
\enddemo

We expect main part of our kernels to live in the distance of 
order $nr$ to the diagonal. %On this set 
In this region we sometimes have another estimate, 
and the following lemma allows us to use it efficiently. 

%\proclaim{(3.6).Lemma}
\p {\be }
{For every $\varepsilon>0$ there exists constant $C$ such
that for every $b$, $\varOmega$, $f:N\mapsto R$, $n\in N$ 
and kernels $A$ and $E$ on $\varOmega$
if $nC_{1}\leq 1$, $E(x,y)=0$ for $d(x,y)>nr$ and 
$$
\| \pro_{i=1}^{n}A_{\alpha(i)}E\|_b\leq f(n)\exp(-\varepsilon\sum\alpha(i))
C_{1}^{|\{i:\alpha(i)>0\}|}
$$
then
$$
\sup_{y\in \varOmega} \inT_{d(x,y)>4nr}|A^{n}E(x,y)|(1+d(x,y))^{b}d\mu(x)
\leq Cf(n)nC_{1}.
$$}
%\endproclaim
\demo{Proof} As $4nr > enr + nr$ we have
$$
\sup_{y\in \varOmega} \inT_{d(x,y)>4nr}|A^{n}E(x,y)|(1+d(x,y))^{b}d\mu(x)
\leq \sm _{ |\alpha|=n \atop \alpha\ne 0} 
\| \pro_{i=1}^{n}A_{\alpha(i)}E\|_b$$
$$\leq f(n)\sm  _{ |\alpha|=n \atop \alpha\ne 0}
C_{1}^{|\{i:\alpha(i)>0\}|}
 \exp(-\varepsilon\sum\alpha(i))$$
$$
\leq f(n)nC_{1}\sm_{|\alpha|=n}\exp(-\varepsilon\sum\alpha(i))(1/n)^
{|\{i:\alpha(i)>0\}|}$$
$$\leq C f(n)nC_{1}.
$$
\enddemo

%\proclaim{(3.7).Lemma}
\p {\bx }
{ Let $m$ be a natural number. For every $n$ and every sequence
$(a_{i})^{n}_{i=1}$ of nonnegative real numbers there exists
subset $I$ of $[1,n]\cap N$ such that
$|I|\leq 2^{m-1}-1$ 
and for every subset $J$ of $[1,n]\cap N$ such that 
$|J|\leq |I|+1$ and $J\cap I=\emptyset$ we have
$$m\sum_{j\in J} a_{j}\leq \sum_{i=1}^{n}a_{i}$$ }
%\endproclaim
\demo{Proof}
Without any loss of generality we assume that the sequence 
is nonincreasing and long enough, otherwise we renumerate it and 
add zeros. 

Next
$$
\sum a_{i}\geq \sum_{k=0}^{m-1}\sum_{2^{k}}^{2^{k+1}-1}a_{i}
$$
so there exists $k\leq m-1$ such that
$$
\sum a_{i}\geq m \sum_{2^{k}}^{2^{k+1}-1}a_{i}.
$$
We take $I=[1,2^{k}-1]\cap N$. Of course the first assertion holds.
Let $J$ be as in the second assertion. Since $(a_{i})$ is nonincreasing
$$
m\sum_{j\in J} a_{j}\leq
m \sum_{2^{k}}^{2^{k+1}-1}a_{i}
$$
which ends the proof.
\enddemo

%\proclaim{(3.8).Lemma}
\p {\bg }
{Let $m$ be a positive natural number, $A$ and $\varOmega$ be as above.
There exists a constant $C$ such that for all $n$
$$
\| \pro_{i=1}^{n}A_{\alpha(i)}\|_b
\leq C r^{(2^{m-1})p}n^{(2^{m-1})(p+1)}\exp(({ {p}\over {m}} -\delta) \sum \alpha (i))
C_{1}^{|\{i:\alpha(i)>0\}|}
$$}
%\endproclaim
\demo{Proof}
Fix $\alpha$. Take $a_{i}=\alpha(i)$ in Lemma \bx and choose 
subset $I$ according to the Lemma.
We write $l=|I|$ and $I=\{i_{j}:j=1,\ldots , l \}$. 
Put $i_{0}=0$ and $i_{l+1}=n+1$. By the Lemma $l\leq 2^{m-1}-1$ and 
$$
m\sm_{j=1}^{l+1}\max_{i_{j-1}<i<i_{j}}\alpha(i)
\leq \sum \alpha(i).
$$
Also, by \bc, 
$$
\| \pro_{i=i_{j-1}+1}^{i_{j}-1}A_{\alpha(i)}\|_b\hfilll $$
$$\hfilll\leq 
Cn^{p+1}\left(r\exp(\max_{i_{j-1}<i<i_{j}}\alpha(i) )\right)^p 
\exp(-\delta\sum_{i_{j-1}<i<i_{j}} \alpha(i))C_1^{|\{i:{i_{j-1}<i<i_{j}},\alpha(i)>0\}|}
$$

Next
$$
\| \pro_{i=1}^{n}A_{\alpha(i)}\|_b \leq
\| \pro_{i=1}^{i_{1}-1}A_{\alpha(i)}\|_b
 \pro_{j=1}^{l}
\left(
\left\| A_{\alpha(i_{j})} \right\|_b
  \left \| \pro_{i=i_{j}+1}^{i_{j+1}-1}A_{\alpha(i)}
\right\|_b
 \right) 
$$
$$
\leq
C(n^{p+1}r^p)^{2^{m-1}} 
\exp(p\sum_{j=1}^{l+1}\max_{i_{j-1}<i<i_{j}}\alpha(i))
\exp( -\delta\sum_{i} \alpha(i))
C_1^{|\{\alpha(i)>0\}|}.
$$
and the lemma follows.
\enddemo
 
  End of the proof of \ba.
We take $m=[p/\delta]+1$ and $r=(n\|A\|_{a})^{1/\delta}$. 
Then \ba holds by \bg and \bd.

%\proclaim{(3.1')}
\t {\bh }
{Let $a>b$, $K$, $\Omega$ satisfy assumptions of \ba and
$$
s > 2^{[(b+q/2)/(a-b)]}((b+q/2)(1+1/(a-b))+1)+1/2.
$$
 There
exists $C$ such that for all $f\in H(s)$
$$
\bnorm f(K)\bnorm_{b}\leq C\|f\|_{H(s)}.
$$}
%\endproclaim
\demo{Proof}
We may assume that
%By assumption
$\|K\|_{L^{2}}\leq 1$ (if not we replace $K$ by $K/\|K\|_{L^{2}}$ and adjust $f$). 
Therefore $\spect K \subset [-1,1]$ and
$f(K)$ does not depend on the values of $f$ outside
$[-1,1]$. %and 
Putting $h=\phi f$ where $\phi \in C^{\infty}_{c}([-2,2])$ 
and $\phi=1$ on $[-1,1]$ we have $f(K)=h(K)$. 
Next
$$
h(K)=\sum \hat h(n)e^{inK}
$$
and
$$
\bnorm h(K)\bnorm_{b}\leq \sum |\hat h(n)|\bnorm e^{inK}\bnorm_{b}\leq 
M\sum |\hat h(n)|(1+|n|)^{\kappa}\leq \sum |\hat h(n)|(1+|n|)^{s}(1+|n|)^{\kappa-s}
$$
$$
\leq M\|h\|_{H(s)}(\sum (1+|n|)^{2(\kappa-s)})^{1/2}
\leq C\|f\|_{H(s)}
$$
where $\kappa$ and $M$ are as in \ba.
\enddemo

\p {\bi }
{Let $I$ and $J$ be closed intervals such that $I\subset \Int\/ J
\subset J \subset (-\pi,\pi)$. Let $B(s)$ be $C(s)$ or $H(s)$, $f\in B(s_{0})$,
%{Let $I$ and $J$ be closed intervals such that $I\subset \Int\/ J
%\subset J \subset (-\pi,\pi)$, $f\in H(s)$,
 $\supp\/f\subset I$, $l>0$. There
exist functions $f_{j}$, $j=0,1,\dots$ satisfying the following conditions
$$f=\sum f_{j}$$ 
$$\supp f_{j}\subset J$$ 
$$|\hat f_{j}(k)|\leq C(s_{0},I,J,l)2^{-s_{0}j}
    (1+\max(0,|k|-2^{j}))^{-l-3}\|f\|_{B(s_{0})}$$ 
$$\|f_{j}\|_{B(0)}\leq C(s_{0},I,J,l)2^{-s_{0}j}\|f\|_{B(s_{0})}$$
where $C(s_{0},I,J,l)$ depend only on $s_{0}$, $I$, $J$, $l$.} 

We choose smooth functions $\varphi$, $\psi$ such that
$\supp \varphi\subset J$, $\varphi|_{I}=1$, $\psi=1$ on 
$[-{1 \over 2},{1 \over 2}]$,
$\supp\psi\subset [-1,1]$ and we put
$$
h_{j}(x)=
\cases
{\sum \psi(k)\hat f(k)e^{ikx} \qquad\text{for $j=0$,}
 \cr
\sum [\psi(2^{-j}k)-\psi(2^{-j+1}k)]\hat f(k)e^{ikx}
\quad\text{otherwise},\cr }
$$
$$
f_{j}=\varphi h_{j}
$$
Third condition holds because
$$
\hat f_{j}(k)=\sum_{r}\hat \varphi(r)\hat h_{j}(k-r)
$$
and $|\hat \varphi(k)| \leq C(1+|k|)^{-l-4}$.


{\bf Remark}. \bi is valid for very general scales of Banach spaces. 
In particular it is valid for Besov and Tribel-Lizorkin spaces.

%\proclaim{(3.10).Theorem}
\t {\bj}
{For every $b\geq 0$, $s>(b+q/2)(1+2/(a-b))$, $a>2b+q/2$ 
%and a closed interval $I$ 
there exist $C$ such that 
%if $\supp f \subset I$, then
$$
\bnorm f(K)K \bnorm_{b} \leq C\|f\|_{C(s)}.
$$}
%\endproclaim
\demo{Proof}
Without any loss of generality we assume that $\supp f\subset (-\pi,\pi)$. 
We decompose $f$ as in \bi (with $l>s$). Fix $k$. We put $n=2^{k+2}$,
 $r=(n^{2}\|A\|_{a})^{(1/(a-b))}$,
 $p=b+q/2$, 
$$
E(x,y)=\cases{ K(x,y)\text{\quad for\ $d(x,y)<nr$} \cr
0 \qquad \text{otherwise}.\cr }$$
Observe that
$$
%\align
\inT_{d(x,y)>4nr}|f_{k}(K)K(x,y)|(1+d(x,y))^b d\mu (y) $$
$$\leq \inT_{d(x,y)>4nr}|f_{k}(K)E(x,y)|(1+d(x,y))^b d\mu (y)$$
$$ \qquad + \|f_{k}(K)\|_{b}\inT_{d(x,y)>nr}|K(x,y)|(1+d(x,y))^b d\mu(y)
=I_{1}+\|f_{k}(K)\|_{b}I_{2}$$
As $I_2 \leq Cn^{-2}$ and $p<\delta$, 
we get estimate for the second term as in \bh. Also
$$
%\inT_{d(x,y)>2nr}|f_{k}(K)E(x,y)&|(1+d(x,y))^b d\mu (y) \\
I_{1}\leq \sm_{|j|\leq 2^{k+2}}|\hat f_{k}(j)| 
\inT_{d(x,y)>4nr}|A^{j}E(x,y)|(1+d(x,y))^b d\mu (y)
+ \sm_{|j|>2^{k+2}}|\hat f_{k}(j)|\thinspace \|A^{j}\|_{b}\|K\|_{b}
\endalign
$$
and by \bb, \be and \bi (note that $C_{1}\leq{1\over n^2}$), this is
$$
\leq Cn^{-s}(nr)^{p}
$$
Next, putting $U=\{x:d(x,y)\leq 4nr \}$, we have
$$
\align
\inT_{U}|f_{k}(K)K(x,y)&|(1+d(x,y))^bd\mu (x) \\
&\aligned
&\leq \|\chi_{U}\|_{L^{2}} \sup_{x\in U} (1+d(x,y))^{b}
 \|f_{k}(K)K(\cdot,y)\|_{L^2}\\
&\leq \|f_{k}\|_{L^{\infty}}\sup_{x\in U} (1+d(x,y))^{b} 
\|\chi_{U}\|_{L^{2}}\|K(\cdot,y)\|_{L^2} \\
&\leq Cn^{-s}(nr)^{p}.
\endaligned
\endalign
$$
Gathering the estimates above we get
$$
\|f_{k}(K)K\|_{b}\leq Cn^{-s}(nr)^{p}\leq C'2^{-k\varepsilon}
$$
for some $\varepsilon>0$,  
which of course implies our claim.
\enddemo



\def\Mr{\kappa}
\def\Ml{\gamma}
\t {\nt} {
Let $I$ and $J$ be closed intervals such that $I\subset \Int\/ J
\subset J$. Assume $a>2b+q/2$. 
If there exist $\Mr$, $\Ml$, $\chi$ such that $\Mr\geq1$, 
$b\geq 0$, $s>(b+q/2)(1+\Mr/(a-b))+2-\Mr$, $s>(b+\Ml/2)(1+\Mr/(a-b))$, 
$s>\chi+b(1+\Mr/(a-b))$ and
for every $f$ such that $\supp f\subset J$ 
$$
\int _{d(x,y)<r}|f(K)|(x,y)dx \leq M(r^{\Ml/2}\|f\|_{L^{2}}+\|f\|_{H(\chi)}),$$
then for every $f$ such that $\supp f\subset I$ 
$$
\|f(K)K\|_{b}\leq M'\|f\|_{H(s)}.$$}

\PP 
The proof is similar to the proof of \bj. We choose 
$r=(n^{\Mr}\|A\|_{a})^{(1/(a-b))}$ to get $C_{1}=n^{-\Mr}$, also $l$ in \bi is made large enough. Then we use the assumption to estimate 
integral over the set $U$: 
$$
\inT_{U}|f_{k}(K)K(x,y)|(1+d(x,y))^{b}d\mu (x)$$
$$\leq (1+4(nr)^b)\inT_{U}|f_{k}(K)K(x,y)|d\mu (x)$$
$$\leq M(1+4(nr)^b)((nr)^{\Ml/2}\|xf_k\|_{L^{2}}+\|xf_k\|_{H(\chi)})$$
$$\leq C(nr)^{b+\Ml/2}n^{-s} +C_1(nr)^b 
\|f_k\|^{\chi/(\chi+b(1+\Mr/(a-b)))}_{H(\chi+b(1+\Mr/(a-b)))}
\|f_k\|^{b(1+\Mr/(a-b))/(\chi+b(1+\Mr/(a-b)))}_{L^{2}}$$
$$\leq C(nr)^{b+\Ml/2}n^{-s} 
+C_2(nr)^b(n^{-s})^{b(1+\Mr/(a-b))/(\chi+b(1+\Mr/(a-b)))}\leq C_3 2^{-k\varepsilon}$$


\leftline{\bf 3. Multiplier theorems.}
\pno=0
\sno=3

   Let $A$  be  a  non-negative  self-adjoint  densely  defined 
operator on $L^{2}(M,\mu)$. Let $E$ be the spectral measure of $A$. 
By the spectral theorem, we write 
$$
Af = \int \lambda dE(\lambda)f
$$
and 
$$
e^{-tA}f = \int e^{-\lambda}dE(\lambda)f. 
$$
We assume that 
$$
e^{-tA}   f(x) = \int e^{-tA}   (x,y)f(y) d\mu(y) 
$$
where the kernels $e^{-tA}   (x,y)$ satisfy the following estimates :

there exist positive numbers $a$, $m$, $\alpha$ and $C$ such that
 for all $t$
$$
\align
 &\sup_{y} \int |e^{-tA}(x,y)|(1+t^{-1/m}d(x,y))^{a} d\mu(x) \leq C \\   
& \sup_{y} 
\mu(B(y,t^{1/m}))\int |e^{-tA}(x,y)|^{2} d\mu(x) \leq C $$

%\proclaim{(4.1) Theorem}
\t {\ca}
  {Assume  that the conditions above  are   satisfied and that 
  $$ \sup_{y,z}\int|e^{-tA}(x,y) - e^{-tA}(x,z)|d\mu(x) \leq Ct^{-\alpha/m} 
d(y,z)^{\alpha}  .
\endalign
$$
     If  
$F \in H(s)_{loc}$, 
$$
s > 2^{[q/(2a)]}((q/2)(1+1/a)+1)+1/2
$$ 
and for a non-zero 
$\varphi \in C^{\infty}_{c}({\Cal R}_{+})$ 
$$
\sup_{t>0} \| F_{t}\varphi \|_{H(s)} \leq M   
$$
where 
$$
F_{t}(\lambda) = F(t\lambda),
$$
then $F(A) = \int F(\lambda)dE(\lambda)$ is of weak type (1,1) and  bounded  on 
$L^{p} (M)$, $1 < p < \infty $.}
\endproclaim
\nic {
\p {\ca}
  {Assume  that the conditions above   are   satisfied.   If $a>q/2$
$F \in C(l)$, $l > (1+2/a)q/2$, and for a non-zero 
$\varphi \in C^{\infty}_{c}({\Cal R}_{+})$ 
$$
 \| F_{t}\varphi \|_{C(l)} \leq M   
$$
where 
$$
F_{t}(\lambda) = F(t\lambda),
$$
then $F(A) = \int F(\lambda)dE(\lambda)$ is of weak type (1,1) and  bounded  
on $L^{p} (M)$, $1 < p < \infty $.}
%\endproclaim
}
%\proclaim{(4.2).Lemma} 
\p {\cb}
{If for some constants $R < 1$, $M$, $a > 
0$, $\alpha > 0$,
a family of  kernels  $\{K_{n}\}_{n=0}^{\infty}$  satisfies
$$
%\gather
\|K_{n} \|_{B((1+R^{n}d)^{a})} \leq M $$
$$\int|K_{n}(x,y)-K_{n}(x,z)|d\mu(x) \leq MR^{n\alpha}
d(y,z)^{\alpha},
%\endgather
$$
then for some $C$ depending only of $R$, $a$, $\alpha$ 
$$
\sup_{z,y}\inT_{d(x,y)>2d(y,z)}
\sm_{n}|K_{n}(x,y)-K_{n}(x,z)|d\mu(x)  \leq MC 
$$
%\endproclaim
}
\demo{Proof} Fix $y$ and $z$. Let 
$$
S = \{ x : d(x,y)>2d(y,z) \}. 
$$
Now 
$$
\inT_{S} |K_{n} (x,y) - K_{n} (x,z)|d\mu(x)\hfil$$
$$ 
\leq \inT_{S} |K_{n} (x,y)|d\mu(x) + \inT_{S} |K_{n} (x,z)|d\mu(x)
\leq 2MR^{-na}d(y,z)^{-a}  . 
$$
Then 
$$
\align
 \sum  \inT_{S} |K_{n} (x,y) - K_{n} (x,z)|&d\mu(x) \\
&\aligned
&\leq \sum \min (MR^{n\alpha}d(y,z)^{\alpha},2MR^{-na}d(y,z)^{-a}) \\
&\leq 2M(1 - R^{a})^{-1}+ M(1 - R^{\alpha})^{-1} \leq CM. 
\endaligned
\endalign
$$
\enddemo

 {{\it Proof of \ca\/}:} First note that if $\mu(M)=\infty$ then second 
assumption about $e^{-tA}$ implies that the spectral measure of $A$ 
has no atom at $0$. If $\mu(M)<\infty$ then this assumption implies 
that the spectral projector corresponding to $0$ is bounded on $L^1$ (
beeing bounded from $L^{1}$ to $L^{2}\subset L^{1}$) and hence by interpolation
and duality on all $L^{p}$, $1\leq p\leq \infty$. In any case we need to 
handle only (strictly) positive part of the spectrum.  
Choose $\varphi\in C^{\infty}_{c}({\Cal R}_{+})$ such that 
$\sum \varphi(2^{mn}\lambda)=1$ for $\lambda>0$.
%on ${\Cal R}_{+}$. 
Let
$$
%\gather
F_{n}(\lambda)=\varphi(\lambda)F(2^{-mn}\lambda),\\
G_{n}(\lambda)=F_{n}(-\log(\lambda)),\\
H_{n}(\lambda)=F_{n}(-\log(\lambda))\lambda^{-1},\\
e_{n}=e^{-2^{mn}A},\\
K_{n}=G_{n}(e_{n})=H_n(e_n)e_{n}.
\endgather
$$
We have
$$
F(A)=\sum F_{n}(2^{mn}A)=\sum K_{n}.
$$
By the assumption,
$$
\gather
\sup_{y} \int |e_{n}(x,y)|(1+2^{-n}d(x,y))^{a}d\mu(x)\leq C \\
\sup_{y} \mu(B(y,2^{n}))\int |e_{n}(x,y)|^{2}d\mu(x)\leq C
\endgather
$$
and of course,
$$
\|G_{n}\|_{H(s)}\leq C'
$$
therefore replacing $d$ by $2^{-n}d$ we may apply \bh to get
$$
\|K_{n}\|_{B((1+2^{-n}d)^{\varepsilon})}\leq M
$$
for sufficiently small $\varepsilon>0$.
Moreover
$$
|K_{n}(x,y)-K_{n}(x,z)| \leq \int |H_{n}(e_{n})(x,s)|
|e_{n}(s,y)-e_{n}(s,z)|d\mu(s)
$$
hence
$$
\int |K_{n}(x,y)-K_{n}(x,z)|d\mu(x) \leq  \|H_{n}(e_{n})\|_{0}
\int |e_{n}(s,y)-e_{n}(s,z)|d\mu(s) \leq M 2^{-\alpha n}d(y,z)^{\alpha}
$$
This means that assumptions of \cb are satisfied.
\enddemo

Now, we use the general theory of Calder\'on-Zygmund operators 
(see for example Coifman-Weiss [\CW]).



\nic{

%\proclaim{(4.3) Theorem}
\t {\cc}
  {Assume  that the conditions above  (4.1) are   satisfied.   If  
$F \in H(s)_{loc}$, 
$$
s > 2^{[q/(2a)]}((q/2)(1+1/a)+3/2)
$$ 
and for a non-zero 
$\varphi \in C^{\infty}_{c}({\Cal R}_{+})$ 
$$
 \| F_{t}\varphi \|_{H(s)} \leq M   
$$
where 
$$
F_{t}(\lambda) = F(t\lambda),
$$
then $F(A) = \int F(\lambda)dE(\lambda)$ is of weak type (1,1) and  bounded  on 
$L^{p} (M)$, $1 < p < \infty $.}
\endproclaim
\demo{Proof}
The same as \ca, only using \bh instead of \bj.
\enddemo
}

\def \Cup{\cup}
\def \dt {\rho_{t} }
\def \Mt {\tilde M}
\def \Hl {M}
\def \Wpl {W_{l}^{p}}
\def \Max {F^{*}}
\def \phi{\varphi}

\def \Ka{K}
\def \Kb{\Psi}
\def \ep{\epsilon}
\t {\dz}
{If $\Ka=\sum \Ka_{k}$, $\Ka$ is bounded on $L^{p}$, $p>1$, $\Ka_{k}\Kb_{j}=0$ 
for $k>j$, $\Ka_{k}\Kb_{j}=\Ka_{k}$ for $k<j$ and
$$
\int |\Ka_{k}(x,y)|(1+2^{-k}d(x,y))^{\ep}\leq C $$
$$
|\Kb_{k}(x,y)|\leq C(1+2^{-k}d(x,y))^{-q-\ep}
(B(x,2^{k})^{-1}+B(y,2^{k})^{-1})$$
then $\Ka$ is of weak type 1-1.}


Put 
$$
\dt(x,y)=(1+t^{-1}d(x,y))^{-q-\varepsilon}(\mu(B(x,t))^{-1} +
\mu(B(y,t))^{-1}),
$$
$$
\Mt f(x)=\sup_{t>0}\int \dt(x,y)|f(y)|dy.
$$

 
\p {\dd} {$$\Mt f(x)\leq C\Hl f(x).$$
In particular $\Mt$ is bounded on $L^{p}$, for all $1<p\leq \infty$.}

\PP  There exists $C$ such that 
$$
\mu(B(x,2d(x,y)))\leq C(1+t^{-1}d(x,y))^{q}\mu(B(x,t)),
$$
$$\mu(B(x,2d(x,y)))\leq \mu(B(y,4d(x,y)))
\leq C(1+t^{-1}d(x,y))^{q}\mu(B(y,t))
$$
for all $x$, $y$, $t>0$. Hence
$$
\dt(x,y)\leq C'(\mu(B(x,\max(t,2d(x,y)))))^{-1}
(1+t^{-1}d(x,y))^{-\varepsilon}.
$$
Fix $t$. We have
$$
\int \dt(x,y)|f(y)|dy \leq
C''\sum_{k=0}^{\infty}2^{-\varepsilon}\mu(B(x,2^{k}t))^{-1}
\int_{B(x,2^{k}t)}|f(y)|dy
$$
$$
\leq C'''Mf(x).
$$ 


%$$ \delta(t)F (x)=F(tx).  $$ 



If $f$ is an arbitrary $L^{1}$ function and $\lambda$ is a positive real 
number, then either $\mu(M)\leq \|f\|_{L^{1}}/\lambda$ and
$$
\mu(\{x: |Kf(x)| >\lambda\})\leq \mu(M)\leq \|f\|_{L^{1}}/\lambda
$$
or we apply to $f$ the Calderon-Zygmund decomposition at height 
$\lambda$ (see for example Coifman-Weiss [\CW], Chapitre 3, Theoreme (2.2) ) 
and obtain balls $B(x_{i},r_{i})$ such that putting $P=\bigcup 
B(x_{i},r_{i})$ we have
$$
\|f\big|_{M-P}\|_{L^{\infty}}\leq C\lambda
$$
$$
\int_{B(x_{i},r_{i})} |f|\leq C\lambda\mu(B(x_{i},r_{i}))
$$
$$
\sum \mu(B(x_{i},r_{i}))\leq C \|f\|_{L^{1}}/\lambda
$$
and every $x\in M$ belongs to at most $C$ different $B(x_{i},r_{i})$.

Put $Q_{i}=B(x_{i},r_{i})$, $Q^{*}_{i}=B(x_{i},2r_{i})$, 
$S_{i}=Q_{i}-\Cup_{j<i} Q_{j}$, $f_{i}=\chi_{S_{i}}f$, 
%We also put  
$k _{i } =  [\log _{2 } (r _{i } )]$. 

\p {\dq} 
{ There exists $C$ such that for all $i$ 
$$
%\int_{(Q^{*}_)^{c}}
\sum_{n}\int_{(Q_{i}^{*})^{c}}
|\Ka_{n}(1- \Kb_{k_{i}})f_{i}|(x)dx \leq C \|f_{i}\|_{L^{1}}.$$ }

\PP 
$$
\int_{(Q_{i}^{*})^{c}}
|\Ka_{n} f_{i}|(x)dx \leq \|f_{i}\|_{L^{1}}\sup_{y \in Q_{i}}
\int_{(Q_{i}^{*})^{c}}|\Ka_{n}(x,y)|dx $$
$$
\leq \|f_{i}\|_{L^{1}}\sup_{y}
\int_{ d(x,y)>2^{k_{i}-1}}|\Ka_{n}(x,y)|dx $$
$$\leq 2^{\ep(n-k_{i}+1)}\|f_{i}\|_{L^{1}}\sup_{y}
\int_{ d(x,y)>2^{k_{i}-1}}|\Ka_{n}(x,y)|(1+2^{-n}d(x,y))^{\ep}dx$$
$$\leq C2^{\ep(n-k_{i})}\|f_{i}\|_{L^{1}}
$$
 By assumption, $\Ka_{n}(1- \Kb_{k_{i}})=0$ for $n>k_{i}$ and  
 $\Ka_{n}(1- \Kb_{k_{i}})=\Ka_{n}$ for $n<k_{i}$. Hence 
$$
\sum_{n}\int_{(Q_{i}^{*})^{c}}
|\Ka_{n}(1- \Kb_{k_{i}})f_{i}|(x)dx 
\leq \|\Ka_{k_{i}}(1-\Kb_{k_{i}})f_{i}\|_{L^{1}}+
\sum_{n < k_{i}}\int_{(Q_{i}^{*})^{c}}
|\Ka_{n}f_{i}|(x)dx $$
$$\leq C \|f_{i}\|_{L^{1}} \sum_{n\leq k_{i}}2^{\ep(n-k_{i})}$$
$$\leq C \|f_{i}\|_{L^{1}}.$$

 
\p {\de} { For every $1\leq p<\infty $ there  exist  $C_{p}$ such  that  
$$\|  \sum   \Kb  _{k _{i }} f _{i } \| ^{p }_{L ^{p }} \leq   
C_{p}\lambda^{p-1} \|f\| _{L ^{1}}.$$}
\PP Put $\tau_{i}=\rho_{2^{k_{i}}}$. It is easy to check 
(using doubling condition) that 
$$
\sup_{y\in Q_{i}}\tau_{i}(x,y)\leq C \inf_{y\in Q_{i}}\tau_{i}(x,y)
$$
with $C$ uniform in $x$ and $i$. 
Fix $i$ and  $x$ and choose $y_{0}\in Q_{i}$.
$$|\Kb_{k_{i}}f_{i}|\leq \int \tau_{i}|f_{i}|(y)d\mu(y)$$
$$\leq C\lambda \mu(Q_{i})\tau_{i}(x,y_{0})$$
$$\leq C'\lambda\tau_{i}\chi_{Q_{i}}.$$
Let $r=p/(p-1)$. If $h\in L^{r}$, $h\geq 0$, then
$$
|(h,\tau_{i}\chi_{Q_{i}})|=|(\tau_{i}h,\chi_{Q_{i}})|\leq
(\Mt h,\chi_{Q_{i}}).
$$
By \dd, 
$$
(h,\sum |\Kb_{k_{i}}f_{i}|)
\leq C(\Mt h,\sum \lambda \chi_{Q_{i}})
\leq C\|h\|_{L^{r}}\|\sum \lambda \chi_{Q_{i}}\|_{L^{p}}
$$
By properties of Calder\'on-Zygmund decomposition 
$\|\sum \lambda \chi_{Q_{i}}\|^{p}_{L^{p}}\leq C\sum\lambda^{p}\mu(Q_{i})
\leq C'\lambda^{p-1}\|f\|_{L^{1}}$, which ends the proof of \de.  



Let 
$$g=f-\sum_{i} (1-\Kb_{k_{i}})f_{i}=f- \sum_{i}f_{i} + 
\sum_{i} \Kb_{k_{i}}f_{i}$$ 
so
$$
\Ka f=\Ka (\sum_{i}(1- \Kb_{k_{i}})f_{i}+g)=
\sum_{n}\sum_{i}\Ka_{n}(1- \Kb_{k_{i}})f_{i}+\Ka g$$
We have
$$\|g\|^{p}_{L^{p}}\leq p(\|f-\sum_{i}f_{i}\|^{p}_{L^{p}}
+\|\sum_{i}\Kb_{k_{i}}f_{i}\|^{p}_{L^{p}})$$
$$\leq p(\|f-\sum_{i}f_{i}\|^{p-1}_{L^{\infty}}\|f-\sum_{i}f_{i}\|_{L^{1}}
+C\lambda^{p-1}\|f\|_{L^{1}})\leq C'\lambda^{p-1}\|f\|_{L^{1}}.$$
Put $E=\Cup_{i} Q^{*}_{i}$. By \dq, 
$$\int_{M-E} |\sum_{n}\sum_{i}\Ka_{n}(1- \Kb_{k_{i}})f_{i}|(x)dx
\leq \sum_{n}\sum_{i}\int_{(Q_{i}^{*})^{c}}
|\Ka_{n}(1- \Kb_{k_{i}})f_{i}|(y)dy
$$
$$
\leq \sum_{i}C\|f_{i}\|_{L^{1}}
\leq C'\|f\|_{L^{1}}.$$
Finally
$$
|\{|Kf|>\lambda\}|\leq |\{|Kg|>\lambda/2\}|+
|\{|\sum_{n}\sum_{i}\Ka_{n}(1- \Kb_{k_{i}})f_{i}|>\lambda/2\}|
$$
$$
\leq {(2\|Kg\|_{L^{p}})^{p} \over \lambda^{p}}+ |E| + 
{2\int_{M-E} |\sum_{n}\sum_{i}\Ka_{n}(1- \Kb_{k_{i}})f_{i}|(x)dx 
\over \lambda }$$
$$
\leq C({\lambda^{p-1}\|f\|_{L^{1}}\over \lambda^{p}}+ 
{\|f\|_{L^{1}} \over \lambda })\leq 2C\lambda^{-1}\|f\|_{L^{1}}$$
which ends the proof of \dz.


 
\t {\da} { If $a>q$, $l >(1+2/a)q/2 +1$, $p>1$ and 
for  a  non-zero  $\varphi  
\in C ^{\infty  }_{c } ({\Cal R }_{+})$,
$$
\int_{0}^{\infty} t^{-1}\|\varphi\delta(t)F\|_{\Wpl}dt<\infty, 
$$
then 
$$
F^{*}f=\sup_{t>0}|\delta(t) F(A)f|
$$
is of weak type (1,1)  and  bounded  on  $L ^{p }$ for  
$1  <  p \leq  \infty $.}

{\bf Remark}. The index $l$ in \da can not be essentially 
lowered. This contrasts with homogeneous multipliers on $R^{n}$, (see [\Se]) 
where one can take $l$ close to $(n+1)/2$ (or use derivatives in $L^{2}$ 
and $l$ close to $n/2$).
 To see this let $A$ be operator on $C^{\infty}_{c}(R^{2})$ 
defined by the formula 
$$\hat {Af} =\psi\hat f$$ where $\hat{}$ denotes the Fourier transform. 
Assume that $\psi$ is homogeneous with respect to anisotropic dilations, 
that is $\psi(tx,t^{2}y)=t^{k}\psi(x,y)$ and that the level 
set $\{(x,y): \psi(x,y)=1\}$ 
contains an interval $I$ not parallel to any of the coordinate axes. 
Let $\supp \hat f$ be contained in small neighbourhood of a point $x$ in 
$I$. Take $F(\lambda)=(1-\lambda)^{(1-\ep)}$. Then 
$$(F(A)f)(x,y)\approx (x^{2}+y^{2})^{-1+\ep/2}$$
when $(x,y)$ lies in a bounded distance from direction normal to $I$ and 
$F(A)f$ is small outside this set. If we take $F(tA)f$ then we get 
similar estimate but the set where $F(tA)f$ is large changes.  
More precise, it  
is obtained applying anisotropic dilation to the set where $F(A)f$ is 
large. Anisotropic dilations act transitively on directions different 
from axes, so 
$$F^{*}(A)f(x,y)\approx (x^{2}+y^{2})^{-1+\ep/2}$$
on a cone with nonempty interior. But then clearly $F^{*}(A)$ is 
not of weak type 1-1. If $k$ is large then $A$ satisfies our 
estimates with large $a$. One can modify this example to 
get differential $A$, (then level set must be tangent of high order to $I$). 


 
\t {\db} { If $a>q$,  $s   >(1+2/a)q/2$ and  for  a  non-zero  $\varphi  
\in C ^{\infty  }_{c } ({\Cal R }_{+})$ and  a  
constant  $C$ 
$$
\sup_{t>0}\|\varphi \delta(t)F \| _{C(s) } \leq   C,
$$ 
then  $F(A)$  is  of  weak  type  (1, 1)  and  bounded  on  $L ^{p }$ for  
$1  <  p <  \infty $. }

{\bf Remark}. With other assumptions as in \db, if
$$
\int_{0}^{\infty} t^{-1}\|\varphi\delta(t)F\|_{C(s)}dt<\infty, $$
then  $F(A)$ is bounded on $L^{1}$.

{\bf Remark}. \db applied to the sublaplacean on Lie group of
polynomial growth gives result essentially equivalent to that 
of G. Aleksopoulos [\Al]. 



\p {\dc} { There exists $C$ such that for all $x$, $y$ and $t$
$$
|e^{tA}(x,y)|\leq C\mu(B(y,t^{1/m}))^{-1}.
$$}

Proof. We are going to prove that for all 
  $\varepsilon>q/2$ there exists $C$ such that
$$
|e^{tA}(x,y)|\leq C(1+t^{-1/m}d(x,y))^{-(a-\varepsilon)}(\mu(B(x,t^{1/m})) 
\mu(B(y,t^{1/m})))^{-1/2}
$$
for all $x$, $y$ and $t$. This easily implies our claim. 

 The estimate above is trivially true for $\varepsilon=a$. We have 
(using Schwartz inequality) 
$$|e^{2tA}(x,y)|(1+t^{-1/m}d(x,y))^{(a-\varepsilon)}
(\mu(B(x,t^{1/m})) 
\mu(B(y,t^{1/m})))^{1/2}=$$
$$\int 
|e^{tA}(x,s)e^{tA}(s,y)|
(\mu(B(x,t^{1/m})) 
\mu(B(y,t^{1/m})))^{1/2}(1+t^{-1/m}d(x,y))^{(a-\varepsilon)}ds
$$
$$
\leq\sup_{y} \mu(B(y,t^{1/m}))\int 
|e^{tA}(s,y)|^{2}(1+t^{-1/m}d(s,y))^{2(a-\varepsilon)}ds
$$
$$
\leq \sup_{y} \int
|e^{tA}(s,y)|(1+t^{-1/m}d(s,y))^{a}ds
$$
$$
\times \sup_{s,y}\mu(B(y,t^{1/m}))
|e^{tA}(s,y)|(1+t^{-1/m}d(x,y))^{(a-2\varepsilon)}
$$
$$
\leq C \sup_{s,y}(\mu(B(x,t^{1/m})) 
\mu(B(y,t^{1/m})))^{1/2}
|e^{tA}(s,y)|(1+t^{-1/m}d(x,y))^{(a-2\varepsilon+q/2)}$$
Repeating this we can get $\varepsilon - q/2$ arbitrarily small and thus 
\dc is proved.  



We  fix  $\varphi  $ and  $\psi  $ such  
that  $\varphi , \psi  $ are  in  $C ^{\infty  } ({\Cal R }),$ 
$\supp  \varphi   \subset   [2^{-m}, 2^{m/2}]$, $(\forall x>0) $ 
$\sum   \varphi (2 ^{mk } x) = 1$, and  $\supp   \psi \subset  [-1,1]$, 
with  $\psi (x)  =  1 $ for  
$x\in [0, 2^{-m/2}]$. 
\def \indk{k}
Let
$$\varphi _{\indk } (\lambda )  =  \varphi (2 ^{m\indk } \lambda ),$$ 
$$\psi  _{\indk } (\lambda )  =  \psi (2 ^{m\indk } \lambda ),$$
$$\Kb_{\indk}=\psi  _{\indk } (A),$$ 
$$F_{\indk}(\lambda)=\varphi_{\indk}(\lambda)F(\lambda),$$
$$G_{\indk}(\lambda)=F_{\indk}(-2^{-m\indk}\log(\lambda))\lambda^{-1},$$
$$e_{\indk}=e^{-2^{m\indk}A},$$
$$W_{\indk}=\psi  _{\indk } (-2 ^{-m\indk }\log(\lambda) )\lambda^{-2},$$
$$K_{\indk}=G_{\indk}(e_{\indk})e_{\indk}.$$
We have
$$
F(A)=\sum F_{\indk}(A)=\sum G_{\indk}(e_{\indk})e_{\indk}=\sum K_{\indk}.
$$
By the assumption,
$$
\sup_{y} \int |e_{\indk}(x,y)|(1+2^{-\indk}d(x,y))^{a}d\mu(x)\leq C 
$$
$$
\sup_{y} \mu(B(y,2^{\indk}))\int |e_{\indk}(x,y)|^{2}d\mu(x)\leq C
$$
and of course,
$$
\|G_{\indk}\|_{C(l)}\leq C'
$$
therefore replacing $d$ by $2^{-\indk}d$ we may apply \bj to get
$$
\|K_{\indk}\|_{B((1+2^{-\indk}d)^{\varepsilon})}\leq M
$$
for sufficiently small $\varepsilon>0$.

Also
$$
\psi_{\indk}(A)=W_{\indk}(e_{\indk})e_{\indk}^{2}$$
and for any $s$
$$\|W_{\indk}\|_{H(s)}\leq C_{s}$$
therefore replacing $d$ by $2^{-\indk}d$ we may apply \bh to get
$$
\int |W_{\indk}(e_{\indk})|(x,y)(1+2^{-k}d(x,y))^{a-\ep} \leq C$$
By \dc (and symmetry) we get
$$\sup_{y} |W_{\indk}(e_{\indk})e_{\indk}|(x,y) \leq \mu(B(x,2^{k}))^{-1}$$
Next
$$|\psi_{\indk}(A)|(x,y) \leq 
\int |W_{\indk}(e_{\indk})e_{\indk}|(x,s)|e_{\indk}|(s,y)ds$$
$$
\leq \int_ {d(x,s)\geq d(x,y)/2}
|W_{\indk}(e_{\indk})e_{\indk}|(x,s)|e_{\indk}|(s,y)ds
+ \int _ {d(y,s)\geq d(x,y)/2}
|W_{\indk}(e_{\indk})e_{\indk}|(x,s)|e_{\indk}|(s,y)ds$$
$$\leq \sup_{s}|e_{\indk}|(s,y)\int_ {d(x,s)\geq d(x,y)/2}
|W_{\indk}(e_{\indk})e_{\indk}|(x,s) ds $$
$$+ \sup_{s}|W_{\indk}(e_{\indk})e_{\indk}|(x,s) \int_ {d(y,s)\geq d(x,y)/2}
|e_{\indk}|(s,y)ds
$$
$$\leq C(1+t^{-1/m}d(x,y))^{-(a-\varepsilon)}(\mu(B(x,t^{1/m}))^{-1} 
+\mu(B(y,t^{1/m}))^{-1})$$
 
In other words
$$|\psi  _{k } (A)|(x, y)  \leq   
C(1+2^{-k}d(x,y))^{-(a-\varepsilon)}(\mu(B(x,2^{-k}))^{-1}
+\mu(B(y,2^{-k}))^{-1})
$$ 
which is the second assumption in \dz. This ends the proof of \db. 

The following lemma is all what is needed to end the proof of \da. 


%Note that by \dd,
%$$\|\sup_{t>0}e^{tA}f\|_{L^{2}}\leq C\|f\|_{L^{2}}.$$


\p {\df} { 
$$
\int \sup_{t>0}|F(tA)\phi_{k}(A)|(x,y)(1+2^{-k}d(x,y))^{\epsilon}dx
\leq C\int_{0}^{\infty} t^{-1}\|\varphi\delta(t)F\|_{\Wpl}dt<\infty. 
$$}

\PP 
We have
$$
\int \sup_{t>0}|F(tA)\phi_{k}(A)|(x,y)(1+2^{-k}d(x,y))^{\epsilon}dx$$
$$\leq \sum_{j}\int \sup_{t>0}|F_{j}(tA)\phi_{k}(A)|(x,y)
(1+2^{-k}d(x,y))^{\epsilon}dx$$
and
$$
\sum_{j} \|\delta(2^{-mj})F_{j}\|_{\Wpl}
\leq C\int_{0}^{\infty} t^{-1}\|\varphi\delta(t)F\|_{\Wpl}dt$$
so it is enough to show that
$$
\int \sup_{t>0}|F_{j}(tA)\phi_{k}(A)|(x,y)(1+2^{-k}d(x,y))^{\epsilon}dx
\leq C\|\delta(2^{-mj})F_{j}\|_{\Wpl}.$$
Without any loss of generality we may assume that $k=0$ and $j=0$. Indeed, 
otherwise we replace $d$ by $2^{-k}d$ and $F$ by $\delta(2^{-mj})F$. 
We write
$$e_{n}=\exp(ine^{-A})e^{-A},$$
$$H_{t}(\lambda)=(\phi_{0})(-\log(\lambda))
F_{0}(-t\log(\lambda))/\lambda.$$

We have
$$F_{0}(tA)\phi(A)=H_{t}(e^{-A})e^{-A}=
\sum_{n}\hat H_{t}(n)e_{n}$$
and $H_{t}=0$ for $t\notin[2^{-2m},2^{2m}]$.
It follows
$$\sup_{t>0}|\phi(A)F_{0}(tA)|(x,y)
\leq\sum_{n}\sup_{2^{2m}\geq t>2^{-2m}}|\hat H_{t}(k)||e_{n}(x,y)|$$
and
$$\int \sup_{t>0}|F_{0}(tA)\phi(A)|(x,y)(1+2^{-k}d(x,y))^{\epsilon}dx$$
$$\leq \sum_{n}
\sup_{2^{2m}\geq t>2^{-2m}}|\hat H_{t}(n)|\int |e_{n}(x,y)|
(1+2^{-k}d(x,y))^{\epsilon}dx $$
$$
\leq C\|F_{0}\|_{\Wpl}
\sum_{n} (1+|n|)^{-l}\|e_{n}\|_{\ep}$$
$$\leq C'\|F_{0}\|_{\Wpl}\sum_{n} (1+|n|)^{-l}(1+|n|)^{(1+2/(a-\ep))q/2}
\leq C''\|F_{0}\|_{\Wpl}.$$


Let $M$ be a homogeneous group (cf. [\FS]) of homogeneous dimension $Q$, 
that is a Lie group equipped with 
one-parameter family $\{\delta_{t}\}_{t>0}$ of automorphisms such that for all 
$x\in M$
$$\lim_{t\rightarrow 0}\delta_{t}x=e$$
where $e$ is the neutral element of $M$ and
for every compact set $A\subset M$
$$
\mu(\delta_{t}(A))=t^{Q}\mu(A)$$
where $\mu$ is the Haar measure. Such a $Q$ is not unique --- as additional 
normalization we require all eigenvalues of $D_{e}\delta_{t}$ 
to be at most $t$ for $t<1$ and 
to have $t$ as one of the eigenvalues. We assume that $A$ is a left-invariant 
hypoelliptic differential operator, homogeneous of 
degree $m>0$, that is (for $f$ in the domain of $A$)
$$
A(f\circ \delta_{t})=t^{m}(Af)\circ \delta_{t}.$$
Such an $A$ is usually called Rockland operator. 
Since $A$ is left-invariant we have
$$
F(A)f=f*H_{F}$$
where $H_{F}$ is a distribution on $M$ called the kernel of $F(A)$. 
In our setting we have a kind of Plancherel formula:
$$
\|H_{F}\|_{L^{2}}=c\int |F(A)|^{2}(x)x^{-1+Q/m}dx.$$
To see this note that as $\|H_{F}\|_{L^{2}}=H_{|F|^{2}}(e)$ the formula above is equivalent to
$$
H_{F}(e)=c\int F(A)(x)x^{-1+Q/m}dx.$$
For $F(x)=e^{-x}$ both 
sides are finite (and nonzero) so one 
can choose $c$ to have equality. Then homogeneity shows that the set of 
$F$ for which equality holds is closed under dilations. Thus the equality 
holds for linear combinations of exponentials - that is on a set dense in $L^{1}(x^{-1+Q/m}dx)$, 
which shows the formula. 
 We fix a riemmanian metric $d$ on $G$. Note that it can happen that 
$q$ (which describes growth of balls for riemmanian metric) is smaller then $Q$.

\t {\dh} { If $A$ and $G$ are as above,   
$s>q/2$ and  for  a  non-zero  $\varphi  
\in C ^{\infty  }_{c } ({\Cal R }_{+})$ and  a  
constant  $C$ 
$$
\sup_{t>0}\|\varphi \delta(t)F \| _{H(s) } \leq   C,
$$ 
then  $F(A)$  is  of  weak  type  (1, 1)  and  bounded  on  $L ^{p }$ for  
$1  <  p <  \infty $. }

{\bf Remark}. If $A$ is a homogeneous sublaplacean this theorem reduces 
to the theorem of M. Christ [\Ch] and G. Mauceri and S. Meda [\MM]. 
If $A$ is nondifferential then possible values of $a$ are bounded and
we must assume that $a>q$ and $s>(1+2/a)q/2$. 
J. Dziuba\'nski 
[\Dz] showed how to apply results like \dh for some nondifferential $A$ 
when $a$ is small to get 
$s=q/2 +\ep$ with arbitrarily small $\ep>0$. 

\PP 
 %First note that 
%It is well known (see [\FS]) that semigroup generated 
% by $A$ satisfies our assumptions with any $a$. 
 First choose $a$ so that $a>Q$ (hence $a>q$), $s>(1+2/a)q/2$.  
Then we proceed as in the proof of \db. The difference is that instead of 
 $2^{-k}d$ we use $d_{k}(x,y)=d(\delta(2^{-k} )x,\delta(2^{-k}) y)$ where 
$d$ is some fixed left-invariant riemmanian metric on $G$. %We 
Also thanks to the Plancherel formula we may apply \nt instead of \bj. 
Indeed, we should estimate $\|K_{k}\|_{B((1+d_k)^{\varepsilon})}$ 
(for $\Kb_{k}$ we directly re-use the proof of \db). 
Using dilations, we reduce to problem to the estimate for 
$\|K_{0}\|_{B((1+d)^{\varepsilon})}$. We write $\tilde f(x)=f(e^{-x})$ so 
$f(e_0)=\tilde f(A)$ (and we assume that $f$ have support in 
some fixed interval $I$ contained 
in positive reals --- we want to be away from $0$). 
Then %Since our kernels are left
$$
\int_{d(x,e)<r}|f(e_0)|(x,e)dx \leq 
|\{x:d(x,e)<r\}|^{1/2}\|f(e_0)\|_{L^{2}}$$
$$\leq
Cr^{q/2}\|\tilde f(A)\|_{L^{2}}=
C_1r^{q/2}(\int |\tilde f(x)|^{2}x^{-1+Q/m}dx)^{1/2}$$
$$\leq C_2 r^{q/2} \|\tilde f\|_{L^{2}}
\leq C_3 r^{q/2}\|f\|_{L^{2}}. $$
Since our kernels are left-invariant the estimate above means that 
assumptinons of \nt are satisfied with $\kappa=2$, $\gamma=q$,
$\chi=0$ so we get uniform bound on 
$\|K_{k}\|_{B((1+d_k)^{\varepsilon})}$. 

To finish the proof choose a homogeneous (dilation invariant) and 
left-invariant metric $\rho$ on $G$ (see for example [\HS]). We have 
$$
(1+\rho(x,y))\leq C(1+d(x,y))$$
and because $\rho$ is homogeneous
$$
(1+2^{-k}\rho(x,y))\leq C(1+d_{k}(x,y)).$$
Hence assumptions of \dz are satisfied with $d$ replaced by $\rho$ 
which ends the proof.



Suppose that $M$ is a smooth compact manifold of dimension $q$ 
without boundary. Assume 
that $A$ is an elliptic differential operator on $M$ of order $m$ which is 
positive definite on $L^{2}$ with respect to a smooth positive density $d\mu$. 
Fix a riemmanian metric $d$ on $M$. 

\p {\dl} { If $M$ and $A$ are as above, $q>1$, $m$ is order of $A$, 
$\supp f \subset [1/4,4]$, $t<1$, then there is $C$ such that
$$
\|f(tA)\|^{2}_{L^{1},L^{2}}\leq C t^{-q/m}(t^{q/m}\|f\|^{2}_{H(q/2)}
 +\|f\|^{2}_{L^{2}}).
$$}
\PP 
We need H\"ormander's estimate on spectral projections (see [\Hos]) for 
$a>1/4$ :
$$
\|E([a,a+a^{(m-1)/m}))\|^{2}_{L^{1},L^{2}}\leq Ca^{(q-1)/m}.$$
Put $a=t^{-1}/4$, $h=t^{-(m-1)/m)}$, $n=15([t^{-1/m}]+1)$. Then, 
using orthogonality and H\"ormander's estimate we get
$$
\|f(tA)\|^{2}_{L^{1},L^{2}} \leq \sum_{k=0}^{n}
\|f(tA)E([a+kh,a+(k+1)h))\|^{2}_{L^{1},L^{2}}$$
$$
\leq Ct^{-(q-1)/m}\sum_{k=0}^{n} \sup_{x\in [(a+kh)t,(a+(k+1)ht)} |f(x)|^{2}.
$$
For any interval $I$ we have 
$$
\sup_{x\in I}|f|^{2}\leq 
C(|I|^{-1}\|f\|^{2}_{L^{2}(I)}+|I|\|\partial_{x}f\|^{2}_{L^{2}(I)})$$
so 
$$
\sum_{k=0}^{n} \sup_{x\in [(a+kh)t,(a+(k+1)ht)} |f(x)|^{2}
\leq C((ht)^{-1}\|f\|^{2}_{L^{2}}+(ht)\|\partial_{x}f\|^{2}_{L^{2}})$$
Also 
$$\|\partial_{x}f\|^{2}_{L^{2}}
\leq C\|f\|^{2(q-2)/q}_{L^{2}}\|f\|^{4/q}_{H(q/2)}
\leq C'((ht)^{-2}\|f\|^{2}_{L^{2}}+(ht)^{q-2}\|f\|^{2}_{H(q/2)}).$$ 
Finally
$$
\|f(tA)\|^{2}_{L^{1},L^{2}}\leq 
Ct^{-(q-1)/m}((ht)^{-1}\|f\|^{2}_{L^{2}}+(ht)^{q-1}\|f\|^{2}_{H(q/2)})$$
$$
\leq Ct^{-q/m}(\|f\|^{2}_{L^{2}}+t^{q/m}\|f\|^{2}_{H(q/2)}).$$


\t {\dr} { If $M$ is a compact riemmanian manifold, 
$q$ is the dimension of $M$, $A$ is an elliptic differential operator 
of order $m$, positive definite on $L^{2}$ with respect to 
a smooth measure $\mu$, $F \in C({\Cal R})$, 
$s>q/2$ and  for  a  non-zero  $\varphi  
\in C ^{\infty  }_{c } ({\Cal R }_{+})$ and  a  
constant  $C$ 
$$
\sup_{t>0}\|\varphi \delta(t)F \| _{H(s) } \leq   C,
$$ 
then  $F(A)$  is  of  weak  type  (1, 1)  and  bounded  on  $L ^{p }$ for  
$1  <  p <  \infty $. }

{\bf Remark}. A variant of \dr is valid on noncompact manifolds. One 
needs bounded geometry assumptions to have global bound in H\"ormander's 
estimate and to control local behaviour of the semigroup. Also conclusion 
asserts only boundedness of operator with kernel restricted to a neighbourhood 
of diagonal.

{\bf Remark}. Theorem remains valid form pseudodifferential operators. 
However, the $a$ one gets depend on $m$ so one must pass to the powers 
of $A$ (like in [\Dz]).  
%As the estimates on $\exp(-tA)$ seem to be not published this wou 
  
\PP 
%We can take $a$ as large as we we wish
%Because the spectrum of $A$ is discrete and corresponding projectors 
%have $L^{\infty}$ kernels any 
We give the proof only for $q>1$. (If $q=1$ one must replace $1/m$ in \dl 
by 
a larger number, and then tediously check that the proofs remain valid).
%For $q=1$ one must a bit enlarge indekses 
%of the spaces, then a variant of \dl is satisfied, 
%and one can tediously check that the changed proof remains valid.

From the local Sobolev lemma we deduce that if 
$F\in C_{c}({\Cal R })$, then $F(A)$ has bounded kernel, hence is 
bounded on all $L^{p}$, $1 \leq p \leq \infty$. Therefore we can assume 
that $\supp F \subset [1,\infty)$. 
The semigroup generated by differential operator $A$ satisfies 
our assumptions with any $a>0$ as long as 
$t$ is bounded (see for example [\Hae]). Because of our 
assumption about support of $F$ estimates 
for large $t$ are not needed. 
 We proceed as in the proof of \db but 
using \dl and \nt instead of \bj. 
Indeed we should estimate $\|K_{k}\|_{B((1+d_k)^{\varepsilon})}$ for $k<0$. 
Fix $k$. Put $d_{k}=2^{-k}d$. We write $\tilde f(x)=f(e^{-x})$ so 
$f(e_{k})=f(e^{-2^{mk}A})=\tilde f(2^{mk}A)$. 
Then
$$
\int_{d_{k}(x,e)<r}|f(e_k)|(x,y)dx \leq
|\{x:d(x,e)<r2^{k}\}|^{1/2}\|f(e_k)(\cdot,y)\|_{L^{2}}$$
$$
\leq Cr^{q/2}2^{kq/2}\|\tilde f(2^{mk}A)\|_{L^{1},L^{2}}$$
$$
\leq C_{1}r^{q/2}2^{kq/2}2^{-kq/2}(2^{kq/2}\|\tilde f\|_{H(q/2)} 
+\|\tilde f\|_{L^{2}})$$
$$
\leq C_{2}r^{q/2}(2^{kq/2}\|f\|_{H(q/2)}+\|f\|_{L^{2}})$$
$$
\leq C_{3}(r^{q/2}\|f\|_{L^{2}}+\|f\|_{H(q/2)}).$$
In the last line we use compactness of our manifold -- we may assume 
that $r2^{k}$ is bounded. Thus, we checked that assumptions of \nt are 
satisfied with $\kappa=2$, $\gamma=q$, $\chi=q/2$. 
\vfil
\eject
{\bf References}
\vskip 0.3cm
\unvbox 2

\end