Ne consider the question whether it is possible oto study s ingular

Ne consider the question whether it is possible oto study s ingular

97
Revista de la
Unión Matemática Arpntina
Volumen 54, 1988.
BOUNDEDNESS OF SINGULAR INTEGRALS
. ON NON NECESSARlLY NORMAUZED
SPACES OF HOMOGENEOUS TYPE
.
.-
R. A. MACIAS and J. L. TORREA
Ne cons i de r the que s t io n whe ther it is po s s ib l e oto
s tudy s in gu l ar i n t e gr a l s o n spac e s of hómo g eneous typ e without
norma l i z in g the me tr ic . An a ff irma t ive answer i s g iven , provi ­
ded the mea s ur e s a t i s f i e s a s moo thne s s c o nd i t i o n wh i ch i s
s tr ic U y l e s s r e s tr i c t ive than d e ones cons ider e d by Aimar [A]
and David , Journ6 and S emmes [ DJS] . To a t t a in t h i s we deve l o p
u s e ful r e s ul t s o n the st ruc ture o f spac e s o f homo geneous type .
ABS TRAC T .
I NTRODUC T I O N
The s tudy o f pro b l ems i n the s e t t ing o f spa c e s o f homo g eneous
type has b e e n prov en t o b e ver y frui t fu l in order to o b t a i n
general r e su l t s wh i c h hav e a w i d e r a n g e o f app l i c a t ions . Fr e ­
quent l y i t s e ems unavo i dab l e to know a quant i t a t ive r e l a t io n
b e tween the me asure o f t h e b a l l and i t s rad ius e s p e c i a l l y i n
order to g e t boundedne s s r esul ts i nvo lv ing int e gra l s . Th i s i s
usual ly a t t a lined a s s uming the spac e to b e normal a s for o ins ­
tanc e , in [A] , [ ew] , [DJS ] and [W] . Es s-e n t i all y a space is nor ­
mal i f the mea sure o f a b a l l i s . c omparab l e to i t s radius . I f
the open b a l l s a r e o p e n s e t s then the s pa c e can b e norma l i z ed
as it is proven in [MS 2 ] . However the r e sul t in g me t r i c i s
1980
Mathemat�c� Su b ject Cla� � � 6 �cat�on .
S econdary 4 2B 2 5
. •
P r im a r y 4 2 B 2 0 .
98
no t e quival ent t o the o r i g inal o n e . Al s o in ma ny ins tanc e s s o ­
me k i nd o f s mo o thne s s ' co nd i t i o n o n the measure o f the b a l l s
needs to b e c o n s id ered a s i n [A] and [DJS ] , s e e [C a ] fo r an
e xamp l e i n a d i ffer ent s i tu a t i o n .
'.
,
We po s e our s e l v e s the que s t i o n up to whi c h point the s e r e s tr i c t io ns ar e nec e s s ary . Tha t i � , fo r ins tanc e , wh e ther i t i s
po ssible t o g ive c o nd i t io ns fo r the boundedn e s s o f s ingu l a r i n ­
t e gr al s o n 12 , fo l l owing Co t l ar l s l emma appr o a c h , pr e s erving
the o r i g i na l metr i c . Wh i c h a l l ows t o keep truc k o f the s hape
of the trunc a t ions . I n a no rma l s pac e c o nd i t i o n s wer e g iven in
[A] , [DJS ] and [W] . We show in The o r em A th a t an a f f i rma t ive
answer can b e g iven c o ns ider ing trunc a t io n s adap t e d to the
measur e of the ba l l s r a ther than to the ir radius , provided we
impo s e a smo o thne s s c ond i t io n on the measure ( s e e ( 1 . 7 ) ) , wh i ch
i s s tr i c t l y l e s s r e s tr i c t ive than the o n e s ado p t e d in [A] and
[ DJS ] . Thr ough s ev er a l examp l e s we d i s cu s s i ts meaning in § 4 .
We s hould pb int out tha t i n [W] no smo o thne s s c o nd i t i o n i s as sumed , however th i s wo r k cOntains s ome s er i ous er r o r s , s e e our
r emark ( 2 . 2 3 ) . In o r d er to a c h i ev e our purpo s e we are l ed to
ma ke . in § 2 a c a r e ful s tudy of int e gra t io n · pr ob l ems , wh ich
we be l i eve t o b e of independent inter e s t and u s e fu l n e s s when
wo rk ing o n s pac e s of homo geneous type .
I
§1 .
MA I N R E S U L TS
Before s ta t ing the r e sul t s we r e c a l l s ome b a s ic fa c t s and d e ­
fin i t i o ns .
A qua s i - d i s tanc e o n a s e t X i s a no n ne ga t ive s ymme tr i c func ­
t i o n , d ( x , y) ' d e f i ned o n X x X such tha t d ( x , y)
O i f and o n l y
i f x·y , and ther e ex i s t s a c o n s tant K s a t i s fying
•
(1 .1)
d ( x , y)
for every x , y and z i n
.;;;
K [d ( x , z ) +d ( z , y) ]
,
X.
As i t is proved i n [MS 2 ] g iven a quas i - d i s t anc e d the r e ex i s t
a qua s i - d i s tanc e d I , un i fo rmly e qu ival ent t o d , a f i n i t e c o ns ­
tant C and a numb er a , O
1 , such tha t
< �, ';;;
·99
l a
a
I d ' ( x , y) - d ' ( x , z ) 1 .;;;; e r - d ' (y , z ) ,
provided d ' ( x , y) .;;;; r and d ' ( x , z ) .;;;; r . The r e fo r e we a s s ume ,
wi thout 1 0 5 5 o f gener a l i ty , the ex i s t e nc e o f o < a .;;;; 1 such
tha t
(1
. 2)
l a
S
I d ( x , y) - d ( x , z ) 1 .;;;; K d ( x , y) - d (y , z ) ,
ho l d s whenever d ( x , y) > d ( x , z ) .
The s e t s { ( x , y) E X x X : d ( x , y) < 1 / n } , n > O , de fi ne a b a s i s
o f a me tr izab l e uni form s truc tur e on X . The b al l s
B ( x , r ) = { y E X : d ( x , y) .;;;; r }
form a b a s is o f ne i ghb orho o d s for the t o po l o gy induc ed b y the
un i form s truc tur e . We c a l l ·t he attent i o n on the fac t tha t
B ( x , r ) i s no t b e ing us ed to d e no t e the s e t
B O (x , r ) . = { y E X : d ( x , y)
<
r} ,
as ' i t i s usual . I t c a n b e s e en tha t B ( x , r ) i s no t in general
the c l o sure of B O ( x , � ) . As w i l l b e c ome appar ent l y the s e t s
B ( x , r ) ar e mor e adj us t e d to our purpo s e s .
A s pa c e o f homo geneous type i s a s e t X endowed w i t h a quas i ­
d i s t anc e d ( x ,y) and a Bo r e l measure 1l s a t i s fy ing a "doub l ing
c o nd i t io n" L e . ther e ex i s t s a c o ns tant A s uc h tha t
1l ( B ( x', 2 Kr ) ) .;;;; All ( B ( x , r ) ) ,
( 1 . 3)
fo r every r > O . We s ay tha t the s pac e o f homo geneous type i s
no rmal i f tber e ex i s t f i n i t e and p o s i t ive c o n s t an ts A 1 , A 2 , K l
and K 2 such that
A r .;;;; ll ( B ( x , r ) ) .;;;; A 2 r
1
X
B (x , r )
B (x, r)
{x}
wh en K 2 1l ( { x } ) .;;;; r .;;;; K l ll ( X)
if r > K ¡ ll (X) and
if
r
<
K 2 1l ( { x } ) .
To a s sume tha t X i s norma l provid e s a knowledge o f the me a s u ­
r e o f a ball given i t s r ad ius . Th i s fac t wa s bas ic , whe n d o ing
100
2
inte gr a t io n in pap er s d eal ing w i th L bound edne s s o f s ingu l ar
integral s l ike [A) , [DJS) and [W) . We want t o avo id any hypo t�
e s i s of th i s t yp e . I Hence we us e a " s rno o thne s s " cond i t io n on
the �easllr e a l t er na t ive t o the ones adop t ed in [A) and [DJS ) .
;;.c-
e o f horno geneous type 'X i s s a id to s a t i s fy pro per ty
The s
H i f ther e ex ists a , O < a � 1 , �.uch tha t
I
, ho l d s fo r every x e x
app ear ing in ( 1 . 2 ) .
and
O <
s � r , whe r e S i s the c o n s t ant
S inc e , fo r ins tance , as surning H d o e s no t e l irnina t e spac e s c og
t a i n ing po int s w i th po s i t ive rne asur e ( 1 . 4 ) i s s tr ic t l y l e s s
�e s tr ic t ive than tho s e cond i t iCins cons id er ed in [A] and [DJS) .
We pr e s ent in §4 sorne exarnp l e s wh ich w i l l h e l p t o c l ar i fy the
rne aning of hypo the s i s H .
The s tandard hypo the s i s o n the s ingul ar integral kernel K ( x , y)
rnus t a l s o be rnod i f íed in order to avo id hypo the s i s o f the no r ­
rna l i z a t io n typ e . We s ha l l s tudy s ingular int e gr a l kerne l s
K ( x , y) s a t i s fy ing the fo l l owing
(K . 1 )
I K ( x , y) I � C p ( B ( x , d ( x , y) ) -
(K . 2)
There exists
a,
O < a � 1.
.
l
, fo r ev er y x�y .
such that for every integer k ;¡;¡' l
kp(B(y,d(x,y)) - 1
I K(x, y) - K(x, z) I + I K(y,x) -K( z ,x) I � CA-a
prov ided
Ak p ( B (y,d (y, z) )
Let
(K . 3)
,
� p ( B ( y , d ( y , x) ) .
O < r < R < óo
K ( x , y) d p
J
J
( y)
O , f or e ve r y x
e r
r<p (B ( x , d ( x , y» ) �R
K ( x , y) d p ( x)
O , for every
r<p (B ( x , d (x , y» ) SR
G iven
O
< r
< R <
K r , Rf ( X)
=
�
, d e f ine
J
r<p (B ( x , d (x , y» ) <R
K ( x ,y) f ( y) dp ( y ) .
y
e X.
101
Th e L
2
r e sul t can be s ta ted as fo l l ows .
THEOREM A . L e t X b e a s p � a e o f hom o g e n e o u s typ e w i t h p r o p e r ty
H. A s s um e t h e k e r n e Z
sat i s f i e s
.1)
and
then t here exi s t s C s u a k tha t
(K , (K . 2)
K(x, y)
O
f o r e 7/e ry O
< r < R,
K r , Rf D 2
(K . 3)
<,;; Cll. f 11 2
a n d f e L 2 ( X) .
( 2 . 2 ) b e l ow i t c an b e s e en tha t c o n (K . 2) is weaker than
( z ,y)
(K . 2) ' I K(x,y) -K(x, z) 1 + I K(y,x) -K ( z ,x) I <,;; d (x,y) .p C(Bd (y,d
(x � y) ) )
provided 2d (z ,y) d (y,x) .
I n order to ' s impl ify the' notation and since no con fus i o n a r i ­
s e s in the following we shall write I E I instead of p (E) and
dx instead of dp (x) .
Hav ing into a c c o unt l emma
d i t ion
<
§2.
T E CH N I CAL RESULTS
LEMMA ( 2 . 1) . ' A s s u m e O < dei,y) < [K(2K) 1 - 13 ] - 13r • Th e n
1 13
1
o � B ( z , r - 5r - 13d(y, z) 13) e B (y,r) e B ( z , r + r - d ( z ,y ) 13) ,
f o r e v e ry 5 s u a h tha t K(2i) 1 - 13 <,;; 5 < r13d(y, z) - a .
l :"'
Pro o f . By the assumptions r -5 r 13 d(y , z) 13 > O . Take x in
B ( z , r - 5 r 1 - 13 d ( y , z ) l3 ) , the n d ( x, y ) < 2Kr . App l ying ( 1 . 2)
d (x,y) <';; d (x, z ) +K( 2Kr) 1 - l3d (y, z) 13 <';; r .
The'n x E B (y,r) . By a,nother application o f ( 1 ; 2) we obtain
t h e rema ining incluss ion .
LEMMA (2 . 2) . " If A j I B (y , s) J < I B (y, r) 1 , t h e n ( 2K) j s < r .
Pr o o f . I t fo l l ows immediately from the doub l ing prop e r ty ( 1 . 3) .
5
./
'
1 02
LEMMA \ ( 2 . 3 ) . A s s um e X s a t i s fi e s prop e r ty H, ( 1 . 4 ) , t h e n e i t h e r
l {x} 1 > fo r e v e r y x E X o r l {x} 1 = O fo r e v e ry E X .
Pr o o f . t f there exists z E X o f measure zero , then given any
E X and 15 > O
l {x } 1 <; I B ( z , d( z ,.x) +c S d ( z , x) I - S ) 1 - I B ( z , d( z ,x) ) - cSd(z ,x ) I - S 1 <;
<; A I B ( z ,d ( z ,x) ) 1 1 - a. B ( z , c ) l a. .
Letting 15 tend to zero , we obtain l {x} 1 =
DEFINITION ( 2 . 4 ) . Let r > O;we denote E(x,r) = {y: IB(x,d(x,y)) I <; r} .
LEMMA ( � . 5 ) : L e t R sup { d (x,y) : y E E (x, r) } , t h e n
E (x,r) e B (x, R) .
· Bo ( x , R)
( 2 . 6)
( 2 . 7)
IE(x,r) I <; r .
(2 . 8)
I B (x,R) I <; Ar .
( 2 . 9 ) If
I B (x,R) I <; r t h e n E(x,r) B (x,R) .
( 2 . 1 0 ) If I B(x, R) I > r t h e n E(x ,r) BO (x,R) .
then is bounded , on
Pro o f . As it is well known if I x l
the other hand if I x l = then I B (x, s) I goes infinity if s
tends to infinity, therefore R
If y belongs to E (x, r) by
definition of R , d (x,y) <; in particular
E (x,r) e B ( x , R) .
Let assume I B(x,R) I <; r . I f E B (x� R) then
I B (x, 4 (x,y) ) I I B (x, R) I <; r .
This says that y E E (x,r) , in particular E (x,r) = B (x,R) . Thus
( 2 . 9 ) is pro ved and ( 2 . 6 ) , ( 2 . 7 ) and ( 2 . 8 ) in this case .
Assume now I B (x, R) I r . Therefore d(x,y) < R for ever y in
E (x, r) and there exists a se quence {Y j } j such that Y j E E (x,r)
and r j d(x'Yj ) increases to R. Then
E (x, r) = j B (x,r . ) = BO (x, R)
O
X
X
o .
=
e
< 00
00
R,
us
>
U
=
1
< oo .
to
y
<;
QO
X
y
J
1 03
and
l B (x, r . ) I .;;;; r .
l E (x, r ) I = lim
j ...¡..oo
J
This proves ( 2 . 1 0 ) and completes the proof of ( 2 . 6 ) , ( 2 . 7 )
and ( 2 . 8 ) .
LEMMA ( 2 . 1 1 ) . I B (y , d(z, y) ) I < A I B ( z ,d ( z ,y) ) I < A 2 I B ( y ,d ( z ,y) ) l .
Pr o o f . I t is irnmediate from the doubling property ( 1 . 3 ) and
(1 .1) .
With the notation of Lernma ( 2 . 5 ) , we have
LEMMA ( 2 . 1 2 ) . If X s a t i s fi e s H t h e n
E (x, r) B (x,R) ,
fo r e v e ry x i n X and r > O .
. Pro o f . Let us assume I {x} I
O . Then given O < n < R , .
1
O .;;;; IB(x,R) I - IBo(x,R) I .;;;; IB(x,R+R H3n S) I - IB(x, R-R - Sn S) I .;;;;
.;;;; AIB(x,R) 1 1 -a IB(x,n) la.
Therefore I B (x, R) 1 .;;;; I B o (x, R) 1 .;;;; r . Thus E (x,r) B (x,R) . Let
us suppose l {x} 1 > 0 . Then by lernma ( 2 . 3 ) , l {y} 1 > 0 for eve­
ry y E X . Thus {y} B (y, E) for sorne E = E (Y) > O (see [MS 2 ] ) .
By ( 2 . 9 ) we can assume I B (x ,R) I > r . I n this case there exists
a se quence {Y j } j such that Y j E E (x,r) and r d(x,y ) in­
creases to R. Let 0 < E j < r j satisfying B (Y jj , E j ) {Yj j } .
Thus
O < I {x} · 1
I B (y . , r . +r l. - a E a.) I .;;;; A I B (y . ,r .) 1 1 "'CX I B (y . , E .) l a .
By ( 2 . 1 1 ) this is les s than
=
=
=
=
=
:<
J
J
J
J
J
J
=
J
J
This implies the existence of an infinite number of �oints
contained in B (x, R) with measure greater than a constant which
104
is impos.s ible .
COROLLARY (2 . 1 3) . If X s a ti s fi e s H a n d ¡ {xl i
E X,
fo r s o me
then
I B ( y, r) I I Bo (y ,r) I
fo r any y E X and r > O .
Let us denote C(x , k) E (x,Ak+ 1 ) \ E (x,Ak ) .
LEMMA ( 2 . 1 4) . A s s um e I x l � Ak+ 1 � ¡ {x} l . The n C(x , k) f 0 fo r
e v e ry x E X .
" < I B (x,r) I AJ"+ 1 } . Assume N f 0 and
{r
Pro o f . Let N
AJ
:
j
j
lét r E N j . Since
B B (x,r) y e: B B (x,d(x ,y) )
there exists y such that
Aj < I B (x,d(x,y) ) I � A j + 1 ,
then C(x, j ) f 0 . Therefore it is enough to prove Nk f 0 . In
order to do this we shall show that Nk 0 impl i es Nj = 0 for
every j > k . Let k l min {j > k : Nj f 0} and r 1 be the infi­
mum of N k I . Thus , r l > O and
Ak 1 � I B (x,r 1 ) ¡ � Ak 1 + 1
If I B (x, r I ) I Ak 1 then r I E Nk 1- 1 which is a contradic�ion .
Therefore r 1 must be long Nk 1 . On the other hand ,
( 2K) - l r 1 rt=. N k<1 ilIip lying
I B (x,r 1 ) I � A I B (x, ( 2K) - l r 1 ) I � AAk � Ak 1 ,
which is again contradiction.
o
I
=
=
=
�
=
=
U
=
=
to
a
In
the following we assume X satisfies property H . Let
RXk sup { d (x,y) : y E E (x, Ak ) } .
x
1 05
E (x,Ak ) = B (x, R� ) . In particular ,
( 2 . 1 5)
C (x, k) = B (x, R� + I ) -B (X, �) .
Moreover by ( 2 . 7 ) and Lemma ( 2 . 1 4 ) , if I x l Ak + l ;> l {x} l ,
By lemma
(2 . 1 2) ,
;>
( 2 . 1 6)
LEMMA
h [ 2K ( 2K) l - a] l / a .
Ah I B (x, d"(x� z) ) 1 < A i - 1 ,
( 2 . 1 7 ) . L e" t U B B u p p o B e
imp Z i e B
(2K)
;>
Then
C(x,i)t.C(z,i) ¿ [B(x,R�+l +S�+ l) -B(z,R�+ l -S�+l )]
[B ( z,R.+S. ) -B ( z,R.-S. ) ] ,
U
x
1.
x
x
1.
1.
U
x
1.
where
By
Pr oo f .
(2 . 1 6)
and the assumption
Thus using Lemma ( 2 . 2 )
[ 2K (2K) l -al l / ad(x, z) <; ( 2K) hd(x,z) < R� <; R�+l .
(2 . 18)
Then we can apply Lemma ( 2 . 1 ) with = ! K ( 2 K) I - a te obtain
B(z,R�-S:C) e B(x,R�) e B(z,R�+S:C) , j = i ,i+1 ,
where
Having into account ( 2 . 1 5 ) if Y E C(x, i) 4C ( z , i ) we have four
possibil i ties
P I y E C (x, i) -B ( z , R � +I ) ' Then y E B(z,F,�+ 1 �S�+l) -B(z,R�+l)
P z : y E C (x, i ) () B ( z , R�) . Then y E B(zI,R�) -B(z,R�-S�)
P 3 : y E C{z , i) -B ( z , R � +I ) : Then y E ( ,R�+I ) -B (z, R� 1 -�+1) '
y E C ( z , i ) () B (x,R �) Then y E B(z,R:C+S :C) -B(z,R�)
eS
J
J
J
J
J
:
•
1.
P4 :
B
•
z
1.
1.
1.
1.
1.
•
1.
•
+
- 1 06 -
is
The proof
c omp l e t e .
COROLLARY ( 2 . 1 9 ) . On t h e a o ndi t i o n s o f the L e mma a b o v e
Mo re o v e r if Y
E
>.
S�) . I A i - 3
C(x, i) llC ( z , i)
I B. ( -1. , R Xi -
�
•
t.b é n
By (2 . 1 8) d ( x ; z ) < R�x a nd 4 ( Rx. - S x. ) > R x. . Thus using
( 2 . 1 6 ) and the doubling condition of the measure .
1
A i - < I B ( x , R X. ) I < A I B ( z , R X. ) I < A 3 1 B ( z , R x. -· S . ) I .
On the other hand if y C(z , i) then I B ( z ,d(z ,y) ) I > A i . I f
C(x, i) - � ( z , i) then either P 1 o� P 2 h o l d . I n the first
case y � . E ( z ,A i+ 1 ) therefore I B ( z ,d(z ,y) ) 1 > A i + 1 . I n the sec.,
ond case d(z ,y) > R� - S � therefore
I B ( z, d ( z ;y) ) 1 > I B ( z , R� - S �) I > A i - 4 .
Pro of.
� --
�
�
�
�
�
�
E
Y e
�
�
�
�
�
(2 . 1 7)
I C(x, i) llC ( z , i) I CA i ( l - a. ) I B ( z , d (x, z) ) l a. .
Property H imp l i e s
COROLLARY ( 2 . 2 0 ) . In t h e a o n d i tions o f L e mma
O;;;;;
Pro o f .
From (2 . 1 8) it fo l l o s
w
tha t
we c a n apply Lemma
( 1 e ) d (x , z) e)
B ( z , R �� ) e B ( x , R �� + � R �� -
(2.1) .
e
e B ( x , 2 R �� ) .
Therefore
t ak ng
i
ne
=
t K ( 2K) 1 - e d ( x , z ) e and
app l ing do ub li n g
1 07
property
1
D.
<
<
<
C I B ( x , R�)
1 1 -a I B ( z , d ( x , z ) ) l a
1
CA i ( l - a ) I B ( z , d ( x , z ) ) l a .
Clearly the same boun� is true for D i + 1 .
Let X i ( x , y) = X C ( x , i ) ( y ) . Observe thÚ
(2 . 2 1 )
LEMMA
X' 1. ( x , y ) < X 1. 1 ( y , x) + X 1. ( y , x) +, X 1. + 1 ( y , x) .
( 2 . 2 2 ) . If A i < I x l , i t f � Z Z OW8
I (i , q , x) J I B ( X , d ( X , y) ) I q x i ( x , y) d y
I' ( i, q,y) = f I B ( x , d ( X , y) ) I qX i ( x , y) dx,
If e i th e � A l
P�o o f ·
�
<
CA i ( l + q )
<
c A l q ! A i (l + q ) .
I x l o � l { x } 1 � A i b o th i n t e g�a Z 8 a�e 2 e �o .
J I B ( X , d ( X , y) ) I � X i ( x , y ) dy
<
CA i q I E ( x , A i + 1 )
I
<
CA i ( l + q )
Let us consider I ' (i , q ,y) , by Lemma ( 2 . 1 1 ) and ( 2 . 2 1 )
I ' (i , q ,y) A l q l [ I B ( y , d ( x , y) i l q X i ( x � y ) dx <
A l q I [ I (i- l , q , y ) + I ( i , q , y H · I ( i l ,q ,y) ]
<
<
+
<
(l + q ) .
l l
< cA q A i
in­
Lemma ( 2 . 2 2 ) seems to be a substitute for
e�ity in Theorem ( 2 . 1 ) of [W] . That ine quality is not
true in general as can be seen taking X = ' { O , 1 , 2 , 3 , ,n, . . . } ;
� (n) = l ; d (n,m) = I n-m i and considering f to be
X [ O ; l] < f < X [ O , 2 ] or X [ � � ) < f < X [ O , � )
REMARK ( 2 . 2 3 ) .
(1)
•
l
'
•
•
1 08
§ 3.
P RO O FS O F T H E O R E MS
We shall use the following
· CO TLAR ' S LEMMA . L e t H, b e a Hi l b e r t s p a c e and
T 1 , T2 ,
, TN a
Le t c :Z
[0 ,00)
fi n i t e o f l i n e a r and c o n t i n u o u s . op e ra to r s o n H .
such that
J
If II T � T . II
�
See [C] .
y c (l) 1 / 2 =
l= _ oo
.;;; c ( i - j )
A <
00
•
•
+
and l e t T � b e t h e a dj o i n t o f T . .
and II T . T � II .;;;
J
�
•
c (i- j )
�
then
N
11 I T . II
i= 1
�
�
.;;; A .
We denote K . (x,y) = K(x,y) X . (x,y) and define. T j f(x) = JK j ( X,y) f(Y) dY, for any function f in L 2 (X) .
Using Lemma ( 2 . 2 2 ) and proceeding as in [A] it follows that
T . is uniformly bounded on L 2 and the kernel of the adj oint
operator T� is K� (x,y) = K . (y ,x) . Moreover
T I T f (x)
J { J K i (y ,X) Kj (y, Z) dy}f( Z} d Z .
j
and
I T�T� . f (x) 1 2 .;;; J I JK �. (y,X) K . (y, Z) dy l l f( Z ) 1 2 dz x
x J I J K (y,X) K j (y, �) dy l dz .
i
Applying again Lemma ( 2 . 2 2 ) and the boundedness of the kernel
the second factor is bounded by a constant independent of. i
and j . AssuJl!.e that , whenever j > i
(3 . 1 )
J l f K ( y ,x) K j ( ) d 1 dz - CAa ( i - j ) .
Then, in this case
n T F / n ; .;;; C Aa ( í- j ) J 1 f ( z) 1 2 [ f 1 f K ( y , x') K ( y , z) d Y 1 dx] d z
j
i
.;;; CAa ( i - j ) lIf 11 ;
J
J
J
J
Pp o o f o f Th e o r e m A .
J
J
�
J
i
J
.
y, Z
y
.....
•
If i > j , using again
(3 . 1 )
.
109
:E;; C f l f ( z) 1 2 [f I JK i (y,X) K j (y , Z) dy l d�1 dZ :E;;
:E;; CAex ( j - i ) 1I f" � .
It is clear that the same estimate holds for " T . T� " 22 . Therefo­
re aplying Cotlar s Lemma the Theorem fol1ows
. Let us prov� ( 3 . 1 ) . �y (K . 3)
1 f I JK i (y ,X) K j ( y, Z) dy l dZ
J l f [K . (y, Z) -K . (X, Z)I K . (y,X)dy l dZ :E;;
:E;; f I K . CY,X) I ff I K . Cy , Z) :-K . (X, Z ) I dz } dy .
Take C 1 = Ah , where ( 2 K) h [ 2 K( 2 K) 1 - 131 1 /13 . Let
1 1 = C 1 I B ( Y , Ar (y , x) ) 1 :?Aj - 1 IK. (y,x) l { f IK. (y,Z) -K. (X,Z) Idz }dy .
iI T F/ " �
l.
I
•
J
.
J
J
l.
J
l.
J
;>
J
l.
By
J
an� Lemma ( 2 . 22 )
f I K j (Y, Z) I dz :E;; C f IB ( y, J( y, Z ) I dz C .
(y )
(K . l )
C
Also
.
:E;;
,j
f I K . (X, z ) I dz :E;; C .
J
.
Therefore using once more ( K . 1 ) and Lemma ( 2 . 2 2 )
X . (y,x) dy :E;;
1 :E;; C
J
B
1
.
(y,d(y,x)
1
1I
C 1 1 B ( Y , d ( Y , x) ) 1 A
Xi (y,X)
:E;; f [ IB (y,dA j(y ,x) ) 1 1 I B (y,d(y,
x)) 1 dy :E;;
�
G
It remains to consider
J-
- a.
l.
.
1 10
1
2
f
=
1
C 1 ! B (y , d (y , x» ! <AJ' -
f
! K . ( y , x) ! { ! K . ( y , Z ) - K . ( X , Z ) ! d z } dy .
J
J
J.
of
I n order to p rove the bound edne s s
121
=
J
1
C 1 ! B (y , d (y , x» ! <AJ. >
•
12
>
let
f
! K . (y, x) l { I K (Y , Z ) -K (x, z) I X . (y, z) d Ú dy .
J
J.
.
>
o>
Ob s erve: tha t i f x E C ( y , i ) , z E C ( y , j ) and Ah I B ( y , d ( x , y) ) I < Aj - 1 ,
l
A ! B ( y , d (x,y) ) I .,¡;; 1 < A - j I B ( y , d ( y , z ) ) ! ,
whe r e
l = max ( h - j + 1 , - i - 1 ) . Then
A j +l ! B (y , d ( x , y ) ) ! < I B ( y , d ( y , z ) ) l .
The r e fore us ing ( K . 2 ) and ( K . 1 )
12
1 .,¡;; C
J
f
X . (y ,x)
X . (y , z)
'
. J.
{ A - ex ( J -i )
J
d
)
(
x)
B
,
,
(
B
))�1 dz } dy .
.., z�
., y.. d(,...
(y
. ,....
-;-;
I y y
I
I ........
Appl ying Lemma ( 2 . 2 2 )
1
21
CA - ex (j - i )
.,¡;;
We need to s how
122
.,¡;;
C
=
a
•
s imi l ar bound fo r
J
C 1 ! B ( y , d ( y , x»
122
J
' _ ! K i (y, X) ! { I Xj (y, z ) - x (X , Z ) ! I K ( x, z) I dz }dy
j
.
! <AJ 1
X . (y, x)
1
J.
. 1 ¡B(y ,d (x ,y) ) I {
I B(x, d (x , z) ) I dz } dy .
C 1 ! B (y , d ( y , x» I <AJ C (y , j ) �C ( x, j j
J
J
>
B y Coro l l ary ( 2 . 1 9 ) we ge t
f x»
c 1 I B ( y , d (y ,
x J.. (y,x)
' _ l I B (y,d(x,y) )
I <AJ
By Co ro l l ary ( 2 . 2 0 ) and Lemma ( 2 . 2 2 )
I
! C (y , j ) �C(x, j ) l dy .
111
Therefore
§4 .
EXAM P L E S
We present some examples in order to understand better proper­
ty H and the relation between the metric and the measure . In
previous works , besides the re quirement for X to be normalized,
the following hypothes is has being imposed
(4 . 1 )
, I B (x, r) I - I B (x, s) I <: C (r-s) Yr l - Y ,
for every O <: s < r and some y > O . The case y = l has being
considered in [A] and y pos itive in [ DJS ] . Observe that the
condition is really meaningful when s is close to r , since
otherwise the normal ization of the space suffices . Thus , it
is clear that (4 . 1 ) is more restrictive as y increases , ' t her�
fore it is enough to consider y <: 1 . Clearly (4 . 1 ) implies H .
The contrary is not true even in the case o f normal ized X; see '
example (4 . 5) . Thus , all the examples cons idered in [A] are
included , as for instance Rn with parabolic metrics and Lebes
gue measure , compact groups with a quasi -metric induced by a
Vitali fami+y, etc .
EXAMPLES OP SPACES THAT DO NOT SATISPY H .
(4 . 2) Let X = [ R - ( 2 , 4) ] { 3 } , d (x,y) the usual distance
and measure defined by ( { 3 } ) = 2 and ( E -. { 3 } ) = , l E I ,
where I E I stands for Lebesgue measure . It is easy to see that
(X, d, ll) is a space of hbmogeneous _ type that does not satis ­
fy the conclusion of Lemma ( 2 . 3 ) .
( 4 . 3 ) Por n > O , let I be the open interval in R centered at
n
J.I
Jl
U
II
1 12
"
2n +
t of
radius ( 2n) - 1 . Le t
.
X =
00
U
n= 1
{ [ 2 n - l , 2 n]
U
I }
n
Take d ( x , y ) the eucl idean d i s tanc e and d e f ine the me a sure l.l
l.l
(E)
=
J
I
1
JXE(x) (X [ 2n- 1 , 2n] (x) +nx1n (x) )
dx .
I t i s c l e ar tha t ( X , d , l.l ) l i s a n spac e o f homo geneous type . Mo ­
reover
ho l d s for every n > O . On the o ther hand
1
1.l ( B ( 2n , Z1 ) ) 1 - 0. 1.l ('B ( 2 n ' 2n
) ) a.
a s n tends , to i nf i n i ty for any O <
O
+
a.
� 1.
E XAMPLES OF S PACE S SATI S FY I NG H .
(4 . 4)
The s p ac e ' cons idered in ( 2 . 2 3 ) s a t i s fi e s H .
L e t ( X , d Y , l.l ) b e a spac e oí homo geneous ' typ e , whe r e d
i s d i s t anc e , y > O and such that
( 4 . 5)
l.l (B ( x , r ) )
I t i s we l l known , s e e for ins tanc e [MS ll , tha t · i f
�
�
= min ( l , )
then d Y s a t i s f i e s ( 1 . 2 ) . We s hal l prove tha t prop e r ty H ii
true fo r a. � min ( l , S / A ) . We ob s erve that it i s s u ff ic i ent to
t ak� r > 2n > O o n prop e r ty H . Let E = r 1 - S n S
D
l.l ( B ( x , r + E ) )
A 1
A ¡:; -
wher e
' .
l.l ( B ( x , r - E ) ) = ( r +. E ) A - ( r - E ) A
2E ,
r - E � ¡:; < r + E . From
r > 2n > O
i t fo l l ow s
1 13
y.
( 4 . 6 ) L e t ( X . , d . � , � . ) , i = 1 , 2 , as in
� "�
�
t aking X = X 1 x X 2 , � = � 8 �
2 and
1
( 4 . 5) .
I t i s c l ear tha t
I
( X , d , � ) b e come s a spa c e o f homo geneous typ e . Mor e ov e r i t can
b e shown tha t
� (B (i, r) )
for d ( i, y) < r ; d ( i , z ) < r and � = min ( � 1 ' � 2 ) ' Pro c e e d i ng a s
in
(4 . 5)
i t fo l l ows tha t ( X , d , � ) s a t i s fi e s H , w i th
S ) = min ( 1 ,
= min e1 , X
CI.
min ( S l ' S 2 )
) .
A 1+A2
( 4 . 7 ) As an imm ed i a t e" appl i c a t i on o f ( 4 . 6 ) we cons ider
R
n
=
R
n1
x
.
•
•
� �. = Leb e s gue me a s ur e i n
The r e fo r e taking A i
ge t
n
xR k ,
R
n .
�
...
......
1
,
n �·
, thus � �. (B ( x �. , r ) )
n i / Y i and S i = min ( 1 , Y i 1 ) , b y
( 4 . 5) ,
we
min ( 1 , y i )
n �.
Then we a r e in the c o nd i t ions o f ( 4 . 6 ) and ( X �d , p ) s a t i s f i e s
H" w i th
min e 1 ; _1_ , . . . , _1_)
Yk
Y1 "
""
114
a.
=
min e 1
,�)
REFE RENC E S
A I MAR , H . , S�nguta� �nteg�at� and app�ox�mate �dent� ­
t�e� on � paee� 06 homogeneou� tifpe, T r a n s . Am e r . Ma th .
Soc . 29 2 ( 1985) , 1 35- 15 3 .
[C a ]
C AL D E R O N , A . P . , Inequat�t�e� 60� the max�mat 6unet�on
�etat�ve to a met�� e, S t u d i a M a t h . 5 7 ( 1 9 7 6 ) , 3 9 7 - 3 0 6 .
[C ]
C O TLAR , M . , A eomb�nato��at �nequat�t if and � t� ap pt�ea ­
t�on� to L 2 � paee� , R e v . Ma t h . C u y a n a 1 ( 1 9 5 5 ) , 4 1 - 5 5 .
[CW]
[D J S ]
[MS 1 ]
C O I FMAN , R , a n d W E I S S , G . , Ext en� �on� 06 Ha� d if � paee� and
B u l l . Ame r . Ma t h � S o c . 8 3 ( 1 9 7 7 ) ,
569-645 .
the�� u� e �n anat if� �� ,
DAV I D , G . , J O URNE , J . L . a n d S EMM E S , S . , Op é�ateu�� de Cat ­
de�6n- Z ifgmund, 60net�on� pa�a-ae e� ét�ve� et �nte� pota ­
t�on, R e v . Ma t . Ib e r o a me r i c a n a , 1 ( 1 9 8 5 ) , 1 - 5 6 .
MAC I A S , R . A . a n d S E G O V I A , C . , S�ng uta� �nteg�at� on gene ­
�at�zed L�p� e h �tz and Ha� d if � paee� , S t u d i a M a t h . 6 5
( 19 79 ) , 55-75 .
[MS 2 ]
[W]
L�p� eh�tz 6unet�on� on � pa ­
ee� 06 homogeneou� t if pe, A d v . i n Ma t h . 3 3 ( 19 79 ) , 2 5 7 - 2 7 0 .
- - - - - - - - - - - - - - - - - - - - - - - - - - ,
W I T TMA N , R . , Ap pt� eat�on 06 a theo�em 06 M . G . K�e�n to
� �nguta� �nteg�at� , T r a n s . Am e r . S o c . 2 9 9 ( 19 8 7 ) , 58 1 -5 9 9 .
R . A . Macías
Pro grama E special de Matemática Apl icada , CONICET
and Univer s idad Nacional del Litoral , Argentina
Current Addre s s : PEMA , Güemes 3450 C . C . 9 1
3000 Santa Fe , Argent ina
J . L . Torrea
Univers idad Autónoma de Madrid
Cantob lanco , 28035 Madrid , España
Rec ib ido en j unio de 1 9 8 9 .