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Le Matematiche (Catania) 49 (1994), no. 1, 149-168 (1995).
Commutators of integral operators
with positive kernels
Marco BRAMANTI
Dipartimento di Matematica. Politecnico di Milano
Via Bonardi 9. 20133 Milano
Abstract. Let O be an integral operator on a homogeneous space (\ß.ß .), defined by a positive, locally
integrable kernel 5, and assume that O is continuous from _: to _; for suitable : and ; ; let + − FQS (\ ).
Here we prove that, if 5 satisfies a "pointwise Ho¨rmander inequality", the operator
G+0ÐBÑ œ
'
\ Öx× kÐx,yÑ
\
± +ÐBÑ +ÐCÑ ± fÐyÑ d.ÐyÑ
satisfies the _: -_; estimate
² G+ 0 ² ; Ÿ - ² + ² ‡ ² 0 ² :
(with :ß
; as above). This estimate in particular implies an analogous one for the commutator of O with +.
0. INTRODUCTION
This note deals with commutators of some integral operators on homogeneous spaces, and it is a
´
continuation of [2] and [3]. In [2] it is proved that, if O is a Calderon-Zygmund
operator on a homogeneous
space Ð\ß. ,.Ñ and + − FQSÐ\Ñ, then the commutator
satisfies the _: -estimate:
GÒOß+Ó0 œ OÐ+0Ñ + † O0
² GÒOß+Ó0 ² : Ÿ - ² + ² ‡ ² 0 ² : ,
Ð0.1Ñ
" : _.
Ð0.2Ñ
(This generalizes a theorem by Coifman-Rochberg-Weiss [10], valid in the euclidean case).
In [3] a similar result is established for fractional integrals. More precisely, if
I ! f Ðx Ñ œ
Ö × k! Ðxß yÑ fÐyÑ d.ÐyÑ
X\ x
G+!0ÐBÑ œ \ Ö ×k!Ðxß yÑ ± +ÐBÑ +ÐCÑ ± fÐyÑ d.ÐyÑ,
then it is proved that for p − Ð1, ! Ñ , ; œ : !,
'
1
1
\
Ð0.3Ñ
that is k! Ðx,yÑ œ .ÐBÐxß dÐxß yÑÑÑ!" , 0 ! ", and, for
Ð\ß. ,.Ñ,
is the fractional integral on
+ − FQSÐ\Ñ, we set
'
x
Ð0.4Ñ
1
² G+!0 ² ; Ÿ - ² + ² ‡ ² 0 ² : .
Ð0.5Ñ
Inequality (0.5), clearly, is a stronger estimate than the analogous one on the commutator of M! ,
proved, in the euclidean case, by Chanillo [9].
The idea of the present paper is that what allows to put the absolute value inside the integral, in the
definition of G+! 0 , is the positivity of the fractional integral kernel 5! , and that for any integral operator
with positive kernel the analogous estimate can be proved:
Theorem 0.1. Let Ð\ß.,.Ñ be a homogeneous space (see §1 for the definition), O an integral operator of
the kind
O fÐ x Ñ œ
'
Ö × kÐx,yÑ fÐyÑ d.ÐyÑ
X\ x
Ð0.6Ñ
5 nonnegative measurable function, such that O is continuous from _:Ð\Ñ into _:Ð\Ñ for every
: − Ð"ß_Ñ. Moreover, assume 5 satisfies a pointwise Ho¨rmander inequality:
with
1
there exist constants -O
0, Q
0, "
1 such that for every B! − \ , <
± kÐx!ß yÑ kÐxß yÑ ± Ÿ
cO
.ÐBÐx!ß.ÐB!ßyÑÑÑ
†
Ð ßÑ
Ð ßÑ
d x! x "
d x! y "
0, B − F< ÐB! Ñ, C Â FQ< ÐB! Ñ,
.
(0.7)
In the following, we will briefly write .ÐBÐx! à yÑÑ for .ÐBÐx! ß.ÐB! ß yÑÑÑ. If \ is bounded, assume that also
5‡ÐB,CÑ œ 5ÐCßBÑ satisfies (0.7). For + − FQSÐ\Ñ, set
G+0ÐBÑ œ
'
\ Öx× kÐx,yÑ
\
± +ÐBÑ +ÐCÑ ± fÐyÑ d.ÐyÑ.
Ð0.8Ñ
Then
² G+0 ² : Ÿ - ² + ² ‡ ² 0 ² : for every : − Ð"ß_Ñ.
Ð0.9Ñ
Example 0.2. Consider the half-space ‘8 œ ‘8" ‚ Ð!ß_Ñ. For B œ ÐBw ,B8 Ñ − ‘8, let B~ œ ÐBw , B8 Ñ be
the "reflected point". The operator
0ÐCÑ
±BC±8 .C
is _: -_: continuous on ‘8 . This operator and its commutator appear in [7] in connection with boundary
estimates for solutions to elliptic equations, and this example was the first motivation for the present study.
Analogous "parabolic" operators are studied in [1].
Note that if we want to bound the kernel ± ~B C ± 8 with some power of ± B C ± , then the best
possible exponent is 8; nevertheless, the kernel is not singular.
Theorem 0.1 implies in particular that the _: -estimate on the commutator of O (see (0.1)-(0.2)) still
~
holds if the kernel 5 is replaced with any other equivalent function 5 , that is
-"5ÐBßCÑ Ÿ ~5ÐBßCÑ Ÿ -# 5ÐBßCÑ
´
for some positive constants -" , -# . Note that this fact cannot be assured for a singular integral of CalderonZygmund type (that is, which is not absolutely convergent).
O0ÐBÑ œ
'
‘8
~
The proof of Theorem 0.1, given in §2, is a variation of the one given in [2] for commutators of
´
Calderon-Zygmund
operators, which makes essentially use of an estimate on the "sharp function" of G+ 0
and is, in turn, based on the analogous result proved in the euclidean case in [16], p.418. The same
technique can be adapted also to fractional integrals: this leads to another proof of the main result in [3]
(see inequality (0.4) above), which holds for a completely general homogeneous space (whereas in [3] a
geometric condition on the space is required). This is discussed in §3. Finally, §4 contains the proof of the
inverse of Theorem 0.1 (and its analog for fractional integrals): 3Þ/Þ, if the operator 0 È G+ 0 is bounded on
_: , then + − FQS.
Acknoledgements. I wish to thank Prof. Filippo Chiarenza for stimulating discussions on the subject of this
paper.
1. SOME BASIC FACTS ABOUT HOMOGENEOUS SPACES
In this section we give precise definitions and recall some known results about homogeneous spaces.
Definition 1.1. Let X be a set. A function d:X ‚ X Ä Ò0,_Ñ is called quasidistance if:
iÑ dÐx,yÑ œ 0 Í x œ y;
iiÑ dÐx,yÑ œ dÐy,xÑ;
iiiÑ there exists a constant c. 1 such that for every x, y, z − X
dÐx,yÑ Ÿ c. ÒdÐx,zÑ . dÐz,yÑÓ.
(1.1)
If ÐX,dÑ is a set endowed with a quasidistance, the "balls" B< ÐxÑ ´ BÐx,rÑ œ y − X: dÐx,yÑ r (for
x − X and r 0) form a base for a complete system of neighborhoods of X, so that X is a Hausdorff space.
Note that the balls are not in general open sets; if they are, then they form a base for the topology of X.
Definition 1.2. We say that ÐX,d,.Ñ is a homogeneous space if:
iÑ X is a set endowed with a quasidistance d, such that the balls are open sets in the topology induced by
d;
š
2
›
iiÑ
. is a positive Borel measure on X, satisfying the doubling condition:
0 .ÐB#< ÐxÑÑ Ÿ c. † .ÐB< ÐxÑÑ _
(1.2)
for every x − X, r 0, some constant c. 1.
The numbers c. , c. in (1.1)-(1.2) will be called "constants of X" and we will write cÐXÑ for a constant
depending on the constants of X.
We start recalling two theorems due to Macias-Segovia (see [15]).
Theorem 1.3.w Let d be a quasidistance on a set X. Then there exists a quasidistance .w on X such that:
ÐiÑ d and .
are equivalent, that is there exist positive constants c" , c# such that for every x, y − X:
c" . w Ðx,yÑ Ÿ dÐx,yÑ Ÿ c# . w Ðx,yÑ;
Ð1.3Ñ
¨
ÐiiÑ .w is "locally Holder
continuous"; more precisely there exist # − Ð0,1Ó and c 0 such that for every x,
y, z − X:
Ð1.4Ñ
± .w Ðx,zÑ .w Ðy,zÑ ± Ÿ c .w Ðx,yÑ# Ò.w Ðx,zÑ . .w Ðy,zÑÓ"#.
Theorem 1.4. Let Ð\ß.ß .Ñ be a homogeneous space. Define:
$ ÐBßCÑ œ inf .(F ): BßC − F , and F is a ball with respect to . ,
if B Á C, and
$ ÐB,BÑ œ 0.
Then:
(3Ñ $ is a quasidistance;
(33Ñ $ and . are topologically equivalent
(note that they are not, in general, equivalent in the sense of (1.3)!);
(333) the space (\ß $ ß .) is normal, that is there exist positive constants -" , -# such that
-"< Ÿ .ÐU(Bß<)) Ÿ -#<
for every B − \ and every < such that .ÐÖB}Ñ < .Ð\Ñ. Here U (Bß<) denotes a ball with respect to $ .
([email protected]) For any $ -ball U there exist two . -balls F" , F# such that F" © U © F# and .(F# ) Ÿ - .ÐF" ), for
some constant - independent of U.
š
›
The definition of standard real analysis tools, such as the maximal function `f, the sharp function f #
and the BMO seminorm ² f ² ‡ , naturally carries over to this context, namely:
œ sup ± fÐyÑ ± d.ÐyÑ ;
x−B
where the sup is taken over all balls containing x and ... œ .Ð Ñ ...
` fÐx Ñ
'
B
'
1 '
B
B
B
;
f # ÐxÑ œ sup B ± fÐyÑ fF ± d.ÐyÑ
x−B
'
where fF
œ f Ð x Ñ d . Ðx Ñ ;
'
B
² f ² ‡ œ sup
f ÐxÑ œ sup ± fÐyÑ fF ± d.ÐyÑ;
x
B
BMO ´ f − _"69- ÐXÑ : ² f ² ‡ _ .
#
'
B
›
š
We recall three results related to these concepts, which have been proved in the context of
homogeneous spaces in [5], [11].
Theorem 1.5. (Maximal Inequality).
For every p − Ð1,_Ó there exists a constant cÐX,pÑ such that for every f − _: :
Theorem 1.6. ÐJohn-Nirenberg LemmaÑ.
² `f ² : Ÿ c ² f ² : .
For every p − Ò1,_Ñ there exists a constant cÐX,pÑ such that for every f − BMO, every ball B:
3
± f Ð x Ñ fF ± : d . Ð xÑ
Š'
B
‹
"Î:
Ÿ c ² f ² ‡.
Theorem 1.7. ÐSharp InequalityÑ.
For every p − Ò1,_Ñ there exists a constant cÐX,pÑ such that for every f − _: :
if .ÐXÑ œ _:
²f²:Ÿc²f ²:
² f f ² : Ÿ c ² f ² :.
#
if .ÐXÑ _:
'
#
X
The following two lemmas about BMO functions can be proved as in the euclidean case:
Lemma 1.8.
integer j :
(See [16], p. 206). Let a − BMO and M
1. Then for every ball B< ÐxÑ and every positive
± aF Q 4 < aF < ± Ÿ c † j ² a ² ‡
where c depends on M and on the dubling constant c. .
Lemma 1.9. Let f − BMO, and let:
fÐxÑ if ± fÐxÑ ± Ÿ n
n
if fÐxÑ n
f8 Ð x Ñ œ
n if fÐxÑ Ÿ n
Ú
Û
Ü
Then f8
− BMO
and:
² f8 ² ‡ Ÿ c ² f ² ‡
with c an absolute constant.
Finally, we will use the following:
Lemma 1.10.
X is bounded.
(See Lemma 1.9 in [2]). Let ÐX,d,.Ñ be a homogeneous space. Then .ÐXÑ _ if and only if
2. _: -ESTIMATES FOR INTEGRAL OPERATORS WITH POSITIVE KERNELS
Here we keep the same notations and assumptions that appear in Theorem 0.1, which we want to
prove in this section. Moreover, let us define the space
W œ š f − __ ÐXÑ ± the support of f is bounded›.
Note that W is dense in _: for every p − Ò"ß_Ñ. For 0
maximal function of 0 , defined in §1. Then:
" , let 0 # , `0 denote the sharp function and
− _69-
Theorem 2.1. For every < − Ð"ß_Ñ there exists a constant -Ð<ßOß\Ñ such that for every + − __Ð\Ñ,
0 − W,
± ÐG+ fÑ Ð–xÑ ± Ÿ c † ² a ² ‡
#
Š
` ŠÒKÐ ± f ± ÑÓ< ‹ ЖxÑ‹
"Î<
.
Proof. Let F$ œ F$ ÐB! ), BßB– − F$ .
ÐG+ 0Ñ(B) œ '\
Š
`Ð ± f ± < ÑЖxÑ‹
"Î<
Ÿ
± +ÐBÑ +ÐCÑ ± 5ÐBßCÑ0ÐCÑ ;FQ ÐCÑ..ÐCÑ .
. \ ± +ÐBÑ +ÐCÑ ± 5ÐBßCÑ0ÐCÑ ;\\FQ ÐCÑ..ÐCÑ ´ TÐBÑ . UÐBÑ.
F ± TÐBÑ TF ± ..ÐBÑ Ÿ 2 F ± TÐBÑ ± ..ÐBÑ Ÿ
Ÿ 2 F ..ÐBÑ FQ ± +ÐBÑ +ÐCÑ ± 5ÐBßCÑ ± 0ÐCÑ ± ..ÐCÑ Ÿ
$
'
$
'
Ð2.1Ñ
'
$
$
'
for every –x − X.
$
'
$
$
Ÿ 2 F ..ÐBÑ FQ ± +ÐBÑ +F ± . ± +ÐCÑ +F ± 5ÐBßCÑ ± 0ÐCÑ ± ..ÐCÑ ´ T" . T#. Ð2.2Ñ
'
'
$
$
Š
$
$
4
‹
'
T" œ 2
F
$
± +ÐBÑ +F ± ..ÐBÑ FQ 5ÐBßCÑ ± 0ÐCÑ ± ..ÐCÑ Ÿ
'
$
$
¨
(by Holder)
Ÿ # B ± aÐxÑ a ± <w d.ÐBÑ
Š'
B$
$
‹
"Î<w
† B ± KÐfÑÐxÑ ± < d.ÐBÑ
Š'
‹
$
"Î<
Ÿ
(by Theorem 1.6)
Ÿ -Ð<ß\Ñ ² + ² ‡
'
T# Ÿ 2
F . .ÐBÑ'FQ
$
$
Š
"Î<
– ‹
`ŠOÐ ± 0 ± Ñ< ‹ ÐBÑ
Ð2.3Ñ
.
± +ÐCÑ +F ± 5ÐBßCÑ ± 0ÐCÑ ± ..ÐCÑ œ
$
œ 2 F O ± + +F ± ± 0 ± ;FQ ..ÐBÑ Ÿ
'
Š
$
$
$
(for every ;
Ÿ#
"
")
"Î;
Ÿ
ÐBÑ
"Î;
± K Ða a Ñf ; Q ÐxÑ ± ; d.ÐBÑ
. ÐB $ Ñ X
Š
‹
'
Š
B$
B
$
‹
‹
(by the _; -continuity of O )
Ÿ cÐqß KÑ
Š
± Ð Ñ a ± ± fÐxÑ ±
" '
. ÐB $ Ñ B Q $ Š a x
B$
‹
;
d.
‹
Ÿ
¨
(choosing q rß by Holder)
Ÿ cÐqß KÑ
†
Š'
(by Lemma 1.8: ± aÐxÑ aB
.
"
Ð B Ñ Š 'B Q
$
$
± fÐxÑ ± < d.ÐBÑ
‹
;Î<
Ð<;ÑÎ<
± aÐxÑ aB ± ;<ÎÐ<;Ñ d.ÐBÑ‹ Ÿ
BQ $
†
"Î;
$
Ÿ
± Ÿ ± aÐxÑ a Q ± . ± a Q a ± Ÿ ± aÐxÑ a Q ± . c ² a ² * )
"Î<
Ÿ cÐrß KÑ .ÐB$ Ñ"Î; BQ ± fÐxÑ ± < d.ÐBÑ †
B
$
B
$
Š'
†
Š'
BQ $
‹
$
B
$
‹
$
± aÐxÑ a Q ± ;<ÎÐ<;Ñ d.ÐBÑ
B
B$
$
Ð<;ÑÎ;<
. ² a ² *.ÐBQ $ ÑÐ<;ÑÎ;< Ÿ
Ÿ
(by Theorem 1.6 and the doubling condition)
Ÿ cÐrß Kß\Ñ BQ ± fÐxÑ ± < d.ÐBÑ
Š'
$
Ÿ -Ð<ßOß\Ñ ² + ² ‡
UÐBÑ œ '\ \FQ
F ± UÐBÑ UF
'
$
$
$
Š
‹
"Î<
† ²a²* Ÿ
– ‹
`Ð ± 0 ± < ÑÐBÑ
"Î<
.
± +ÐBÑ +ÐCÑ ± 5ÐBßCÑ0ÐCÑ..ÐCÑ.
± ..ÐBÑ Ÿ 2 F ± UÐBÑ - ± ..ÐB) œ
'
$
(with - to be chosen later)
5
Ð2.4Ñ
œ 2 F l \\FQ ± +F +ÐCÑ ± 5ÐBßCÑ0ÐCÑ..ÐCÑ - .
'
'
$
.
'
\ \F Q
$
Š
± +ÐBÑ +ÐCÑ ± ± +F +ÐCÑ ± 5ÐBßCÑ0ÐCÑ..ÐCÑl..ÐBÑ Ÿ
'
$
'
\ \F Q
$
k
‹
$
Ÿ 2 F ..ÐBÑ
.
$
$
¸'
\ \F Q
$
± +F +ÐCÑ ± 5ÐBßCÑ0ÐCÑ..ÐCÑ - .
¸
$
± +ÐBÑ +ÐCÑ ± ± +F +ÐCÑ ± 5ÐBßCÑ ± 0ÐCÑ ± ..ÐCÑ ´ U" . U#.
k
$
In U" , choose - œ '\ \FQ
'
U" Ÿ 2
F . .ÐBÑ '\ \FQ
$
$
$
Ÿ
± +F +ÐCÑ ± 5ÐB!ßCÑ0ÐCÑ..ÐCÑ.
$
Ð2.5Ñ
Then:
± +F +ÐCÑ ± ± 5ÐB!ßCÑ 5ÐBßCÑ ± ± 0ÐCÑ ± ..ÐCÑ.
$
By (0.7), the inner integral in U" is bounded by:
±aÐyÑaF ±±fÐyѱ
cO d x! x " 'X\BQ $ .ÐBÐx!àyÑц$dÐx!,yÑ" d. y
† Ð ßÑ
Ÿ
cO
†
ÐÑŸ
†
$"
Now:
'
œ
Ÿ
_
!'
4œ"
X \B Q $
X \B Q $
"4
4œ" ÒM $Ó
$
"Î<w
"
†
"Î<
±fÐyѱ<
.ÐBÐx!àyÑцdÐx!,yÑ d.ÐyÑ .
(2.6)
"
±fÐyѱ<
X\BQ $ .ÐBÐx!àyÑцdÐx!,yÑ" d. y
ÐÑœ
Ð ÑŸ
"
† .ÐBÐx!", M 4 $ÑÑ
(by the doubling condition)
Analogously:
'
Ÿ cÐ\ß MÑ †
±aÐyÑaF ±<w
.ÐBÐx!àyÑцdÐx!,yÑ d.ÐyÑ
±fÐyѱ<
M 4 $ŸdÐx!ßyÑ%M 4" $ .ÐBÐx!àyÑцdÐx!,yÑ" d. y
_
!
'
'
¨
(by Holder)
± fÐyÑ ± < d.ÐyÑ Ÿ
Ÿ cÐ\ß Mß "Ñ † $" † `Ð ± f ± < ÑЖxÑ.
"
$
_
"4
4œ" ÒM $Ó
BÐx!, M 4" $Ñ
±aÐyÑaF ±<w
.ÐBÐx!àyÑцdÐx!,yÑ d.ÐyÑ
X \B Q $
!
'
"
"
(2.7)
Ÿ
† BÐx!ßM 4" $Ñ ± aÐyÑ aF ± <w d.ÐyÑ Ÿ
'
$
(by Theorem 1.6 and Lemma 1.8)
Ÿ
_
" !
cÐrß Kß XÑ
4œ"
$"
² a ² <‡w
Š
"j<w ‹
M "4
6
œ cÐrß Kß XÑ † $" † ² a ² ‡<w .
"
(2.8)
From (2.7)-(2.8):
"Î<
–‹ .
U" Ÿ -Ð<ßOß\Ñ ² + ² ‡ Š`Ð ± 0 ± < ÑÐBÑ
'
U# Ÿ 2
F
$
± +ÐBÑ +F ± ..ÐBÑ
$
'
Ÿ 2 F ± O ± 0 ± ;\\FQ ÐBÑ ± < ..ÐBÑ
'
Š
$
$
‹
\ \F Q
"Î<
$
5ÐBßCÑ ± 0ÐCÑ ± ..ÐCÑ Ÿ
† F ± +ÐBÑ +F ± ..ÐBÑ
(by Theorem 1.6, and since O Š ± 0 ± ;\ \FQ
Ÿ -Ð<ß\Ñ ² + ² ‡
Š
Ð2.9Ñ
$
'
$
$
‹
<w
"Î<w
Ÿ
ÐBÑ Ÿ O ± 0 ± ÐBÑ aB)
Š
‹
"Î<
–‹ .
`Ð[O ( ± 0 ± )]< ÑÐBÑ
Ð2.10Ñ
Collecting inequalities (2.1)-(2.10), we get the theorem.
By the _: -continuity of O and the maximal inequality (Theorem 1.5), we get from Theorem 2.1 the
following
Corollary 2.2. Under the same assumptions of Theorem 2.1, for every : − Ð"ß_Ñ there exists a constant
-Ð:ßOß\Ñ such that for every + − __Ð\Ñ, 0 − W,
² ÐG+ 0Ñ# ² : Ÿ - ² + ² ‡ ² 0 ² :.
Proof of Theorem 0.1. In view of Lemma 1.9 and the density of W in _: for every : _, it is enough to
prove the theorem for + − __ and 0 − W.
Taking + − __ and 0 − W, we have that G+ 0 − _: for every : − Ð"ß_Ñ.
Now, if .Ð\Ñ œ _, that is \ is unbounded (see Lemma 1.10), the theorem follows from Corollary
2.2 by the "sharp inequality" (Theorem 1.7). If .Ð\Ñ _, from Corollary 2.2 and Theorem 1.7 we get:
² G+ 0 ' \ G+ 0 ² : Ÿ - ² + ² ‡ ² 0 ² :.
Ð2.11Ñ
Then it is enough to prove that
± '\ G+ 0ÐBÑ..ÐBÑ ± Ÿ -.Ð\Ñ"Î: ² + ² ‡ ² 0 ² :.
w
Ð2.12Ñ
This can be done repeating the proof of Theorem 3.2 in [2]. For convenience of the reader, the proof is
included below.
Let us consider:
ÐG+ fÑÐxÑ d.ÐxÑ œ 'X d.ÐxÑ'X kÐx,yÑÒaÐxÑ aÐyÑÓ fÐyÑ d.ÐyÑ.
Since a, f are bounded functions and .ÐXÑ _, the integral on the right hand side of the last equation
'
X
converges absolutely and equals
'X fÐyÑ d.ÐyÑ'X kÐx,yÑÒaÐyÑ aÐxÑÓ d.ÐxÑ œ 'X fÐyÑG+‡ "ÐyÑ d.ÐyÑ.
Here we denote by G+‡ the operator defined as in (0.8), with 5ÐBßCÑ replaced by 5‡ ÐBßCÑ ´ 5ÐCßBÑ.
Hence:
± 'X ÐG+ fÑÐxÑ d.ÐxÑ ± Ÿ ² f ² : ² G+‡ " ² : .
Since also k ‡ satisfies (0.7), (2.11) holds for G+‡ , too, and
² G+‡ " ² : Ÿ c ² a ² ‡ .ÐXÑ"Î: . .ÐXÑ"Î: ± 'X ÐG+‡ "ÑÐxÑ d.ÐxÑ ± .
w
w
w
Again by Fubini's theorem, we get:
7
(2.13)
(2.14)
'
X
ÐG+‡ "ÑÐxÑ d.ÐxÑ œ 'X ÐKaÑÐyÑ d.ÐyÑ 'X ÐK‡aÑÐxÑ d.ÐxÑ.
(2.15)
¨
By Holder
and the _: estimate on K:
± 'X ÐKaÑÐyÑ d.ÐyÑ ± Ÿ ² Ka ² : † .ÐXÑ"Î: Ÿ c ² a ² : † .ÐXÑ"Î: .
w
Without loss of generality, we can assume
(2.16)
w
Ð Ñ Ð Ñœ0
' a x d. x
X
since the commutator we are estimating is not affected by adding a constant to a. We claim that
² a ² : Ÿ c: ² a ² ‡ .ÐXÑ"Î:.
Ð2.17Ñ
To see this, recall that, by Lemma 1.10, X is bounded, so it coincides with some ball B. Then, by Theorem
1.6,
'
X
± aÐxÑ ± : d.ÐxÑ œ .ÐXÑ † ' B ± aÐxÑ aF ± : d.ÐxÑ Ÿ c .ÐXÑ ² a ² :‡ .
Therefore from (2.16) we get:
± 'X ÐKaÑÐyÑ d.ÐyÑ ± Ÿ c ² a ² ‡.ÐXÑ
An analogous estimate holds for the second term in (2.15) and from (2.13) (2.18) the result follows.
(2.18)
3. FRACTIONAL INTEGRALS
As we said in the Introduction, Theorem 0.1 can be rephrased for fractional integrals on
homogeneous spaces; this leads to another proof of the results in [3].
Define the fractional maximal function:
`" 0ÐBÑ œ sup
B−F
1
. (F )
"
"
'
F
± 0ÐCÑ ± ..ÐCÑ
for 0 Ÿ " 1
Ð3.1Ñ
(3Þ/Þ, the sup is taken for all the balls containing B). When " œ 0, we have the standard maximal function.
reasoning as in [16], p.153, the following can be proved:
Proposition 3.1. If 0 Ÿ " 1, 1 : Ÿ "1 , ;1 œ 1: " , then there exists - œ -Ð" ß:) 0 such that for
every 0
− _:
² `" 0 ² ; Ÿ - ² 0 ² : .
Ð3.2Ñ
Moreover, we know that
Proposition 3.2. (See [13]). If
ÐÑœ
I! f x
'
Ð ÑÐÑ ÐÑ
X\Öx× k! x,y f y d. y
is the fractional integral on Ð\ß. ,.Ñ, that is k! Ðx,yÑ œ .ÐBÐx,dÐx,yÑÑÑ!" , 0 ! ", and 1 : 1
1
:
; œ : !, then there exists - œ -Ð!ß:) 0 such that for every 0 − _
² M! 0 ² ; Ÿ - ² 0 ² : .
1
!
,
Ð3.3Ñ
¨
Theorem 3.3. Let 5! be as in Proposition 3.2 and assume that it satisfies the following "Hormander
inequality":
there exist constants -
0, "
0, Q
1 such that for every B! − \ , <
± k!Ðx!ß yÑ k!Ðxß yÑ ± Ÿ
. ŠBÐx
c
!
ß.ÐB ßyÑÑ‹
!
!
"
0, B − F< ÐB! Ñ, C Â FQ< ÐB! Ñ,
† ddÐÐxx ßßxyÑÑ
!
!
"
"
.
If \ is bounded, assume that also 5!‡ ÐB,CÑ œ 5! ÐCßBÑ satisfies (3.4). For + − FQSÐ\Ñ, set
G+!0ÐBÑ œ '\\Öx×k!Ðx,yÑ ± +ÐBÑ +ÐCÑ ± fÐyÑ d.ÐyÑ.
Then, if :ß; are as in Proposition 3.2,
8
Ð3.4Ñ
Ð3.5Ñ
² G+!0 ² ; Ÿ - ² + ² ‡ ² 0 ² :.
Ð3.6Ñ
Sketch of the proof. We claim that:
for every ! − Ð!ß"Ñ, < − Ð0, !1 Ñ there exists - œ -Ð<ß !ß\Ñ such that for every x − X
"Î<
± ÐG+! fÑ#ÐxÑ ± Ÿ c † ² a ² ‡ŠM ŠÒM!Ð ± f ± ÑÓ< ‹ ÐxÑ‹ . ŠM!< Ð ± f ± < ÑÐxÑ‹
"Î<
Ÿ.
Ð3.7Ñ
(3.7) can be proved similarly to Theorem 2.1, using (3.3) and (3.4), with a careful choice of the exponents.
We do not give account of the details. By Theorem 1.5, Proposition 3.1 and Proposition 3.2, inequality (3.7)
implies:
² (G+!0 )# ² ; Ÿ - ² + ² ‡ ² 0 ² :.
Then Theorem 3.3 follows from (3.8) like Theorem 0.1 follows from Corollary 2.2.
Ð3.8Ñ
The previous Theorem requires that the kernel 5! satisfies inequality (3.4). This is a "geometric"
assumption on the space Ð\ß.ß .Ñ, which can be false, for instance when the space has "atoms": in this case
the function B È .(F< (B)) is discontinuous, while (3.4) requires its continuity. Moreover, even in nonpathological cases, the proof of property (3.4) can be difficult: for instance, if \ is a manifold with variable
curvature, the function B È .(F< (B)) has not a simple explicit form. On the other hand, a non-trivial
example where (3.4) can be proved is ‘R with the euclidean distance and a weighted measure . . œ AÐBÑ
.B, where A (A − _"69-, A 0) is such that . satisfies a doubling condition (for instance, A is an E: weight
of Muckenhoupt). These homogeneous spaces are studied in [12], in relation with fractional integrals; in
particular, they prove that, under the above assumptions,
for every 6 <
can be proved.
""
"
.ŠFÐBß6 . <Ñ‹ .ŠFÐBß6Ñ‹ Ÿ - .ŠFÐBß<Ñ‹ † .ŠFÐBß6Ñ‹
0, B − ‘R , some - 0 and " − Ð!ß"Ñ. From this result, with some computation, (3.4)
However, it is possible to remove assumption (3.4) without loss of generality, using some known
results on homogeneous spaces: what follows is an explaination of this extension.
Let (\ß. ,.) be any homogeneous space; let us construct the quasidistance $ as in Theorem 1.4; then,
starting from $ , let us construct the quasidistance $ w as in Theorem 1.3. The fractional integral kernel 5! , in
the space (\ß $ w ß .), can be defined as:
5!w (BßC) œ $w (BßC)!".
¨
Lemma 3.6. The kernel 5!w defined as in (3.9) satisfies a Hormander
inequality (3.4).
Proof.
± 5!w (BßC) 5!w (B!ßC) ± Ÿ
Ÿ - ±$ ÐB,CÑ$ ÐB ,CÑ$ ÐB ,CÑ
!
w
"
w
w
!
!Ñ
#Ð"
!
!
"
±
Ð3.9Ñ
(for $ w ÐB! ,CÑ 2$ w ÐB,B! Ñ)
$ ÐB ,Cѱ
Ÿ - ±$ ÐB$,CÑ
Ÿ
ÐB ,CÑ
w
w
w
!
!
!
"
!Ñ
#Ð"
(by (1.4))
Ÿ - $$ ÐBÐB ,,BÑCÑ
w
w
!
!Ñ
Ÿ - $ $ÐBÐB,CÑ,BÑ
w
w
Š$ w
! Ñ#
Ð"
#Ð"
!
!
!
ÐB,CÑ . $w ÐB!,CÑ‹
! Ñ#
Ð"
! ÑÐ"#Ñ
Ð"
Ð"!ÑÐ"#Ñ
œ - 5!w ÐB!,CÑ † Š $$ ÐBÐB ,,BÑCÑ ‹
w
w
!
Ÿ
"
!
with " œ (1 !)# .
Theorem 3.7. Let (\ß.ß .) be any homogeneous space. Then the conclusion of Theorem 3.3 holds.
Proof. By Theorem 3.3 and Lemma 3.6, estimate (3.6) holds in (\ß $ w ,.). By definition of $ and since $ and
$ w are equivalent, we see that
9
5!ÐBßCÑ œ .ÐFÐBß.ÐBßCÑÑÑ!" and 5!w (BßC) œ $w (BßC)!"
are equivalent (where F denotes a . -ball). Therefore we can write:
² G+!0 ² ; Ÿ - ² + ² FQSÐ\ß$ ,.Ñ ² 0 ² :
with G+! 0 defined as in (0.4) by 5! or 5!w . We now claim that:
² + ² FQSÐ\ß$ ,.Ñ Ÿ - ² + ² FQSÐ\ß.,.Ñ.
w
w
Ð3.10Ñ
Ð3.11Ñ
To prove (3.11), it is enough to show that
² + ² FQSÐ\ß$,.Ñ Ÿ - ² + ² FQSÐ\ß.,.Ñ ,
Ð3.12Ñ
since $ and $ w are equivalent. By Theorem 1.4, ([email protected]), we know that for any $ -ball U there exist two . -balls
F", F# such that F" © U © F# and .(F#) Ÿ -.ÐF" ), for some constant - independent of U. From this fact
and Theorem 1.6:
1
'
. (U ) U
± +ÐBÑ +U ± # ..ÐBÑ Ÿ
(for any - − ‘)
Ÿ .(F- ) 'F ± +ÐBÑ - ± # ..ÐBÑ Ÿ
#
#
Ÿ .(1U) 'U ± +ÐBÑ - ± # ..ÐBÑ Ÿ
(for - œ .(F" ) 'F +ÐCÑ. .ÐCÑ)
#
#
Ÿ - ² + ² #FQSÐ\ß.,.Ñ.
So (3.12) is proved. From (3.10)-(3.11), the Theorem follows.
Remark 3.8. Theorem 3.7 improves the analogous result in [2], where a geometrical property of the space
(\ß.ß .) is assumed. Let us call (T ) this assumption: it means, roughly speaking, that \ has not too many
empty spherical shells. Condition (T ) could be removed also from the proof given in [2] noting that,
whatever the space (\ß.ß .) is, the space (\ß $ ß .) satisfies always condition (T ): this fact was proved by
[6]. Therefore the result holds for (\ß $ ß .) and so, reasoning as in the proof of Theorem 3.7, also for
(\ß.ß .).
4. THE INVERSE THEOREM
We conclude pointing out that the inverse theorems of Theorem 0.1 and Theorem 3.3 hold. Namely:
Theorem 4.1. (Inverse of Theorem 3.3). Let Ð\ß. ,.Ñ be a homogeneous space, + − _"69- (\ ), G+! 0 be
defined as in Theorem 2.3, :ß;ß ! as in Proposition 2.2 and assume that following inequality holds
² G+!0 ² ; Ÿ L ² 0 ² :
Ð4.1Ñ
for some positive constant L , every 0 − _: . Then + − FQS and
² + ² ‡ Ÿ - † L,
Ð4.2Ñ
for some constant - independent of +.
Theorem 4.2. (Inverse of Theorem 0.1). Let Ð\ß. ,.Ñ be a homogeneous space, + − _"69- (\ ), G+ 0 be
defined as in Theorem 0.1 and assume that the kernel 5 satisfies the growth condition:
± 5ÐBßCÑ ± Ÿ . ŠFÐBß.ÐBßCÑÑ
.
Ð4.3Ñ
‹
If, for some : − Ð"ß_Ñ, the following inequality holds
² G+ 0 ² : Ÿ L ² 0 ² :
for some positive constant L , every 0 − _: , then + − FQS and
² + ² ‡ Ÿ - † L,
for some constant - independent of +.
Theorem 4.1 has been proved in [4]; the proof of Theorem 4.2 can be carried out similarly, setting
! œ 0 in the proof of Theorem 4.1 and using assumption (4.3). For convenience of the reader, the proof of
Theorem 4.1 is included below.
10
Proof of Theorem 4.1. Let 0
œ ;F with F any fixed ball. Then
G+!0ÐBÑ œ 'F\Öx×k!Ðx,yÑ ± +ÐBÑ +ÐCÑ ± d.ÐyÑ.
If BßC − F ,
. ŠFÐBß.ÐBßCÑÑ‹ Ÿ -Ð\Ñ † .(F ),
then
± 5!ÐBßCÑ ± - .(F)!",
and
G+!0ÐBÑ - .(F)!"'F ± +ÐBÑ +ÐCÑ ± ..ÐCÑ - .(F)! ± +ÐBÑ ' F +ÐCÑ..ÐCÑ ± .
Raising to the ; both sides of (4.6) and integrating on F :
- .(\)!; 'F ± +ÐBÑ +F ± ; ..ÐBÑ Ÿ 'F ± G+!0 (BÑ ± ; ..ÐBÑ Ÿ
Ð4.4Ñ
Ð4.5Ñ
Ð4.6Ñ
(by (4.1))
Ÿ L; ² ;F ² ;: œ L; .(\);Î:.
Then from the relation between :ß; and ! we get
"Î;
F ± +ÐBÑ +F ± ; ..ÐBÑ‹ Ÿ -O.
Š'
Since F is generic, by Theorem 1.6, (4.2) follows.
11
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Marco Bramanti
Dipartimento di Matematica
Politecnico di Milano
Via Bonardi 9
20133 Milano.
12
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