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Maximal Functions over Hypersurfaces with Flat Points
Journal of Mathematical Analysis and Applications 260, 70᎐82 Ž2001. doi:10.1006rjmaa.2001.7436, available online at http:rrwww.idealibrary.com on
Maximal Functions over Hypersurfaces with Flat Points Kyung Soo Rim School of Mathematics, Korea Institute for Ad¨ anced Study, Seoul 130-012, Korea E-mail: [email protected] Submitted by William F. Ames Received June 15, 1999
For n G 4, let S be a flat hypersurface in ⺢ n , and let d s d , where g C⬁0 Ž⺢ n . and is the surface area measure on S. Then the maximal functions Mf associated to S and by Mf Ž x . s sup t ) 0 < HS f Ž x y t . d Ž .<, f g S Ž⺢ n ., are bounded on certain Orlicz spaces L⌽ Ž⺢ n .. 䊚 2001 Academic Press Key Words: maximal function; flat hypersurface.
1. INTRODUCTION For n G 2, let S be a surface in ⺢ n, and put d s d , where g C0⬁Ž⺢ n . and is the surface area measure on S. Then the maximal function M can be defined by Mf Ž x . s sup < Ž f ) t . Ž x . < s sup t)0
t)0
HS f Ž x y t . d Ž .
,
for f g S Ž⺢ n ., the Schwartz space. When S is a sphere in ⺢ n Ž n G 3., Stein proved that M is bounded on pŽ n. L ⺢ if and only if p ) nrŽ n y 1. w8, 10x. Later Bourgain obtained the corresponding result for the circular maximal function over convex curves in ⺢ 2 w2x. Beginning with two articles the problem has been extensively studied to the general manifolds S. We say that S is of finite type if at every point x g S, every one dimensional tangent line to S at x makes finite order of contact with S; i.e., the Gaussian curvature vanishes at most finite order at every point of S. We say that S is flat if S contains at least one point at which the Gaussian curvature vanishes to infinite order. 70 0022-247Xr01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
MAXIMAL FUNCTIONS OVER HYPERSURFACES
71
For a finite type S, it is known that M is bounded on L p Ž⺢ n . Ž n G 3. for a sufficiently large p w5᎐7, 9x. However, for a flat surface S, there is no L p-continuity of M for any p w9x. A few years ago Bak proved that only for n s 3 is M equipped with a flat hypersurface S bounded on L⌽ Ž⺢ 3 ., the Orlicz space associated to a Young’s function ⌽ given by an exponential r type ⌽ Ž t . s e t w1x. Bak and many other authors have guessed that his result is true in higher dimension G4. In this paper we will show the extended version of Bak’s result to n G 4.
2. PRELIMINARIES We state the Orlicz interpolation theorem which was proved by Bak in w1, Lemma 1.1x. Let Ž X, M1 , . and Ž Y, M2 , . be measure spaces, where and are positive -finite measures, and let T be a sublinear operator defined on a suitable linear space of functions f on X such that Tf is a measurable function on Y. For Young’s function ⌽ the Orlicz space L⌽ Ž d . is equipped with the norm 5 f 5 L⌽ Ž d . s inf s ) 0 :
½
H⌽ Ž < f Ž x .
The inverse of ⌽ is defined for t g w0, ⬁. by ⌽y1 Ž t . s inf s ) 0 : ⌽ Ž s . ) t 4 . 2.1. INTERPOLATION LEMMA. Let r g w1, ⬁.. Assume that there exists a constant C such that for all t ) 0 r
Ž x : < Tf Ž x . < ) t 4 . F Ž C 5 f 5 rrt . , 5 Tf 5 ⬁ F C 5 f 5 ⬁ . Suppose a Young’s function ⌽ is gi¨ en by ⌽ Ž s . s H0s Ž t . dt, where : w0, ⬁. ª w0, ⬁. is a nondecreasing function such that Ž t . s 0 for 0 F t F 1, and Ž t . ) 0 for t ) 1 such that there exist constants c ) 1, C0 , and C1 such that u
Ž 2.1.
H1 Ž t .
t r dt F C0 Ž u . ru ry1
for u ) 1,
and for e¨ ery ) 1
Ž 2.2.
C1 Ž t . r Ž t . G Ž .
for t G c.
72
KYUNG SOO RIM
Then there exists a constant C s C Ž ⌽, r . such that 5 Tf 5 L⌽ Ž d . F CB⌽y1 Ž Ž ArB .
r
.5 f 5L
⌽
Žd. .
2.2. COROLLARY. Let T and Tk be sublinear operators such that < Tf Ž x .< F Ý⬁ks 1 < Tk f Ž x .< a.e. If for some r g w1, ⬁. and k s 1, 2, . . . r
Ž x : < Tk f Ž x . < ) t 4 . F Ž A k 5 f 5 rrt . ,
for e¨ ery t ,
5 Tk f 5 ⬁ F Bk 5 f 5 ⬁ , then 5 Tf 5 L⌽ Ž d . F C 5 f 5 L⌽ Ž d . , where C s Ý⬁ks 1 Bk⌽y1 ŽŽ A krBk . r . - ⬁. Now we state the result of Bruna et al. w4x: 2.3. LEMMA. Suppose S ; ⺢ n is a compact hypersurface of finite type bounding a con¨ ex domain. Let g ⺢ n with < < s 1, and let x 0 g S with orthogonal to Tx 0 , the affine tangent line to S at x 0 . Let g C⬁Ž⺢ n . be supported in a small neighborhood of the origin. Put H Ž x 0 , . s
HS e
i ² x , :
Ž x y x0 . d Ž x . ,
where is the induced surface area measure on S. Then there are constants C j depending only on the surface S such that H Ž x 0 , . s e i ² x 0 , : F Ž . , where < F Ž j. Ž . < F C j yj B˜Ž x 0 , <
ž
/
where B˜Ž x 0 , <
Ž 3.1.
␥ ⬘Ž t . t Ž ny1.r2
is nondecreasing for f ) 0,
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MAXIMAL FUNCTIONS OVER HYPERSURFACES
and also we may assume ␥ ⬘Ž1. s 1. The hypersurface S ; ⺢ n is defined by S s Ž x, l q ␥ Ž< x <.. : x g ⺢ ny 14 for some l g ⺢. Suppose ␥ satisfies Ž3.1. and let m F Ž n y 1.r2 Ž m may be negative.. Then for s G t ) 0, we get
␥ ⬘Ž s . s
m
y
␥ ⬘Ž t . t
m
s
␥ ⬘Ž s . s
Ž ny1.r2
y
␥ ⬘Ž t . m Ž ny1.rŽ2 m.
s t
G
␥ ⬘Ž s . s
Ž ny1.r2
y
␥ ⬘Ž t . t Ž ny1.r2
G 0.
Thus ␥ ⬘Ž t .rt m is also nondecreasing for all m F Ž n y 1.r2. Let us note that Ž3.1. implies ␥ ⬙ Ž t . G Ž n y 1.r2␥ ⬘Ž t .rt ) 0. The fact that the Gaussian curvature K Ž x . of S at x is a constant times ␥ ⬙ Ž< x <.Ž␥ ⬘Ž< x <.r< x <. ny 2 yields that the Gaussian curvature of S can only vanish at the origin. To prove the main theorem stated below, the following estimate is very crucial. In fact, many authors have guessed the decaying order Ž n y 1.r2. Nevertheless, the result has a meaning because the proof of the main theorem mainly depends on the exact constants C and aŽ ny1.r2y1r ␥ ⬘Ž ar2.Ž ny1.r2 . 3.1. THEOREM. Let n G 4 and let be a nonnegati¨ e number. Suppose g C01 ŽŽ0, ⬁.. is a nonnegati¨ e function compactly supported in the inter¨ al Ž a, ⬁. such that a F . Let ␥ and S be as abo¨ e. Let be the measure on the surface S such that d Ž x, ␥ Ž< x <.. s Ž< x <. dx. Then for a multi-index ␣ with < ␣ < F 1 there is a constant C depending only on such that
⭸␣ ⭸ ␣
ˆ Ž . F C C
aŽ ny1.r2y1
␥ ⬘ Ž ar2.
Ž ny1 .r2
Ž 1 q < < . y ny1 r2 , Ž
.
where C F 5 5 ⬁ q 5 ⬘ 5 1 if < ␣ < s 0, and C F 5 5 ⬁ q 5 5 1 q 5 ⬘ 5 1 if < ␣ < s 0. The proof of Theorem 3.1 will appear in the last section. The proof of the following lemma closely follows that of Theorem 3 in w9x. 3.2. LEMMA. Let n G 3. Let S be as abo¨ e and let d Ž x . s Ž< x <. dx where g C0⬁Ž⺢. and assume that there is a constant A such that for all g ⺢n
⭸␣ ⭸ ␣
ˆ Ž . F A Ž 1 q < < .
y Ž ny1 .r2
,
whene¨ er < ␣ < F 1. Then for e¨ ery small ␦ ) 0 and r ) 0 with Ž n y 1.rn ) 1rr ) Ž n y 1.rn y ␦r2 there exists constant C independent of A such that 5 Mf 5 Lr Ž⺢ n . F CA b 5 f 5 Lr Ž⺢ n . , where Mf Ž x . s sup t ) 0 <Ž f ) t .Ž x .< and b s Ž n y 1.rn q ␦ .
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KYUNG SOO RIM
Now we will construct a Young’s function ⌽ which will be used later: We point out that this construct was found by Bak in w1, Theorem 3.1x. Let ␦ ) 0 be a sufficiently small number as in Lemma 3.2, and let r g Ž nrŽ n y 1., 2 nrŽ2Ž n y 1. y n ␦ .. In addition let d ) 1rŽ n y 1.. We take  ) 1 so that for all such r r
Ž 3.2. y
Ž 3.3.
2n ny1
ž
ny1 n
q
q ␦ F ,
/
2r d  Ž n y 1.
F
Ž n y 2. 2
.
We define a nondecreasing function on w0, ⬁. such that
¡0
Ž t. s
~
if 0 F t F 1 if 1 - t
strictly positive t d n y1
¢␥ ⬘ Ž t
yd
.
ny 1 2
if t is sufficiently large.

Then, for 0 - 2 rŽ d  Ž n y 1.. F 1r2 there is a constant c such that
Ž t. t ry1q
s
tyd
ny 1 2
2n 2r 2 .  Žy ny 1 q d  Ž ny 1 . q d  Ž ny 1 .
␥ ⬘ Ž tyd .
ny 1 2
is nondecreasing for t G c,

which follows from Ž3.1. with Ž3.3.. Consequently, satisfies the condition Ž2.1.. Moreover, assume that for r ) 1
Ž 3.4.
␥ ⬘ Ž rt . ␥ ⬘Ž t .
is nondecreasing for t ) 0.
Then also satisfies the condition Ž2.2.. Now thus we can define a Young’s function ⌽ as
Ž 3.5.
⌽ Ž u. s
u
H0 Ž t . dt.
For simplicity, S ' Ž x, ␥ Ž< x <.. : x g ⺢ ny 14 , and let d Ž x . s Ž< x <. dx where g C0⬁Ž⺢. is a nonnegative function with Ž t . s 1 if < t < F 1, and Ž t . s 0 if < t < G 2. 3.3. MAIN THEOREM. For n G 4 let S be gi¨ en as in Theorem 3.1 and assume Ž3.4.. If we consider ⌽ as Ž3.5., then there is a constant C such that 5 Mf 5 L⌽ Ž⺢ n . F C 5 f 5 L⌽ Ž⺢ n . , where f g S Ž⺢ n ..
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MAXIMAL FUNCTIONS OVER HYPERSURFACES
Proof. Let 0 g C0⬁ŽŽ2, 8.. be a nonnegative function such that 0 Ž2 k t . s 1 for t ) 0. Put k Ž t . s 0 Ž2 k t . and d k Ž x . s k Ž< x <. d Ž x . s k Ž< x <. Ž< x <. dx. Define
Ý⬁ks y⬁
Mk f Ž x . s sup < Ž f ) k , t . Ž x . < s sup t)0
t)0
HS f Ž x y t . d Ž . k
.
Then d s Ý⬁ks 1 d k , and Mf Ž x . F Ý⬁ks1 Mk f Ž x .. Since supp k ; Ž a k , 4 a k ., where a k s 2yk q1 , we can take s 1 in Theorem 3.1. Thus by Theorem 3.1 for < ␣ < F 1
⭸␣ $ y Ž ny1 .r2 Ž . F A k Ž 1 q < <. , ⭸ ␣ k where A k s CaŽkny1.r2y1r␥ ⬘Ž a kr2.Ž ny1.r2 . Here the constant C is independent of k since 5 kX 5 1 s 5 0X 5 1. By Lemma 3.2 we have 5 Mk f 5 Lr Ž⺢ n . F CAbk 5 f 5 Lr Ž⺢ n . ,
Ž 3.6.
where b s Ž n y 1.rn q ␦ and Ž n y 1.rn ) 1rr ) Ž n y 1.rn y ␦r2. Further since Cn, b ' 2 Ž ny1.yŽ ny3. b r2 ) 1, we have 5 Mk f 5 ⬁ F Ca kny 1 5 f 5 ⬁ F CCn , b a kny 1 5 f 5 ⬁ .
Ž 3.7.
We may assume that two constants C of Ž3.6. and Ž3.7. are equal to each other. We write G Ž t . s t Ž ny1.␥ ⬘Ž t .Ž ny1.r2 . Then Ž t . s ty1 G Ž tyd .y if t is sufficiently large. In this notation we use Interpolation Lemma 2.1. We ˜ Ž u. ' u Ž u. s GŽ uyd .y  for large u. Since is increasing, define ⌽ ˜ ˜ Ž u.. So ⌽ Ž u. f ⌽ ˜ Ž u. for large u. Consequently, ⌽ Ž ur2. F ⌽ Ž u. F ⌽ y1 Ž . y1 Ž . y1 Ž y1r  .y1r d ˜ ⌽ u f⌽ u sG u for large u. Further we define G␦ Ž t . s t Ž ny1.r byŽ ny3.r2␥ ⬘Ž t .Ž ny1.r2 . Then by Ž3.2., G␦ Ž t . b r r  G G␦ Ž t . G G Ž t . if t F 1. By Lemma 2.1, we obtain 5 Mk f 5 L⌽ F Ca kny1 ⌽y1
ž Ž A rŽ C b k
ny1 n , b ak
F C⬘a kny1 Gy1 Ž G␦ Ž a kr2 . F C⬘a kny1 Gy1 Ž G Ž a kr2 . .
r
..
/5 f 5
b rr  y1 rd
.
y1 rd
L⌽
5 f 5 L⌽
5 f 5 L⌽
F C⬘a kny 1y1r d 5 f 5 L⌽ . Therefore 5 Mf 5 L⌽ F
⬁
Ý ks0
5 Mk f 5 L⌽ F C
⬁
Ý 2yk Ž ny1y1r d . 5 f 5 L
⌽
F C 5 f 5 L⌽ ,
ks1
where the last convergence follows from d ) 1rŽ n y 1. of Ž3.2..
76
KYUNG SOO RIM
For example, for ␥ satisfying the hypothesis, the curves of exponential b type may be considered. The typical example of such a ␥ is ey1 r t , b ) 0, bd u near the origin. In this case the Young’s function is ⌽ Ž u. s e . There are also many other examples of ␥ Ž t . whose curvature vanishes to infinite order at the origin Žrefer to w1, Example 3.3x..
4. THE PROOF OF THEOREM 3.1 In this section we prove Theorem 3.1. Proof of Theorem 3.1. First we prove the case < ␣ < s 0. So we may take s with ) 0 Ž , 0, . . . , 1., where 0 - is large enough so that we can choose G a, ␥ ⬘Ž . s . Then < < f . Here the symbol f means the ratio of the expressions on either side is bounded between two positive constants depending only on . We may assume l s 0 for notational simplicity. Using polar coordinates we have
ˆ Ž . s
HS e
s
H⺢
s
H0 e
s
H0 e
i ² , x :
ny1
⬁
⬁
d Ž x .
e i ² , Ž x , ␥ Ž < x <..: Ž < x < . d Ž x .
i ␥ Ž r . ny2
Ž r.
r
HS
ny2
e i r x 1 d ny 2 Ž x . dr
$
i ␥ Ž r . ny2
Ž r . ny 2 Ž r Ž 1, 0, . . . , 0 . . dr ,
r
where d ny 2 is the standard Lebesgue measure on S ny 2 . From Lemma 2.3, the partition of unity yields $
ny 2 Ž . s H Ž Ž 1, 0, . . . , 0 . , . q H Ž Ž y1, 0, . . . , 0 . , y . s eyi F1 Ž . q e i F2 Ž . , where Fk Ž k s 1, 2. satisfies the estimates
Ž 4.1.
< FkŽ j. Ž . < F C j
Ž 4.2.
< FkŽ j. Ž . < F C j ym ,
for j G 0, m F Ž n y 2.r2 q j. Thus ˆ Ž . s I q II, where Is
⬁
H0 e
i ␥ Ž r . ny2
r
Ž r . eyi r F1 Ž r . dr ,
77
MAXIMAL FUNCTIONS OVER HYPERSURFACES
and II s
⬁
H0 e
i ␥ Ž r . ny2
Ž r . e i r F2 Ž r . dr .
r
Estimate of I. We write F s F1 and let Ž r . s ␥ Ž r . y ␥ Ž . y ␥ ⬘Ž . Ž r y .. Then Ž . s ⬘Ž . s 0 and ⬘Ž r . ) 0 for r ) . The assumption Ž3.1. implies that
⬘Ž r . r Ž ny1.r2
is strictly increasing for r )
and that if r F 2 , then since ␥ ⬘Ž r . G ␥ ⬘Ž . rr G 2␥ ⬘Ž . r Ž ny1.r2
Ž 4.3.
s
⬘Ž r .
r Ž ny1.r2
F
2 r Ž ny1.r2
␥ ⬘Ž r .
.
F Ž r . r ny2 Ž r . dr s
H0
␥ ⬘Ž r . y ␥ ⬘Ž .
Put e i Ž ␥ ⬘Ž . y ␥ Ž .. I s
⬁
H0 e
i Ž r .
2
q
⬁
H2 .
We estimate the second term. Set  s max a, 2 4 . By integration by parts ⬁
s
⬁
H2 H s
1 i y
s
e
1 i
i Ž r .
1
⬁
⬁
H
d dr
Ž e i Ž r . . F Ž r .
F Ž r .
He i 
i Ž r .
d dr
r Ž ny1.r2
⬘Ž r .
ž
r Ž ny1.r2
⬘Ž r .
r Ž ny3.r2 Ž r . dr
⬁
r
Ž r.
Ž ny3.r2
F Ž r .
r Ž ny1.r2
⬘Ž r .

r Ž ny3.r2 Ž r . dr
/
' I1 q I2 . By Ž4.2. with m s Ž n y 3.r2, j s 0, and Ž4.3., Ž3.1. < I1 < F C0 yŽ ny1.r2yŽ ny3.r2
 Ž ny1.r2 ⬘Ž  .
F 2C0 5 5 ⬁ yŽ ny1.r2␥ ⬘ Ž . F 2C0 5 5 ⬁ yŽ ny1.r2
Ž .
y Ž ny3 .r2
␥ ⬘ Ž a.
aŽ ny1.r2
␥ ⬘ Ž a.
aŽ ny1.r2
Ž ny1 .r2
,
78
KYUNG SOO RIM
where the last equality follows from ␥ ⬘Ž a. F ␥ ⬘Ž .. We decompose I2 into three parts as < I2 < F
1
⬁
H
q q q
< F⬘ Ž r . <
1
⬁
⬁
< F Ž r . <
H 
ny3 1
⬁
H 
2
r Ž ny3.r2 Ž r . dr
⬘Ž r .
d r Ž ny1.r2
< F Ž r . <
H  1
r Ž ny1.r2
r Ž ny3.r2 Ž r . dr
dr ⬘ Ž r . r Ž ny1.r2
⬘Ž r .
r Ž ny3.r2 ⬘ Ž r . dr
r Ž ny1.r2
< F Ž r . <
⬘Ž r .
r Ž ny3.r2y1 Ž r . dr
' I3 q I4 q I5 q I6 . By Ž4.2. with m s Ž n y 2.r2, j s 1, and Ž4.3., Ž3.1. I3 F
 Ž ny1.r2
␥ ⬘Ž .
Ž ny1.r2
F 2 5 5⬁
⬁
Ž ny3 .r2
H < F⬘ Ž t . < t ⬘ Ž  . 
aŽ ny1.r2
␥ ⬘ Ž a.
Ž ny1.r2
F 2C1 5 5 ⬁ F 4C1 5 5 ⬁
⬁
Ž ny1 .r2
H < F⬘ Ž t . < t
aŽ ny1.r2
␥ ⬘ Ž a.
Ž ny1.r2
ž
Ž ny1 .r2
aŽ ny1.r2
Ž ny1.r2␥ ⬘ Ž a .
1q
⬁
Ž tr Ž . . dt
Ž ny3.r2
Ž ny3.r2
y3 r2
H1 t
dt
dt
/
.
Ž ny1 .r2
By Ž4.2. with m s Ž n y 3.r2, j s 0, and Ž4.3., Ž3.1. I4 F C0 5 5 ⬁ s C0 5 5 ⬁ s C0 5 5 ⬁
1
Ž ny1.r2␥ ⬘ Ž a .
⬁
Ž ny3 .r2
1
Ž ny1.r2␥ ⬘ Ž a .
H
dr ⬘ Ž r . ⬁
Ž ny3 .r2
d r Ž ny1.r2
H
d r Ž ny1.r2 dr ⬘ Ž r .
 Ž ny1.r2 Ž ny1.r2␥ ⬘ Ž a .
s 2C0 5 5 ⬁
Ž ny3 .r2
⬘Ž  .
aŽ ny1.r2
Ž ny1.r2␥ ⬘ Ž a .
Ž ny1 .r2
,
dr
dr
MAXIMAL FUNCTIONS OVER HYPERSURFACES
79
where the equalityrinequality follows from the fact that r Ž ny1.r2r ⬘Ž r . is strictly monotone in Ž  , ⬁.. Likewise we can estimate I5 and I6 , I5 F 2C0 5 ⬘ 5 1
aŽ ny1.r2
Ž ny1.r2␥ ⬘ Ž a .
,
Ž ny1 .r2
and I6 F 2C0 5 5 1
aŽ ny1.r2y1 Ž ny1 .r2
Ž ny1.r2␥ ⬘ Ž a .
.
Thus aŽ ny3.r2
< I < F CC
␥ ⬘ Ž a.
Ž ny1 .r2
yŽ ny1.r2 ,
where C s 5 5 ⬁ q 5 ⬘ 5 1. Next we estimate H02 . We assume a - 2 since otherwise the integral vanishes. We put LŽ r . s
r
Ha e
i Ž s.
ds.
By differentiation of Ž3.1., ⬙ Ž r . s ␥ ⬙ Ž r . G Ž n y 1.r2␥ ⬘Ž r .rr G Ž n y 1.r2␥ ⬘Ž a.ra for r G a. Van der Corput’s lemma yields
Ž 4.4.
< LŽ r . < F C
(
a
␥ ⬘ Ž a.
y1 r2
Ž r G a. .
By integration by parts 2
Ha
2
s L Ž r . F Ž r . r
Ž r.
y
ny 2
a
2
Ha
LŽ r .
d dx
Ž F Ž r . r ny2 Ž r . . dr
' I 7 q I8 . By Ž4.2. with m s Ž n y 2.r2, j s 0, and Ž4.3., Ž3.1. < I7 < F CC0 5 5 ⬁
( (
F C⬘C0 5 5 ⬁ s C⬘C0 5 5 ⬁
Ž ny2.r2
a
␥ ⬘ Ž a . Ž ny2.r2␥ ⬘ Ž . Ž ny 2.r2 aŽ ny2.r2
a
␥ ⬘ Ž a . Ž ny2.r2␥ ⬘ Ž ar2. Ž ny 2.r2 aŽ ny1.r2
␥ ⬘ Ž ar2.
Ž ny1 .r2
yŽ ny1.r2 .
80
KYUNG SOO RIM
Similarly < I8 < F Ž n y 2 . q
2
H0
q
2
H0
(
FC
2
H0
< L Ž r . F Ž r . < r ny 3 Ž r . dr
< L Ž r . F Ž r . < r ny 2 < ⬘ Ž r . < dr < L Ž r . F⬘ Ž r . < r ny 2 Ž r . dr a
␥ ⬘ Ž a .
ž
5 5⬁ q
2
H0
2
H0
Ž r . y
(
␥ ⬘ Ž a .
ž
5 5⬁
Ž Ž r . y Ž ny2.r2 r ny2 . H0
< ⬘ Ž r . < dr
/
Ž ny2.r2 Ž ny2.r2 ␥ ⬘ Ž .
q5 ⬘ 5 1
Ž ny 2 .r2
Ž ny2.r2 Ž ny2.r2␥ ⬘ Ž .
aŽ ny1.r2
␥ ⬘ Ž ar2.
Ž ny1 .r2
Ž ny2 .r2
Ž ny2.r2
q2r Ž Ž n y 2 . . 5 5 ⬁ s C⬙ Ž 5 5 ⬁ q 5 ⬘ 5 1 .
dr
2
rg Ž0, 2 .
a
r
< L Ž r . F Ž r . < r ny 2 < ⬘ Ž r . < dr
q sup
F C⬘
Ž ny2 .r2 ny3
Ž ny2.r2␥ ⬘ Ž .
Ž ny2 .r2
/
yŽ ny1.r2 .
This completes the estimate of I. Estimate of II. We only sketch the estimate of II since it is very similar to I. We split II into two parts as that of I,
II s
2
H0
q
⬁
H2 .
As the estimate of I we integrate the second term with the notation
81
MAXIMAL FUNCTIONS OVER HYPERSURFACES
 s max a, 2 4 . Then ⬁
H2
s
s
1 i 1 i q
⬁
H e
d
Ž e i Ž␥ Ž r .q r . . F Ž r .
dr
i Ž ␥ Ž r .q r .
1
⬁
He i 
F Ž r .
␥ ⬘Ž r . q
␥ ⬘Ž r . q
dr
ž
F Ž r .
dr
⬁
r ny2 Ž r .
d
i Ž ␥ Ž r .q r .
r ny 2 Ž r .

r ny 2 Ž r .
␥ ⬘Ž r . q
/
dr
' II1 q II2 . The term II1 can be estimated as before, whereas II2 has a non-monotone function, r Ž ny1.r2rŽ␥ ⬘Ž r . q .. However, it is written as the product of two monotone functions r Ž ny 1.r2 r ⬘Ž r . and 1 y 2 r Ž ␥ ⬘Ž r . q . s ⬘Ž r .rŽ␥ ⬘Ž r . q .. So by the same arguments for I1 and I2 , we can proceed to the goal. On the other hand, defining LŽ r . s
r
H0 e
i Ž ␥ Ž s.q r .
ds,
we obtain the desired estimate for the first term H02 with the methods for I7 and I8 . Now suppose < ␣ < s 1. Let s with s Ž1 , . . . , ny1 , 1. ' Ž ⬘, 1. and < ⬘ < s . If ␣ s Ž1, 0, . . . , 0. or Ž0, . . . , 0, 1. Žit is enough., then
Ž 4.5. Ž 4.6.
⭸ ⭸ 1 ⭸ ⭸ n
⬁
ˆ Ž . s i
¨ˆŽ
H0 e
i ␥ Ž r . ny2
r
Ž r.
HS
ny2
e i r² ⬘r , x : rx 1 d ny2 Ž x . dr ,
⬁
. s iH e i ␥ Ž r . r ny2␥ Ž r . Ž r . H ny2e i r² ⬘r , x : d ny 2 Ž x . dr . 0
S
Considering the new cutoff function as r Ž r . and the surface area measure x 1 d ny2 Ž x . s d Ž x ., we estimate Ž4.5. by analogous ways for I, II. Also we estimate Ž4.6. with the new cutoff function ␥ Ž r . Ž r ..
REFERENCES 1. J.-G. Bak, Averages over surfaces with infinitely flat points, J. Funct. Anal. 129 Ž1995., 455᎐470. 2. J. Bourgain, Averages in the plane over convex curves and maximal operators, J. Anal. Math. 47 Ž1986., 69᎐85.
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KYUNG SOO RIM
3. J.-G. Bak, D. McMichael, J. Vance, and S. Wainger, Fourier transforms of surface area measure on convex surfaces in ⺢ 3 , Amer. J. Math. 111 Ž1989., 633᎐668. 4. J. Bruna, A. Nagel, and S. Wainger, Convex hypersurfaces and Fourier transforms, Ann. Math. 127 Ž1988., 333᎐365. 5. M. Cowling and G. Mauceri, Inequalities for some maximal functions, II, Trans. Amer. Math. Soc. 296 Ž1986., 341᎐365. 6. A. Greenleaf, Principal curvature and harmonic analysis, Indiana Math. J. 30 Ž1981., 519᎐537. 7. A. Nagel, A. Seeger, and S. Waingeri, Averages over convex hypersurfaces, Amer. J. Math. 115 Ž1993., 903᎐927. 8. E. M. Stein, Maximal functions: Spherical means, Proc. Natl. Acad. Sci. U.S. A. 73 Ž1976., 2174᎐2175. 9. C. Sogge and E. M. Stein, Averages of functions over hypersurfaces in ⺢ n , In¨ ent. Math. 82 Ž1985., 543᎐556. 10. E. M. Stein and S. Wainger, Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc. 84 Ž1978., 1239᎐1295. 11. A. Zygmund, ‘‘Trigonometric Series,’’ 2nd ed., Vol. 2, Cambridge Univ. Press, Cambridge, UK, 1968.
Maximal Functions over Hypersurfaces with Flat Points Kyung Soo Rim School of Mathematics, Korea Institute for Ad¨ anced Study, Seoul 130-012, Korea E-mail: [email protected] Submitted by William F. Ames Received June 15, 1999
For n G 4, let S be a flat hypersurface in ⺢ n , and let d s d , where g C⬁0 Ž⺢ n . and is the surface area measure on S. Then the maximal functions Mf associated to S and by Mf Ž x . s sup t ) 0 < HS f Ž x y t . d Ž .<, f g S Ž⺢ n ., are bounded on certain Orlicz spaces L⌽ Ž⺢ n .. 䊚 2001 Academic Press Key Words: maximal function; flat hypersurface.
1. INTRODUCTION For n G 2, let S be a surface in ⺢ n, and put d s d , where g C0⬁Ž⺢ n . and is the surface area measure on S. Then the maximal function M can be defined by Mf Ž x . s sup < Ž f ) t . Ž x . < s sup t)0
t)0
HS f Ž x y t . d Ž .
,
for f g S Ž⺢ n ., the Schwartz space. When S is a sphere in ⺢ n Ž n G 3., Stein proved that M is bounded on pŽ n. L ⺢ if and only if p ) nrŽ n y 1. w8, 10x. Later Bourgain obtained the corresponding result for the circular maximal function over convex curves in ⺢ 2 w2x. Beginning with two articles the problem has been extensively studied to the general manifolds S. We say that S is of finite type if at every point x g S, every one dimensional tangent line to S at x makes finite order of contact with S; i.e., the Gaussian curvature vanishes at most finite order at every point of S. We say that S is flat if S contains at least one point at which the Gaussian curvature vanishes to infinite order. 70 0022-247Xr01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
MAXIMAL FUNCTIONS OVER HYPERSURFACES
71
For a finite type S, it is known that M is bounded on L p Ž⺢ n . Ž n G 3. for a sufficiently large p w5᎐7, 9x. However, for a flat surface S, there is no L p-continuity of M for any p w9x. A few years ago Bak proved that only for n s 3 is M equipped with a flat hypersurface S bounded on L⌽ Ž⺢ 3 ., the Orlicz space associated to a Young’s function ⌽ given by an exponential r type ⌽ Ž t . s e t w1x. Bak and many other authors have guessed that his result is true in higher dimension G4. In this paper we will show the extended version of Bak’s result to n G 4.
2. PRELIMINARIES We state the Orlicz interpolation theorem which was proved by Bak in w1, Lemma 1.1x. Let Ž X, M1 , . and Ž Y, M2 , . be measure spaces, where and are positive -finite measures, and let T be a sublinear operator defined on a suitable linear space of functions f on X such that Tf is a measurable function on Y. For Young’s function ⌽ the Orlicz space L⌽ Ž d . is equipped with the norm 5 f 5 L⌽ Ž d . s inf s ) 0 :
½
H⌽ Ž < f Ž x .
The inverse of ⌽ is defined for t g w0, ⬁. by ⌽y1 Ž t . s inf s ) 0 : ⌽ Ž s . ) t 4 . 2.1. INTERPOLATION LEMMA. Let r g w1, ⬁.. Assume that there exists a constant C such that for all t ) 0 r
Ž x : < Tf Ž x . < ) t 4 . F Ž C 5 f 5 rrt . , 5 Tf 5 ⬁ F C 5 f 5 ⬁ . Suppose a Young’s function ⌽ is gi¨ en by ⌽ Ž s . s H0s Ž t . dt, where : w0, ⬁. ª w0, ⬁. is a nondecreasing function such that Ž t . s 0 for 0 F t F 1, and Ž t . ) 0 for t ) 1 such that there exist constants c ) 1, C0 , and C1 such that u
Ž 2.1.
H1 Ž t .
t r dt F C0 Ž u . ru ry1
for u ) 1,
and for e¨ ery ) 1
Ž 2.2.
C1 Ž t . r Ž t . G Ž .
for t G c.
72
KYUNG SOO RIM
Then there exists a constant C s C Ž ⌽, r . such that 5 Tf 5 L⌽ Ž d . F CB⌽y1 Ž Ž ArB .
r
.5 f 5L
⌽
Žd. .
2.2. COROLLARY. Let T and Tk be sublinear operators such that < Tf Ž x .< F Ý⬁ks 1 < Tk f Ž x .< a.e. If for some r g w1, ⬁. and k s 1, 2, . . . r
Ž x : < Tk f Ž x . < ) t 4 . F Ž A k 5 f 5 rrt . ,
for e¨ ery t ,
5 Tk f 5 ⬁ F Bk 5 f 5 ⬁ , then 5 Tf 5 L⌽ Ž d . F C 5 f 5 L⌽ Ž d . , where C s Ý⬁ks 1 Bk⌽y1 ŽŽ A krBk . r . - ⬁. Now we state the result of Bruna et al. w4x: 2.3. LEMMA. Suppose S ; ⺢ n is a compact hypersurface of finite type bounding a con¨ ex domain. Let g ⺢ n with < < s 1, and let x 0 g S with orthogonal to Tx 0 , the affine tangent line to S at x 0 . Let g C⬁Ž⺢ n . be supported in a small neighborhood of the origin. Put H Ž x 0 , . s
HS e
i ² x , :
Ž x y x0 . d Ž x . ,
where is the induced surface area measure on S. Then there are constants C j depending only on the surface S such that H Ž x 0 , . s e i ² x 0 , : F Ž . , where < F Ž j. Ž . < F C j yj B˜Ž x 0 , <
ž
/
where B˜Ž x 0 , <
Ž 3.1.
␥ ⬘Ž t . t Ž ny1.r2
is nondecreasing for f ) 0,
73
MAXIMAL FUNCTIONS OVER HYPERSURFACES
and also we may assume ␥ ⬘Ž1. s 1. The hypersurface S ; ⺢ n is defined by S s Ž x, l q ␥ Ž< x <.. : x g ⺢ ny 14 for some l g ⺢. Suppose ␥ satisfies Ž3.1. and let m F Ž n y 1.r2 Ž m may be negative.. Then for s G t ) 0, we get
␥ ⬘Ž s . s
m
y
␥ ⬘Ž t . t
m
s
␥ ⬘Ž s . s
Ž ny1.r2
y
␥ ⬘Ž t . m Ž ny1.rŽ2 m.
s t
G
␥ ⬘Ž s . s
Ž ny1.r2
y
␥ ⬘Ž t . t Ž ny1.r2
G 0.
Thus ␥ ⬘Ž t .rt m is also nondecreasing for all m F Ž n y 1.r2. Let us note that Ž3.1. implies ␥ ⬙ Ž t . G Ž n y 1.r2␥ ⬘Ž t .rt ) 0. The fact that the Gaussian curvature K Ž x . of S at x is a constant times ␥ ⬙ Ž< x <.Ž␥ ⬘Ž< x <.r< x <. ny 2 yields that the Gaussian curvature of S can only vanish at the origin. To prove the main theorem stated below, the following estimate is very crucial. In fact, many authors have guessed the decaying order Ž n y 1.r2. Nevertheless, the result has a meaning because the proof of the main theorem mainly depends on the exact constants C and aŽ ny1.r2y1r ␥ ⬘Ž ar2.Ž ny1.r2 . 3.1. THEOREM. Let n G 4 and let be a nonnegati¨ e number. Suppose g C01 ŽŽ0, ⬁.. is a nonnegati¨ e function compactly supported in the inter¨ al Ž a, ⬁. such that a F . Let ␥ and S be as abo¨ e. Let be the measure on the surface S such that d Ž x, ␥ Ž< x <.. s Ž< x <. dx. Then for a multi-index ␣ with < ␣ < F 1 there is a constant C depending only on such that
⭸␣ ⭸ ␣
ˆ Ž . F C C
aŽ ny1.r2y1
␥ ⬘ Ž ar2.
Ž ny1 .r2
Ž 1 q < < . y ny1 r2 , Ž
.
where C F 5 5 ⬁ q 5 ⬘ 5 1 if < ␣ < s 0, and C F 5 5 ⬁ q 5 5 1 q 5 ⬘ 5 1 if < ␣ < s 0. The proof of Theorem 3.1 will appear in the last section. The proof of the following lemma closely follows that of Theorem 3 in w9x. 3.2. LEMMA. Let n G 3. Let S be as abo¨ e and let d Ž x . s Ž< x <. dx where g C0⬁Ž⺢. and assume that there is a constant A such that for all g ⺢n
⭸␣ ⭸ ␣
ˆ Ž . F A Ž 1 q < < .
y Ž ny1 .r2
,
whene¨ er < ␣ < F 1. Then for e¨ ery small ␦ ) 0 and r ) 0 with Ž n y 1.rn ) 1rr ) Ž n y 1.rn y ␦r2 there exists constant C independent of A such that 5 Mf 5 Lr Ž⺢ n . F CA b 5 f 5 Lr Ž⺢ n . , where Mf Ž x . s sup t ) 0 <Ž f ) t .Ž x .< and b s Ž n y 1.rn q ␦ .
74
KYUNG SOO RIM
Now we will construct a Young’s function ⌽ which will be used later: We point out that this construct was found by Bak in w1, Theorem 3.1x. Let ␦ ) 0 be a sufficiently small number as in Lemma 3.2, and let r g Ž nrŽ n y 1., 2 nrŽ2Ž n y 1. y n ␦ .. In addition let d ) 1rŽ n y 1.. We take  ) 1 so that for all such r r
Ž 3.2. y
Ž 3.3.
2n ny1
ž
ny1 n
q
q ␦ F ,
/
2r d  Ž n y 1.
F
Ž n y 2. 2
.
We define a nondecreasing function on w0, ⬁. such that
¡0
Ž t. s
~
if 0 F t F 1 if 1 - t
strictly positive t d n y1
¢␥ ⬘ Ž t
yd
.
ny 1 2
if t is sufficiently large.

Then, for 0 - 2 rŽ d  Ž n y 1.. F 1r2 there is a constant c such that
Ž t. t ry1q
s
tyd
ny 1 2
2n 2r 2 .  Žy ny 1 q d  Ž ny 1 . q d  Ž ny 1 .
␥ ⬘ Ž tyd .
ny 1 2
is nondecreasing for t G c,

which follows from Ž3.1. with Ž3.3.. Consequently, satisfies the condition Ž2.1.. Moreover, assume that for r ) 1
Ž 3.4.
␥ ⬘ Ž rt . ␥ ⬘Ž t .
is nondecreasing for t ) 0.
Then also satisfies the condition Ž2.2.. Now thus we can define a Young’s function ⌽ as
Ž 3.5.
⌽ Ž u. s
u
H0 Ž t . dt.
For simplicity, S ' Ž x, ␥ Ž< x <.. : x g ⺢ ny 14 , and let d Ž x . s Ž< x <. dx where g C0⬁Ž⺢. is a nonnegative function with Ž t . s 1 if < t < F 1, and Ž t . s 0 if < t < G 2. 3.3. MAIN THEOREM. For n G 4 let S be gi¨ en as in Theorem 3.1 and assume Ž3.4.. If we consider ⌽ as Ž3.5., then there is a constant C such that 5 Mf 5 L⌽ Ž⺢ n . F C 5 f 5 L⌽ Ž⺢ n . , where f g S Ž⺢ n ..
75
MAXIMAL FUNCTIONS OVER HYPERSURFACES
Proof. Let 0 g C0⬁ŽŽ2, 8.. be a nonnegative function such that 0 Ž2 k t . s 1 for t ) 0. Put k Ž t . s 0 Ž2 k t . and d k Ž x . s k Ž< x <. d Ž x . s k Ž< x <. Ž< x <. dx. Define
Ý⬁ks y⬁
Mk f Ž x . s sup < Ž f ) k , t . Ž x . < s sup t)0
t)0
HS f Ž x y t . d Ž . k
.
Then d s Ý⬁ks 1 d k , and Mf Ž x . F Ý⬁ks1 Mk f Ž x .. Since supp k ; Ž a k , 4 a k ., where a k s 2yk q1 , we can take s 1 in Theorem 3.1. Thus by Theorem 3.1 for < ␣ < F 1
⭸␣ $ y Ž ny1 .r2 Ž . F A k Ž 1 q < <. , ⭸ ␣ k where A k s CaŽkny1.r2y1r␥ ⬘Ž a kr2.Ž ny1.r2 . Here the constant C is independent of k since 5 kX 5 1 s 5 0X 5 1. By Lemma 3.2 we have 5 Mk f 5 Lr Ž⺢ n . F CAbk 5 f 5 Lr Ž⺢ n . ,
Ž 3.6.
where b s Ž n y 1.rn q ␦ and Ž n y 1.rn ) 1rr ) Ž n y 1.rn y ␦r2. Further since Cn, b ' 2 Ž ny1.yŽ ny3. b r2 ) 1, we have 5 Mk f 5 ⬁ F Ca kny 1 5 f 5 ⬁ F CCn , b a kny 1 5 f 5 ⬁ .
Ž 3.7.
We may assume that two constants C of Ž3.6. and Ž3.7. are equal to each other. We write G Ž t . s t Ž ny1.␥ ⬘Ž t .Ž ny1.r2 . Then Ž t . s ty1 G Ž tyd .y if t is sufficiently large. In this notation we use Interpolation Lemma 2.1. We ˜ Ž u. ' u Ž u. s GŽ uyd .y  for large u. Since is increasing, define ⌽ ˜ ˜ Ž u.. So ⌽ Ž u. f ⌽ ˜ Ž u. for large u. Consequently, ⌽ Ž ur2. F ⌽ Ž u. F ⌽ y1 Ž . y1 Ž . y1 Ž y1r  .y1r d ˜ ⌽ u f⌽ u sG u for large u. Further we define G␦ Ž t . s t Ž ny1.r byŽ ny3.r2␥ ⬘Ž t .Ž ny1.r2 . Then by Ž3.2., G␦ Ž t . b r r  G G␦ Ž t . G G Ž t . if t F 1. By Lemma 2.1, we obtain 5 Mk f 5 L⌽ F Ca kny1 ⌽y1
ž Ž A rŽ C b k
ny1 n , b ak
F C⬘a kny1 Gy1 Ž G␦ Ž a kr2 . F C⬘a kny1 Gy1 Ž G Ž a kr2 . .
r
..
/5 f 5
b rr  y1 rd
.
y1 rd
L⌽
5 f 5 L⌽
5 f 5 L⌽
F C⬘a kny 1y1r d 5 f 5 L⌽ . Therefore 5 Mf 5 L⌽ F
⬁
Ý ks0
5 Mk f 5 L⌽ F C
⬁
Ý 2yk Ž ny1y1r d . 5 f 5 L
⌽
F C 5 f 5 L⌽ ,
ks1
where the last convergence follows from d ) 1rŽ n y 1. of Ž3.2..
76
KYUNG SOO RIM
For example, for ␥ satisfying the hypothesis, the curves of exponential b type may be considered. The typical example of such a ␥ is ey1 r t , b ) 0, bd u near the origin. In this case the Young’s function is ⌽ Ž u. s e . There are also many other examples of ␥ Ž t . whose curvature vanishes to infinite order at the origin Žrefer to w1, Example 3.3x..
4. THE PROOF OF THEOREM 3.1 In this section we prove Theorem 3.1. Proof of Theorem 3.1. First we prove the case < ␣ < s 0. So we may take s with ) 0 Ž , 0, . . . , 1., where 0 - is large enough so that we can choose G a, ␥ ⬘Ž . s . Then < < f . Here the symbol f means the ratio of the expressions on either side is bounded between two positive constants depending only on . We may assume l s 0 for notational simplicity. Using polar coordinates we have
ˆ Ž . s
HS e
s
H⺢
s
H0 e
s
H0 e
i ² , x :
ny1
⬁
⬁
d Ž x .
e i ² , Ž x , ␥ Ž < x <..: Ž < x < . d Ž x .
i ␥ Ž r . ny2
Ž r.
r
HS
ny2
e i r x 1 d ny 2 Ž x . dr
$
i ␥ Ž r . ny2
Ž r . ny 2 Ž r Ž 1, 0, . . . , 0 . . dr ,
r
where d ny 2 is the standard Lebesgue measure on S ny 2 . From Lemma 2.3, the partition of unity yields $
ny 2 Ž . s H Ž Ž 1, 0, . . . , 0 . , . q H Ž Ž y1, 0, . . . , 0 . , y . s eyi F1 Ž . q e i F2 Ž . , where Fk Ž k s 1, 2. satisfies the estimates
Ž 4.1.
< FkŽ j. Ž . < F C j
Ž 4.2.
< FkŽ j. Ž . < F C j ym ,
for j G 0, m F Ž n y 2.r2 q j. Thus ˆ Ž . s I q II, where Is
⬁
H0 e
i ␥ Ž r . ny2
r
Ž r . eyi r F1 Ž r . dr ,
77
MAXIMAL FUNCTIONS OVER HYPERSURFACES
and II s
⬁
H0 e
i ␥ Ž r . ny2
Ž r . e i r F2 Ž r . dr .
r
Estimate of I. We write F s F1 and let Ž r . s ␥ Ž r . y ␥ Ž . y ␥ ⬘Ž . Ž r y .. Then Ž . s ⬘Ž . s 0 and ⬘Ž r . ) 0 for r ) . The assumption Ž3.1. implies that
⬘Ž r . r Ž ny1.r2
is strictly increasing for r )
and that if r F 2 , then since ␥ ⬘Ž r . G ␥ ⬘Ž . rr G 2␥ ⬘Ž . r Ž ny1.r2
Ž 4.3.
s
⬘Ž r .
r Ž ny1.r2
F
2 r Ž ny1.r2
␥ ⬘Ž r .
.
F Ž r . r ny2 Ž r . dr s
H0
␥ ⬘Ž r . y ␥ ⬘Ž .
Put e i Ž ␥ ⬘Ž . y ␥ Ž .. I s
⬁
H0 e
i Ž r .
2
q
⬁
H2 .
We estimate the second term. Set  s max a, 2 4 . By integration by parts ⬁
s
⬁
H2 H s
1 i y
s
e
1 i
i Ž r .
1
⬁
⬁
H
d dr
Ž e i Ž r . . F Ž r .
F Ž r .
He i 
i Ž r .
d dr
r Ž ny1.r2
⬘Ž r .
ž
r Ž ny1.r2
⬘Ž r .
r Ž ny3.r2 Ž r . dr
⬁
r
Ž r.
Ž ny3.r2
F Ž r .
r Ž ny1.r2
⬘Ž r .

r Ž ny3.r2 Ž r . dr
/
' I1 q I2 . By Ž4.2. with m s Ž n y 3.r2, j s 0, and Ž4.3., Ž3.1. < I1 < F C0 yŽ ny1.r2yŽ ny3.r2
 Ž ny1.r2 ⬘Ž  .
F 2C0 5 5 ⬁ yŽ ny1.r2␥ ⬘ Ž . F 2C0 5 5 ⬁ yŽ ny1.r2
Ž .
y Ž ny3 .r2
␥ ⬘ Ž a.
aŽ ny1.r2
␥ ⬘ Ž a.
aŽ ny1.r2
Ž ny1 .r2
,
78
KYUNG SOO RIM
where the last equality follows from ␥ ⬘Ž a. F ␥ ⬘Ž .. We decompose I2 into three parts as < I2 < F
1
⬁
H
q q q
< F⬘ Ž r . <
1
⬁
⬁
< F Ž r . <
H 
ny3 1
⬁
H 
2
r Ž ny3.r2 Ž r . dr
⬘Ž r .
d r Ž ny1.r2
< F Ž r . <
H  1
r Ž ny1.r2
r Ž ny3.r2 Ž r . dr
dr ⬘ Ž r . r Ž ny1.r2
⬘Ž r .
r Ž ny3.r2 ⬘ Ž r . dr
r Ž ny1.r2
< F Ž r . <
⬘Ž r .
r Ž ny3.r2y1 Ž r . dr
' I3 q I4 q I5 q I6 . By Ž4.2. with m s Ž n y 2.r2, j s 1, and Ž4.3., Ž3.1. I3 F
 Ž ny1.r2
␥ ⬘Ž .
Ž ny1.r2
F 2 5 5⬁
⬁
Ž ny3 .r2
H < F⬘ Ž t . < t ⬘ Ž  . 
aŽ ny1.r2
␥ ⬘ Ž a.
Ž ny1.r2
F 2C1 5 5 ⬁ F 4C1 5 5 ⬁
⬁
Ž ny1 .r2
H < F⬘ Ž t . < t
aŽ ny1.r2
␥ ⬘ Ž a.
Ž ny1.r2
ž
Ž ny1 .r2
aŽ ny1.r2
Ž ny1.r2␥ ⬘ Ž a .
1q
⬁
Ž tr Ž . . dt
Ž ny3.r2
Ž ny3.r2
y3 r2
H1 t
dt
dt
/
.
Ž ny1 .r2
By Ž4.2. with m s Ž n y 3.r2, j s 0, and Ž4.3., Ž3.1. I4 F C0 5 5 ⬁ s C0 5 5 ⬁ s C0 5 5 ⬁
1
Ž ny1.r2␥ ⬘ Ž a .
⬁
Ž ny3 .r2
1
Ž ny1.r2␥ ⬘ Ž a .
H
dr ⬘ Ž r . ⬁
Ž ny3 .r2
d r Ž ny1.r2
H
d r Ž ny1.r2 dr ⬘ Ž r .
 Ž ny1.r2 Ž ny1.r2␥ ⬘ Ž a .
s 2C0 5 5 ⬁
Ž ny3 .r2
⬘Ž  .
aŽ ny1.r2
Ž ny1.r2␥ ⬘ Ž a .
Ž ny1 .r2
,
dr
dr
MAXIMAL FUNCTIONS OVER HYPERSURFACES
79
where the equalityrinequality follows from the fact that r Ž ny1.r2r ⬘Ž r . is strictly monotone in Ž  , ⬁.. Likewise we can estimate I5 and I6 , I5 F 2C0 5 ⬘ 5 1
aŽ ny1.r2
Ž ny1.r2␥ ⬘ Ž a .
,
Ž ny1 .r2
and I6 F 2C0 5 5 1
aŽ ny1.r2y1 Ž ny1 .r2
Ž ny1.r2␥ ⬘ Ž a .
.
Thus aŽ ny3.r2
< I < F CC
␥ ⬘ Ž a.
Ž ny1 .r2
yŽ ny1.r2 ,
where C s 5 5 ⬁ q 5 ⬘ 5 1. Next we estimate H02 . We assume a - 2 since otherwise the integral vanishes. We put LŽ r . s
r
Ha e
i Ž s.
ds.
By differentiation of Ž3.1., ⬙ Ž r . s ␥ ⬙ Ž r . G Ž n y 1.r2␥ ⬘Ž r .rr G Ž n y 1.r2␥ ⬘Ž a.ra for r G a. Van der Corput’s lemma yields
Ž 4.4.
< LŽ r . < F C
(
a
␥ ⬘ Ž a.
y1 r2
Ž r G a. .
By integration by parts 2
Ha
2
s L Ž r . F Ž r . r
Ž r.
y
ny 2
a
2
Ha
LŽ r .
d dx
Ž F Ž r . r ny2 Ž r . . dr
' I 7 q I8 . By Ž4.2. with m s Ž n y 2.r2, j s 0, and Ž4.3., Ž3.1. < I7 < F CC0 5 5 ⬁
( (
F C⬘C0 5 5 ⬁ s C⬘C0 5 5 ⬁
Ž ny2.r2
a
␥ ⬘ Ž a . Ž ny2.r2␥ ⬘ Ž . Ž ny 2.r2 aŽ ny2.r2
a
␥ ⬘ Ž a . Ž ny2.r2␥ ⬘ Ž ar2. Ž ny 2.r2 aŽ ny1.r2
␥ ⬘ Ž ar2.
Ž ny1 .r2
yŽ ny1.r2 .
80
KYUNG SOO RIM
Similarly < I8 < F Ž n y 2 . q
2
H0
q
2
H0
(
FC
2
H0
< L Ž r . F Ž r . < r ny 3 Ž r . dr
< L Ž r . F Ž r . < r ny 2 < ⬘ Ž r . < dr < L Ž r . F⬘ Ž r . < r ny 2 Ž r . dr a
␥ ⬘ Ž a .
ž
5 5⬁ q
2
H0
2
H0
Ž r . y
(
␥ ⬘ Ž a .
ž
5 5⬁
Ž Ž r . y Ž ny2.r2 r ny2 . H0
< ⬘ Ž r . < dr
/
Ž ny2.r2 Ž ny2.r2 ␥ ⬘ Ž .
q5 ⬘ 5 1
Ž ny 2 .r2
Ž ny2.r2 Ž ny2.r2␥ ⬘ Ž .
aŽ ny1.r2
␥ ⬘ Ž ar2.
Ž ny1 .r2
Ž ny2 .r2
Ž ny2.r2
q2r Ž Ž n y 2 . . 5 5 ⬁ s C⬙ Ž 5 5 ⬁ q 5 ⬘ 5 1 .
dr
2
rg Ž0, 2 .
a
r
< L Ž r . F Ž r . < r ny 2 < ⬘ Ž r . < dr
q sup
F C⬘
Ž ny2 .r2 ny3
Ž ny2.r2␥ ⬘ Ž .
Ž ny2 .r2
/
yŽ ny1.r2 .
This completes the estimate of I. Estimate of II. We only sketch the estimate of II since it is very similar to I. We split II into two parts as that of I,
II s
2
H0
q
⬁
H2 .
As the estimate of I we integrate the second term with the notation
81
MAXIMAL FUNCTIONS OVER HYPERSURFACES
 s max a, 2 4 . Then ⬁
H2
s
s
1 i 1 i q
⬁
H e
d
Ž e i Ž␥ Ž r .q r . . F Ž r .
dr
i Ž ␥ Ž r .q r .
1
⬁
He i 
F Ž r .
␥ ⬘Ž r . q
␥ ⬘Ž r . q
dr
ž
F Ž r .
dr
⬁
r ny2 Ž r .
d
i Ž ␥ Ž r .q r .
r ny 2 Ž r .

r ny 2 Ž r .
␥ ⬘Ž r . q
/
dr
' II1 q II2 . The term II1 can be estimated as before, whereas II2 has a non-monotone function, r Ž ny1.r2rŽ␥ ⬘Ž r . q .. However, it is written as the product of two monotone functions r Ž ny 1.r2 r ⬘Ž r . and 1 y 2 r Ž ␥ ⬘Ž r . q . s ⬘Ž r .rŽ␥ ⬘Ž r . q .. So by the same arguments for I1 and I2 , we can proceed to the goal. On the other hand, defining LŽ r . s
r
H0 e
i Ž ␥ Ž s.q r .
ds,
we obtain the desired estimate for the first term H02 with the methods for I7 and I8 . Now suppose < ␣ < s 1. Let s with s Ž1 , . . . , ny1 , 1. ' Ž ⬘, 1. and < ⬘ < s . If ␣ s Ž1, 0, . . . , 0. or Ž0, . . . , 0, 1. Žit is enough., then
Ž 4.5. Ž 4.6.
⭸ ⭸ 1 ⭸ ⭸ n
⬁
ˆ Ž . s i
¨ˆŽ
H0 e
i ␥ Ž r . ny2
r
Ž r.
HS
ny2
e i r² ⬘r , x : rx 1 d ny2 Ž x . dr ,
⬁
. s iH e i ␥ Ž r . r ny2␥ Ž r . Ž r . H ny2e i r² ⬘r , x : d ny 2 Ž x . dr . 0
S
Considering the new cutoff function as r Ž r . and the surface area measure x 1 d ny2 Ž x . s d Ž x ., we estimate Ž4.5. by analogous ways for I, II. Also we estimate Ž4.6. with the new cutoff function ␥ Ž r . Ž r ..
REFERENCES 1. J.-G. Bak, Averages over surfaces with infinitely flat points, J. Funct. Anal. 129 Ž1995., 455᎐470. 2. J. Bourgain, Averages in the plane over convex curves and maximal operators, J. Anal. Math. 47 Ž1986., 69᎐85.
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KYUNG SOO RIM
3. J.-G. Bak, D. McMichael, J. Vance, and S. Wainger, Fourier transforms of surface area measure on convex surfaces in ⺢ 3 , Amer. J. Math. 111 Ž1989., 633᎐668. 4. J. Bruna, A. Nagel, and S. Wainger, Convex hypersurfaces and Fourier transforms, Ann. Math. 127 Ž1988., 333᎐365. 5. M. Cowling and G. Mauceri, Inequalities for some maximal functions, II, Trans. Amer. Math. Soc. 296 Ž1986., 341᎐365. 6. A. Greenleaf, Principal curvature and harmonic analysis, Indiana Math. J. 30 Ž1981., 519᎐537. 7. A. Nagel, A. Seeger, and S. Waingeri, Averages over convex hypersurfaces, Amer. J. Math. 115 Ž1993., 903᎐927. 8. E. M. Stein, Maximal functions: Spherical means, Proc. Natl. Acad. Sci. U.S. A. 73 Ž1976., 2174᎐2175. 9. C. Sogge and E. M. Stein, Averages of functions over hypersurfaces in ⺢ n , In¨ ent. Math. 82 Ž1985., 543᎐556. 10. E. M. Stein and S. Wainger, Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc. 84 Ž1978., 1239᎐1295. 11. A. Zygmund, ‘‘Trigonometric Series,’’ 2nd ed., Vol. 2, Cambridge Univ. Press, Cambridge, UK, 1968.