Uniform boundedness of oscillatory singular integrals on Hardy spaces

Publicacions Ma em`a iques, Vol. 44 (2000), 605–611
UNIFORM BOUNDEDNESS OF OSCILLATORY
SINGULAR INTEGRALS ON HARDY SPACES
Leslie C. Cheng and Yibiao Pan
Abs ac
We p o e he uni o m H1boundedness o oscilla o y singula in-
eg als wi h degene a e phase unc ions.
1. In oduc ion
In a ecen pape [1], Ca be y, Ch is , and W igh es ablished a se-
ies o impo an es ima es o oscilla o y in eg als and suble el se s. As
one o he applica ions hey also ob ained he uni o m boundedness on
Lpspaces o oscilla o y singula in eg als wi h possibly fla phase unc-
ions. The pu pose o his pape is o apply hei esul s o he s udy o
he uni o m boundedness o oscilla o y singula in eg als on he Ha dy
space H1(Rn).
Oscilla o y singula in eg al ope a o s a e singula in eg al ope a o s
which ca y oscilla o y ac o s in hei ke nels. They ha e a isen in many
p oblems in ha monic analysis and ela ed a eas and ha e been s udied
ex ensi ely ([10], [11]). Fo p e ious wo k on he boundedness on Lp
and Ha dy spaces o such ope a o s, see Phong-S ein [7], Ricci-S ein [8],
S ein [10], Pan [5], [6], and Ca be y-Ch is -W igh [1].
Le n∈N,ϕ∈C∞
0(Rn), and Φ ∈C∞(Rn) be a eal- alued unc ion
sa is ying ∇Φ(0) = 0. Le K(·) be a Calde ´on-Zygmund ke nel on Rn,
i.e. |K(x)|+|x||∇K(x)|≤A|x|−nand a<|x|<b K(x)dx = 0 o b>a>0.
Fo each λ∈Rwe define he (localized) oscilla o y singula in eg al
ope a o Tλon Rnby
(Tλ )(x) = p. . Rn
eiλΦ(x−y)K(x−y)ϕ(x−y) (y)dy.(1)
The ollowing was p o ed in [6]:
606 L. C. Cheng, Y. Pan
Theo em A. Le K,ϕ,Φ, and Tλbe gi en as abo e. I DαΦ(0) =0 o
some mul i-index αwi h |α|≥2, hen he e exis s a posi i e cons an C
such ha
Tλ H1(Rn)≤C H1(Rn)
(2)
o ∈H1(Rn)and λ∈R.
Since i ollows easily om he s anda d heo y o singula in eg als
ha Tλis a bounded ope a o on H1(Rn) o each λ∈R, he impo -
ance o Theo em A lies in he uni o mi y o bounds on he ope a o
no ms TλH1→H1. I was also shown in [6] ha (2) may no hold i
DαΦ(0) anishes o e e y α.
Theo em B. Le x, y ∈R. Define
(Tλ )(x)=p. . |x−y|≤1
eiλΦ(x−y) (y)
x−ydy.(3)
The e exis s a noncons an unc ion Φ∈C∞(R)wi h Φ(k)(0)=0 o
all k=0,1,2,..., such ha
sup
λ∈R
TλH1→H1=∞.(4)
A phase unc ion Φ is said o be fla a 0 i DαΦ(0) = 0 o all α. Can
he uni o m boundedness o Tλ’s on H1(Rn) s ill hold when he phase
unc ion Φ is fla a 0? Inspi ed by he wo k o Ca be y, Ch is , and
W igh we add ess his issue by es ablishing he ollowing:
Theo em C. Le K,ϕ,Φ, and Tλbe gi en as in Theo em A. Le αbe
a mul i-index wi h |α|≥3. Suppose ha a leas one coo dina e o αis
s ic ly g ea e han 1and he e exis δ, A > 0such ha
max
|β|=|α|sup
|x|≤s
|DβΦ(x)|≤Ain
s≤|x|≤δ|DαΦ(x)|(5)
holds o all s∈(0,δ). Then he e exis s a posi i e cons an Csuch ha
Tλ H1(Rn)≤C H1(Rn)
(6)
o ∈H1(Rn)and λ∈R.
Condi ion (5) is simply Condi ion (8.1) o [1] in he con olu ional
se ing. As poin ed ou in [1], (5) is always sa isfied i DαΦ(0) =0
and i may also be sa isfied e en i Φ is fla a 0. Fo example, when
n=1,Φ(x)=η(x) exp(ω(x)/x2) sa isfies (5) i ηand ωa e smoo h and
η(0) =0,ω(0) <0.
Th oughou he es o he pape he le e Cwill s and o a cons an
bu no necessa ily he same one in each occu ence.
Oscilla o y Singula In eg als 607
2. Some Lemmas
Lemma 2.1. Le K,ϕ,Φ, and Tλbe gi en as in Theo em A. Le αbe
a mul i-index wi h |α|≥3. Suppose ha a leas one coo dina e o αis
s ic ly g ea e han 1and ha (5) holds o some δ, A > 0, and e e y
s∈(0,δ). Then o e e y p∈(1,∞) he e exis s a posi i e cons an Cp
such ha
Tλ Lp(Rn)≤Cp Lp(Rn)
(7)
o ∈Lp(Rn)and λ∈R.
P oo : Le ψ∈C∞
0(Rn) such ha m∈Znψm≡1 whe e ψm(x)=
ψ(x−m). Le Tλ,m( )=Tλ(ψm ). I ollows om Theo em 8.2 in [1]
ha he ope a o s {Tλ,m :λ∈R,m∈Zn}a e uni o mly bounded on
Lp(Rn) o 1 <p<∞. One hen ob ains (7) by obse ing ha he se s
{supp(Tλ,m( )) : m∈Zn}ha e he fini e o e lapping p ope y.
Le Qn=[−1/2,1/2]n. The ollowing esul is due o Ca be y, Ch is ,
and W igh ([1]).
Lemma 2.2. Fo n=n+n, each 1≤p, q ≤∞(p=1,q=∞) and
each β∈(N∪{0})nwi h a leas one nonze o en y in each o {1,... ,n
}
and {n+1,... ,n
}and wi h a leas one en y g ea e han 1, he e
exis ε=ε(n, β, p, q)>0and C=C(ε, n, β, p, q)such ha o e e y
eal- alued, in eg able unc ion usa is ying Dβu≥1on Qn, o e e y
λ∈R, he ope a o
(Sλ )(x)=Qn
eiλu(x,x ) (x)dx
(8)
sa isfies
Sλ Lq(Qn)≤Cλ−ε Lp(Qn ).(9)
3. P oo o Theo em C
Le K,ϕ, Φ, and Tλbe gi en as in Theo em C. By he a omic de-
composi ion o H1(Rn) and he cha ac e iza ion o H1(Rn) using Riesz
ans o ms (see [2]-[4], [11]), i suffices o show ha Tλ is in L1(Rn)
uni o mly in λ o all a oms .
608 L. C. Cheng, Y. Pan
Le αbe he mul i-index such ha (5) holds and se
J(s)=s|α|−1in
s≤|x|≤δ|DαΦ(x)|.
Le ρ>0 be small (say, ρ<δ/2), be an a om which is suppo ed in
ρQnand sa isfies  ∞≤ρ−nand Rn (x)dx = 0. Le λ>0.
We shall fi s p o e ha
|x|≥2ρ
|Tλ (x)|dx ≤C(10)
holds o some cons an Cindependen o λ,ρ, and .
I λρ ≤(J(δ))−1, hen (10) ollows om
|Tλ (x)|≤ A
|x|nρQn
|eiλΦ(x−y)−eiλΦ(x)||ϕ(x−y) (y)|dy +|T0 (x)|
≤A(λρ)
|x|n−1ρQn
|ϕ(x−y) (y)|dy +|T0 (x)|
o |x|≥2ρ. Thus we may assume ha λρ>(J(δ))−1.
Le = max{J−1(1/(λρ)),2ρ}.Fo j≥1 define he ope a o Qjby
(Qj )(x)=χ[1,2](|x|)Qn
eiλΦ(2j x−ρy)ϕ(2j x −ρy)g(y)dy.(11)
By ou assump ion he e exis mul i-indices α(1),α(2) such ha α=
α(1) +α(2),|α(2)|= 1, and a leas one en y o α(1) is g ea e han 1.
Thus
|Dα(1)
xDα(2)
y[Φ(2j x −ρy)]|≥[2j(|α|−1)ρ]J( )
holds o 1 ≤|x|≤2 and y∈Qn. Since he ac o ϕ(2j x −ρy)is
ha mless, i ollows om Lemma 2.2 ha
1≤|x|≤2
|Qjg(x)|2dx1/2
≤C[2j(|α|−1)λρJ( )]−εgL2(Qn)
(12)
Oscilla o y Singula In eg als 609
o some posi i e εindependen o α,ρ,jand g. By le ing ρ(x)=
ρn (ρx) and escaling, we ha e
|x|≥2
|Tλ (x)|dx ≤C|x|≥2 Rn
eiλΦ(x−y)ϕ(x−y) (y)dy
dx
|x|n
+|x|≥2ρρQn
|K(x−y)−K(x)|| (y)|dydx
≤C
j≥1
Qj( ρ)2+C
≤C
1+(λρJ( ))−ε
j≥1
2−ε(|α|−1)j
≤C.
(13)
Le
P(x)= 
|β|≤|α|−1
(β!)−1DβΦ(0)xβ
and Ψ(x)=Φ(x)−P(x). By Theo em A,
RnRn
eiλP (x−y)K(x−y)ϕ(x−y) (y)dy
dx ≤C.(14)
Thus, by (14)
2ρ≤|x|<2
|Tλ (x)|dx
≤2ρ≤|x|≤ /2ρQn
|eiλ(Ψ(x−y)−Ψ(x)) −1||K(x−y)ϕ(x−y) (y)|dydx+C
≤Cλ( max
|β|=|α|sup
|x|≤
|DβΦ(x)|)
×2ρ≤|x|≤ /2
|x||α|−1−ndxρQn
|y|| (y)|dy+C
≤CλρJ( )+C≤C,
(15)
whe e we ook =J−1(1/(λρ)) (o he wise =2ρand he in eg al o
|Tλ (x)|on {2ρ≤|x|<2 }is i ially bounded). By combining (13)
and (15) we see ha (10) holds.

610 L. C. Cheng, Y. Pan
By H¨olde ’s inequali y and Lemma 2.1,
|x|<2ρ
|Tλ (x)|dx ≤Cρn/2 2≤C.(16)
I ollows om (10) and (16) ha
Tλ 1≤C(17)
holds o hose a oms desc ibed a he beginning o ou p oo . By
using ansla ion and (16) ( o la ge ρ’s) we see ha (17) holds o an
a bi a y a om . Theo em C is p o ed.
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Oscilla o y Singula In eg als 611
[11] E. M. S ein,“Ha monic analysis: eal- a iable me hods, o hog-
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Leslie C. Cheng:
Depa men o Ma hema ics
B yn Maw College
B yn Maw , PA 19010
U.S.A.
E-mail add ess:[email p o ec ed]
Yibiao Pan:
Depa men o Ma hema ics
Uni e si y o Pi sbu gh
Pi sbu gh, PA 15260
U.S.A.
E-mail add ess:[email p o ec ed]
P ime a e si´o ebuda el 21 de desemb e de 1999,
da e a e si´o ebuda el 31 de gene de 2000.