Volume 10 (2009), Issue 3, Article 78, 15 pp.
ON CERTAIN ROUGH INTEGRAL OPERATORS AND EXTRAPOLATION
HUSSAIN AL-QASSEM1 , LESLIE CHENG, AND YIBIAO PAN
D EPARTMENT OF M ATHEMATICS AND P HYSICS
Q ATAR U NIVERSITY
D OHA -Q ATAR
husseink@yu.edu.jo
D EPARTMENT OF M ATHEMATICS
B RYN M AWR C OLLEGE
B RYN M AWR , PA 19010, U.S.A.
lcheng@brynmawr.edu
D EPARTMENT OF M ATHEMATICS
U NIVERSITY OF P ITTSBURGH
P ITTSBURGH , PA 15260, U.S.A.
yibiao@pitt.edu
Received 24 January, 2009; accepted 31 August, 2009
Communicated by S.S. Dragomir
A BSTRACT. In this paper, we obtain sharp Lp estimates of two classes of maximal operators
related to rough singular integrals and Marcinkiewicz integrals. These estimates will be used to
obtain similar estimates for the related singular integrals and Marcinkiewicz integrals. By the
virtue of these estimates and extrapolation we obtain the Lp boundedness of all the aforementioned operators under rather weak size conditions. Our results represent significant improvements as well as natural extensions of what was known previously.
Key words and phrases: Maximal operator, Rough kernel, Singular integral, Orlicz spaces, Block spaces, Extrapolation, Lp
boundedness.
2000 Mathematics Subject Classification. Primary 42B20; Secondary 42B15, 42B25.
1. I NTRODUCTION AND M AIN R ESULTS
Throughout this paper, let Rn , n ≥ 2, be the n-dimensional Euclidean space and Sn−1 be the
unit sphere in Rn equipped with the normalized Lebesgue surface measure dσ. Also, we let ξ 0
denote ξ/ |ξ| for ξ ∈ Rn \{0} and p0 denote the exponent conjugate to p, that is 1/p + 1/p0 = 1.
Let KΩ,h (x) = Ω(x0 )h(|x|) |x|−n , where h : [0, ∞) −→ C is a measurable function and Ω
is a function defined on Sn−1 with Ω ∈ L1 (Sn−1 ) and
Z
(1.1)
Ω (x) dσ (x) = 0.
Sn−1
1 Permanent
025-09
Address: Department of Mathematics, Yarmouk University, Irbid-Jordan.
2
H USSAIN A L -Q ASSEM , L ESLIE C HENG ,
AND
Y IBIAO PAN
For 1 ≤ γ ≤ ∞, let ∆γ (R+ ) denote the collection of all measurable functions h : [0, ∞) −→
R
1/γ
R
C satisfying sup R1 0 |h(t)|γ dt
< ∞.
R>0
Define the singular integral operator SΩ,h , the parametric Marcinkiewicz integral operator
(γ,∗)
(γ,∗)
MΩ,h and their related maximal operators SΩ and MΩ by
Z
(1.2)
SΩ,h f (x) = p.v.
f (x − u)KΩ,h (u)du,
Rn
∞
Z
MΩ,h f (x) =
(1.3)
0
(γ,∗)
SΩ
(1.4)
(γ,∗)
MΩ
(1.5)
2 ! 12
dt
,
f (x − u) |u| KΩ,h (u)du
t
|u|≤t
Z
−ρ
t
ρ
f (x) = sup |SΩ,h f (x)| ,
h
f (x) = sup |MΩ,h f (x)| ,
h
where f ∈ S(Rn ), ρ = σ + iτ (σ, τ ∈ R with σ > 0) and the supremum is taken over the set
of all radial functions h with h ∈ Lγ (R+ , dt/t) and khkLγ (R+ ,dt/t) ≤ 1.
If h(t) ≡ 1 and ρ = 1, we shall denote SΩ,h by SΩ and MΩ,h by MΩ , which are respectively
the classical singular integral operator of Calderón-Zygmund and the classical Marcinkiewicz
integral operator of higher dimension.
The study of the mapping properties of SΩ , MΩ and their extensions has a long history.
Readers are referred to [7], [3], [4], [16], [18], [5], [10], [22], [24], [25] and the references
therein for applications and recent advances on the study of such operators.
Let us now recall some results which will be relevant to our current study. We start with the
following results on singular integrals:
Theorem 1.1. If Ω satisfies one of the following conditions, then SΩ,h is bounded on Lp (Rn )
for 1 < p < ∞.
(a) Ω ∈ L(log L)(Sn−1 ) and h ∈ ∆γ (R+ ) for some γ > 1. Moreover, the condition
Ω ∈ L(log L)(Sn−1 ) is an optimal size condition for the Lp boundedness of SΩ (see
[13] in the case h(t) ≡ 1 and see [10]).
0
(b) Ω ∈ L(log L)1/γ (Sn−1 ) and h ∈ Lγ (R+ , dt/t) for some γ > 1 (see [2] and see [8] in
the case γ = 2).
(0,0)
(c) Ω ∈ Bq (Sn−1 ) for some q > 1 and h ∈ ∆γ (R+ ) for some γ > 1. Moreover, the
(0,0)
condition Ω ∈ Bq (Sn−1 ) is an optimal size condition for the Lp boundedness of SΩ
(see [4] and [5]).
(0,−1/2)
(d) Ω ∈ Bq
(Sn−1 ) for some q > 1 and h ∈ L2 (R+ , dt/t) ([9]).
α
Here L(log L) (Sn−1 ) (for α > 0) is the class of all measurable functions Ω on Sn−1 which
satisfy
Z
α
kΩkL(log L)α (Sn−1 ) =
|Ω(y)| log (2 + |Ω(y)|)dσ(y) < ∞
Sn−1
(0,υ)
and Bq (Sn−1 ), υ > −1, is a special class of block spaces whose definition will be recalled
in Section 2.
J. Inequal. Pure and Appl. Math., 10(3) (2009), Art. 78, 15 pp.
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ROUGH I NTEGRAL O PERATORS AND E XTRAPOLATION
3
Theorem 1.2. If Ω satisfies one of the following conditions, then MΩ,h is bounded on Lp (Rn )
for 1 < p < ∞.
(a) Ω ∈ L(log L)1/2 (Sn−1 ) and h = 1. Moreover, the exponent 1/2 is the best possible
([26], [7]).
(0,−1/2)
(b) Ω ∈ Bq
(Sn−1 ) for some q > 1 and h = 1. Moreover, the condition Ω ∈
(0,−1/2)
Bq
(Sn−1 ) is an optimal size condition for the L2 boundedness of MΩ ([3]).
0
(c) Ω ∈ L(log L)1/γ (Sn−1 ), h ∈ Lγ (R+ , dt/t), 1 < γ ≤ 2 and γ 0 ≤ p < ∞ or Ω ∈
L(log L)1/2 (Sn−1 , h ∈ Lγ (R+ , dt/t), 2 < γ < ∞ and 2 ≤ p < ∞ ([6]).
(γ,∗)
On the other hand, the study of the related maximal operator SΩ
of many authors. See for example, [14], [1], [5], [6], [8], [9], [27].
has attracted the attention
(γ,∗)
Theorem 1.3. If Ω satisfies one of the following conditions, then SΩ is bounded on Lp (Rn ).
(a) Ω ∈ C(Sn−1 ), (nγ)/(nγ − 1) < p < ∞ and 1 ≤ γ ≤ 2. Moreover, the range of p is
the best possible [14].
(0,−1/2)
(b) Ω ∈ Bq
(Sn−1 ) for some q > 1, γ = 2, 2 ≤ p < ∞. Moreover, the condition
(0,−1/2)
(2,∗)
Ω ∈ Bq
(Sn−1 ) is an optimal size condition for the L2 boundedness of SΩ to
hold ([1], [9]).
1/γ 0
(c) Ω ∈ L(log L) (Sn−1 ), γ 0 ≤ p < ∞. Moreover, the exponent 1 /2 is the best possible
(2,∗)
for the L2 boundedness of SΩ to hold (see [8] for γ = 2 and [2] for γ 6= 2).
In view of the above results, the following question is very natural:
Problem 1.1. Is there any analogue of Theorem 1.1(d) and Theorem 1.3(b) in the case γ 6= 2?
Is there an analogy of Theorem 1.2(c)? Is there any room for improvement of the range of p in
both Theorem 1.2(c) and Theorem 1.3(c)?
The purpose of this paper is two-fold. First, we answer the above questions in the affirmative.
Second, we present a unified approach different from the ones employed in previous papers (see
(γ,∗)
(γ,∗)
for example, [1], [2], [6], [8], [9]) in dealing with the operators SΩ,h , MΩ,h , SΩ and MΩ
(0,υ)
when the kernel function Ω belongs to the block space Bq (Sn−1 ) (for υ > −1) or Ω belongs
α
to the class L(log L) (Sn−1 ) (for α > 0). This approach will mainly rely on obtaining some
delicate sharp Lp estimates and then applying an extrapolation argument.
Now, let us state our main results.
Theorem 1.4. Suppose that Ω∈ Lq (Sn−1 ) for some 1 < q ≤ ∞. Then
1/γ 0
q
(γ,∗) (1.6)
kΩkLq (Sn−1 ) kf kLp (Rn )
SΩ (f ) p n ≤ Cp
q−1
L (R )
holds for (nαγ 0 )/(γ 0 n + nα − γ 0 ) < p < ∞ and 1 ≤ γ ≤ 2, where α = max{2, q 0 }. Moreover,
the exponent 1/γ 0 is the best possible in the case γ = 2.
Theorem 1.5. Let α and Ω be as in Theorem 1.4 and let 1 ≤ γ < ∞. Then
1/β
q
(γ,∗) (1.7)
kΩkLq (Sn−1 ) kf kLp (Rn )
MΩ (f ) p n ≤ Cp
q−1
L (R )
holds for (nαβ)/(βn + nα − β) < p < ∞, where β = max{2, γ 0 }. Moreover, the exponent
1/β is the best possible in the case γ = 2.
J. Inequal. Pure and Appl. Math., 10(3) (2009), Art. 78, 15 pp.
http://jipam.vu.edu.au/
4
H USSAIN A L -Q ASSEM , L ESLIE C HENG ,
AND
Y IBIAO PAN
Theorem 1.6. Suppose that Ω∈ Lq (Sn−1 ), 1 < q ≤ 2, and h ∈ Lγ (R+ , dt/t) for some 1 <
γ ≤ ∞. Then
(1.8)
0
kSΩ,h (f )kLp (Rn ) ≤ Cp (q − 1)−1/γ khkLγ (R+ ,dt/t) kΩkLq (Sn−1 ) kf kLp (Rn )
for 1 < p < ∞ and
(1.9)
kMΩ,h (f )kLp (Rn ) ≤ Cp (q − 1)−1/β khkLγ (R+ ,dt/t) kΩkLq (Sn−1 ) kf kLp (Rn )
for (nβq 0 )/(βn + nq 0 − β) < p < ∞.
By the conclusions from Theorems 1.4 – 1.6 and applying an extrapolation method, we get
the following results:
Theorem 1.7.
1/γ 0
(γ,∗)
(a) If Ω ∈ L(log L) (Sn−1 ) and 1 < γ ≤ 2, the operator SΩ is bounded on Lp (Rn )
for 2 ≤ p < ∞;
(0,1/γ 0 −1)
(γ,∗)
(b) If Ω ∈ Bq
(Sn−1 ) and 1 < γ ≤ 2, the operator SΩ is bounded on Lp (Rn ) for
2 ≤ p < ∞;
1/γ 0
(c) If Ω ∈ L(log L) (Sn−1 ) and h ∈ Lγ (R+ , dt/t) for some 1 < γ ≤ ∞, the operator
SΩ,h is bounded on Lp (Rn ) for 1 < p < ∞;
(0,1/γ 0 −1)
(d) If Bq
(Sn−1 ) for some q > 1 and h ∈ Lγ (R+ , dt/t) for some 1 < γ ≤ ∞, the
operator SΩ,h is bounded on Lp (Rn ) for 1 < p < ∞.
Theorem 1.8. Let 1 < γ < ∞ and β = max{2, γ 0 }.
(γ,∗)
1/β
(a) If Ω ∈ L(log L) (Sn−1 ) and 1 < γ < ∞, the operator MΩ is bounded on Lp (Rn )
for 2 ≤ p < ∞;
(0,1/β−1)
(γ,∗)
(b) If Ω ∈ Bq
(Sn−1 ) and 1 < γ < ∞, the operator MΩ is bounded on Lp (Rn )
for 2 ≤ p < ∞.
1/β
(c) If Ω ∈ L(log L) (Sn−1 ) and h ∈ Lγ (R+ , dt/t) for some 1 < γ < ∞, the operator
MΩ,h is bounded on Lp (Rn ) for 2 ≤ p < ∞;
(0,1/β−1)
(d) If Bq
(Sn−1 ) for some q > 1 and h ∈ Lγ (R+ , dt/t) for some 1 < γ < ∞, the
operator MΩ,h is bounded on Lp (Rn ) for 2 ≤ p < ∞.
Remark 1.
(1) For any q > 1, 0 < α < β and −1 < υ, the following inclusions hold and are proper:
β
α
Lq (Sn−1 ) ⊂ L(log L) (Sn−1 ) ⊂ L(log L) (Sn−1 ),
[
Lr (Sn−1 ) ⊂ Bq(0,υ) (Sn−1 ) for any − 1 < υ,
r>1
Bq(0,υ2 ) (Sn−1 ) ⊂ Bq(0,υ1 ) (Sn−1 ) for any − 1 < υ1 <
Lγ (R+ , dt/t) ⊂ ∆γ (R+ ) for 1 ≤ γ < ∞.
(0,υ−1)
υ2 ,
υ
The question regarding the relationship between Bq
and L(log L) over Sn−1 (for
υ > 0) remains open.
(γ,∗)
(2) The Lp boundedness of SΩ for (nαγ 0 )/(γ 0 n + nα − γ 0 ) < p < ∞ and Ω∈ Lq (Sn−1 )
was proved in [1] only in the case γ = 2, but the importance of Theorem 1.4 lies in
the fact that the estimate (1.6) in conjunction with an extrapolation argument will play
a key role in obtaining all our results and will allow us to obtain the Lp boundedness of
(γ,∗)
SΩ under optimal size conditions on Ω.
J. Inequal. Pure and Appl. Math., 10(3) (2009), Art. 78, 15 pp.
http://jipam.vu.edu.au/
ROUGH I NTEGRAL O PERATORS AND E XTRAPOLATION
5
(γ,∗)
(3) We point out that it is still an open question whether the Lp boundedness of SΩ holds
for 2 < γ < ∞. In the case γ = ∞, the authors of [14] pointed out that the maximal
(∞,∗)
operator SΩ (f ) is not bounded on all Lp spaces for 1 ≤ p ≤ ∞. On the other hand,
(γ,∗)
we notice that Theorem 1.5 gives that MΩ is bounded on Lp for any 1 ≤ γ < ∞.
(4) Theorem 1.7 (a)(c) and Theorem 1.8 (c) represent an improvement over the main results
in [8] and improves the range of p in [2]. Also, Theorem 1.7 (b)(d) and Theorem 1.8 (d)
represent an improvement over the main results in [9] and [1].
Throughout the rest of the paper the letter C will stand for a constant but not necessarily the
same one in each occurrence.
2. D EFINITIONS AND S OME BASIC L EMMAS
Block spaces originated from the work of M.H. Taibleson and G. Weiss on the convergence
of the Fourier series in connection with developments of the real Hardy spaces. Below we
shall recall the definition of block spaces on Sn−1 . For further background information about
the theory of spaces generated by blocks and its applications to harmonic analysis, one can
(0,υ)
consult the book [20]. In [20], Lu introduced the spaces Bq (Sn−1 ) with respect to the study
of singular integral operators.
Definition 2.1. A q-block on Sn−1 is an Lq (1 < q ≤ ∞) function b(x) that satisfies
(i) supp(b) ⊂ I;
0
(ii) kbkLq ≤ |I|−1/q , where |·| denotes the product measure on Sn−1 , and I is an interval
on Sn−1 , i.e., I = {x ∈ Sn−1 : |x − x0 | < α} for some α > 0 and x0 ∈ Sn−1 .
(0,υ)
(0,υ)
The block space Bq
= Bq (Sn−1 ) is defined by
(
)
∞
X
Bq(0,υ) = Ω ∈ L1 (Sn−1 ) : Ω =
λµ bµ , Mq(0,υ) {λµ } < ∞ ,
µ=1
where each λµ is a complex number; each bµ is a q-block supported on an interval Iµ
on Sn−1 , υ > −1 and
∞
o
n
X
(0,υ)
λµ 1 + log(υ+1) Iµ −1 .
Mq
{λµ } =
µ=1
Let
kΩkBq(0,υ) (Sn−1 ) = Nq(0,υ) (Ω) = inf Mq(0,υ) {λµ } ,
where the infimum is taken over all q-block decompositions of Ω.
Definition 2.2. For arbitrary θ ≥ 2 and Ω : Sn−1 → R, we define the sequence of measures
∗
{σΩ,h,k : k ∈ Z} and the corresponding maximal operator σΩ,h,θ
on Rn by
Z
Z
Ω(u0 )
(2.1)
f dσΩ,h,θ,k =
h(|u|)
f (u) du;
|u|n
Rn
θk ≤|u|<θk+1
(2.2)
∗
σΩ,h,θ
(f ) = sup ||σΩ,h,θ,k | ∗ f | .
k∈Z
We shall need the following lemma which has its roots in [15] and [5]. A proof of this lemma
can be obtained by the same proof (with only minor modifications) as that of Lemma 3.2 in [5].
We omit the details.
J. Inequal. Pure and Appl. Math., 10(3) (2009), Art. 78, 15 pp.
http://jipam.vu.edu.au/
6
H USSAIN A L -Q ASSEM , L ESLIE C HENG ,
AND
Y IBIAO PAN
Lemma 2.1. Let {σk : k ∈ Z} be a sequence of Borel measures on Rn . Suppose that for some
a ≥ 2, α, C > 0, B > 1 and p0 ∈ (2, ∞) the following hold for k ∈ Z, ξ ∈ Rn and for
arbitrary functions {gk } on Rn :
± α
(i) |σ̂k (ξ)| ≤ CB ak ξ log(a) ; 12 12 P
P
2
2
(ii) |σk ∗ gk |
|gk |
≤ CB .
k∈Z
k∈Z
p0
p0
Then for p00 < p < p0 , there exists a positive constant Cp independent of B such that
X
σk ∗ f ≤ Cp B kf kp
k∈Z
p
p
n
holds for all f in L (R ).
∗
Lemma 2.2. Let θ and σΩ,h,θ
be given as in Definition 2.2. Suppose h ∈ Lγ (R+ , dt/t) for some
1 < γ ≤ ∞ and Ω ∈ L1 (Sn−1 ). Then
∗
σΩ,h,θ (f ) ≤ Cp (log θ)1/γ 0 khk γ
(2.3)
L (R+ ,dt/t) kΩkL1 (Sn−1 ) kf kp
p
for γ 0 < p ≤ ∞ and f ∈ Lp (Rn ), where Cp is independent of Ω, θ and f.
Proof. By Hölder’s inequality we have
1/γ
|σΩ,h,θ,k ∗ f (x)| ≤ khkLγ (R+ ,dt/t) kΩkL1 (Sn−1 )
Z
θk+1
Z
dt
|Ω(y)| |f (x − ty)| dσ(y)
t
Sn−1
×
θk
By Minkowski’s inequality for integrals we get
∗
1/γ
σΩ,h,θ (f ) ≤ (log θ)1/γ 0 khk γ
L (R+ ,dt/t) kΩkL1 (Sn−1 )
p
Z
γ0 ×
|Ω(y)| My (|f | )
p/γ 0
Sn−1
where
My f (x) = sup R
−1
R>0
!1/γ 0
γ0
Z
.
1/γ 0
dσ(y)
,
R
|f (x − sy)| ds
0
is the Hardy-Littlewood maximal function of f in the direction of y. By the fact that the operator
My is bounded on Lp (Rn ) for 1 < p < ∞ with a bound independent of y, by the last inequality
we get (2.3). Lemma 2.2 is thus proved.
By following an argument similar to that in [17] and [2], we obtain the following:
Lemma 2.3. Let θ and Ω be as in Lemma 2.2. Assume that h ∈ Lγ (R+ , dt/t) for some γ ≥ 2.
Then for γ 0 < p < ∞, there exists a positive constant Cp which is independent of θ such that
! 21 ! 12 X
X
2
2
1/γ 0
|σΩ,h,θ,k ∗ gk |
kΩkL1 (Sn−1 ) |gk |
≤ Cp (log θ)
k∈Z
k∈Z
p
p
n
holds for arbitrary measurable functions {gk } on R .
Now, we need the following simple lemma.
J. Inequal. Pure and Appl. Math., 10(3) (2009), Art. 78, 15 pp.
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ROUGH I NTEGRAL O PERATORS AND E XTRAPOLATION
7
Lemma 2.4. Let q > 1 and β = max{2, q 0 }. Suppose that Ω ∈ Lq (Sn−1 ). Then for some
positive constant C, we have
Z
2
Z
min{2,q}
|Ω(ξ)|max{0,2−q} |f (ξ)|2 dσ(ξ)
(2.4)
Ω(ξ)f (ξ)dσ(ξ) ≤ kΩkq
Sn−1
Sn−1
and
Z
max{0,2−q}
|Ω(ξ)|
(2.5)
|f (x − tξ)| dσ(ξ) ≤ C
Sn−1
kΩkmax{0,2−q}
q
β/2
MSph |f |
β2
(x)
for all positive real numbers t and x ∈ Rn and arbitrary functions f, where MSph is the
spherical maximal operator defined by
Z
MSph f (x) = sup
|f (x − tu)| dσ(u).
t>0
Sn−1
Proof. Let us first prove (2.4). First if q ≥ 2, by Hölder’s inequality we have
Z
2
Z
20
q
0
2
q
≤ kΩk
Ω(ξ)f
(ξ)dσ(ξ)
|f
(ξ)|
dσ(ξ)
q
n−1
n−1
S
Z S
≤ kΩk2q
|f (ξ)|2 dσ(ξ),
Sn−1
which is the inequality (2.4) in the case q ≥ 2. Next, if 1 < q < 2, the inequality (2.4) follows
by Schwarz’s inequality.
To prove (2.5) we need again to consider two cases. First if q ≥ 2, we notice
that
(2.5) is
obvious. Next if 1 < q < 2, (2.5) follows by Hölder’s inequality on noticing that
The lemma is thus proved.
q
2−q
0
= q 0 /2.
3. P ROOF OF THE M AIN R ESULTS
We begin with the proof of Theorem 1.4.
Proof. Since Lq (Sn−1 ) ⊆ L2 (Sn−1 ) for q ≥ 2, Theorem 1.4 is proved once we prove that
0
(γ,∗) (3.1)
SΩ (f ) p n ≤ Cp (q − 1)−1/γ kΩkLq (Sn−1 ) kf kLp (Rn )
L (R )
0 0
holds for 1 < q ≤ 2, (nq γ )/(γ 0 n + nq 0 − γ 0 ) < p < ∞ and 1 ≤ γ ≤ 2.
We shall prove (3.1) by first proving (3.1) for the cases γ = 1 and γ = 2 and then we use the
idea of interpolation to cover the case 1 < γ < 2.
Proof of (3.1) for the case γ = 1. By duality we have
Z
(1,∗)
SΩ f (x) = f
(x
−
tu)Ω(u)dσ(u)
Sn−1
Z
=
Sn−1
L∞ (R+ ,dt/t)
f (x − tu)Ω(u)dσ(u)
L∞ (R+ ,dt)
Z
≤ sup
t>0
|f (x − tu)| |Ω(u)| dσ(u).
Sn−1
Using Hölder’s inequality we get
(1,∗)
SΩ f (x)
q0
≤ kΩkq MSph |f |
J. Inequal. Pure and Appl. Math., 10(3) (2009), Art. 78, 15 pp.
1/q0
(x)
.
http://jipam.vu.edu.au/
8
H USSAIN A L -Q ASSEM , L ESLIE C HENG ,
AND
Y IBIAO PAN
By the results of E.M. Stein [23] and J. Bourgain [12] we know that MSph (f ) is bounded on
Lp (Rn ) for n ≥ 2 and p > n/(n − 1). Thus by using the last inequality we get (3.1) for the
case γ = 1.
Proof of (3.1) for the case γ = 2. By Hölder’s inequality we obtain
(2,∗)
SΩ
∞
Z
f (x) ≤
0
2 ! 12
dt
.
Ω(u)f (x − tu)dσ(u)
t
Sn−1
Z
Let θ = 2q and let {ϕk }∞
−∞ be a smooth partition of unity in (0, ∞) adapted to the interval
−k−1
−k+1
[θ
,θ
]. Specifically, we require the following:
X
ϕk ∈ C ∞ , 0 ≤ ϕk ≤ 1,
ϕk (t) = 1,
0
k∈Z
s
d ϕk (t) Cs
dts ≤ ts ,
supp ϕk ⊆ [θ−k−1 , θ−k+1 ],
where Cs is independent of the lacunary sequence {θk : k ∈ Z}. Define the multiplier operators
n
d
ˆ
Tk in Rn by (T
k f )(ξ) = ϕk (|ξ|)f (ξ). Then for any f ∈ S(R ) and k ∈ Z we have f (x) =
P
j∈Z (Tk+j f )(x). Thus, by Minkowski’s inequality we have

2  12
Z θk+1 X Z
dt
X
(2,∗)


SΩ f (x) ≤
Ω(u)Tk+j f (x − tu)dσ(u)
t
k
n−1
j∈Z S
k∈Z θ
X
≤
Xj f (x),
j∈Z
where
Xj f (x) =
XZ
k∈Z
θk+1
θk
2 ! 12
dt
Ω(u)Tk+j f (x − tu)dσ(u)
.
t
Sn−1
Z
We notice that to prove (3.1) for the case γ = 2, it is enough to prove that the inequality
(3.2)
1
kXj (f )kp ≤ Cp (q − 1)− 2 kΩkq 2−δp |j| kf kp
holds for 1 < q ≤ 2, (nq 0 γ 0 )/(nγ 0 + nq 0 − γ 0 ) < p < ∞ and for some positive constants Cp and
δp . This inequality can be proved by interpolation between a sharp L2 estimate and a cruder Lp
estimate. We prove (3.2) first for the case p = 2. By using Plancherel’s theorem we have
Z
2 dt
X
2
dξ,
(3.3)
kXj (f )k2 =
(Hk (ξ))2 fˆ(ξ)
t
∆k+l k∈Z
where ∆k = ξ ∈ Rn : θ−k−1 ≤ |ξ| ≤ θ−k+1 and
Z
(3.4)
θk+1
Hk (ξ) =
θk
2 ! 12
dt
Ω(x)e−itξ·x dσ(x)
.
t
Sn−1
Z
We claim that
(3.5)
n − λ λo
k q0 k+1 q0
|Hk (ξ)| ≤ C(log θ) kΩkq min θ ξ
, θ ξ
,
1
2
where C is a constant independent of k, ξ and q.
J. Inequal. Pure and Appl. Math., 10(3) (2009), Art. 78, 15 pp.
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ROUGH I NTEGRAL O PERATORS AND E XTRAPOLATION
9
Let us now turn to the proof of (3.5). First, by a change of variable and since
2 Z
Z
k tξ·x
k
−iθ
Ω(x)Ω(y)e−iθ tξ·(x−y) dσ(x)dσ(y),
Ω(x)e
dσ(x) =
Sn−1
Sn−1 ×Sn−1
we obtain
Z
Z
2
(Hk (ξ)) ≤
(3.6)
θ
Ω(x)Ω(y)
Sn−1 ×Sn−1
−iθk tξ·(x−y) dt
e
dσ(x)dσ(y).
t
1
Using an integration by parts,
Z θ
dt
k
−iθ tξ·(x−y)
≤ C(log θ) θk ξ −1 |ξ 0 · (x − y)|−1 .
e
t
1
R
θ −iθk tξ·(x−y) dt By combining this estimate with the trivial estimate 1 e
≤ (log θ), we get
t Z
θ
k −α 0
θ ξ |ξ · (x − y)|−α
≤
C(log
θ)
t
−iθk tξ·(x−y) dt e
1
for any 0 < α ≤ 1.
Thus, by the last inequality, (3.6) and Hölder’s inequality we get
− α
|Hk (ξ)| ≤ C(log θ) kΩkq θk ξ 2q0
1
2
Z
!
dσ(x)dσ(y)
Sn−1 ×Sn−1
|ξ 0 · (x − y)|αq
0
1
2q 0
.
By choosing α so that αq 0 < 1 we obtain that the last integral is bounded in ξ 0 ∈ Sn−1 and hence
− α
1
(3.7)
|Hk (ξ)| ≤ C(log θ) 2 kΩkq θk ξ 2q0 .
Secondly, by the cancellation conditions on Ω we obtain
2 ! 12
−itξ·x
dt
Hk (ξ) ≤
|Ω(x)| e
− 1 dσ(x)
t
θk
Sn−1
1
≤ C(log θ) 2 kΩk1 θk+1 ξ .
Z
θk+1
Z
By combining the last estimate with the estimate |Hk (ξ)| ≤ C(log θ)1/2 kΩk1 , we get
α
1
(3.8)
|Hk (ξ)| ≤ C(log θ) 2 kΩk1 θk+1 ξ 2q0 .
By combining the estimates (3.7)–(3.8), we obtain (3.5).
Now, by (3.4) and (3.5) we obtain
1
kXj (f )k2 ≤ C2−λ|j| (q − 1)− 2 kΩkq kf k2 .
(3.9)
Let us now estimate kXj (f )kp for (nq 0 γ 0 )/(γ 0 n + nq 0 − γ 0 ) < p < ∞ with p 6= 2. Let us
0
first consider the case p > 2. By duality, there is a nonnegative function g in L(p/2) (Rn ) with
kgk(p/2)0 ≤ 1 such that
kXj (f )k2p
Z
=
XZ
Rn k∈Z
θk+1
θk
2
dt
g(x)dx.
Ω(u)Tk+j f (x − tu)dσ(u)
t
Sn−1
Z
J. Inequal. Pure and Appl. Math., 10(3) (2009), Art. 78, 15 pp.
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10
H USSAIN A L -Q ASSEM , L ESLIE C HENG ,
AND
Y IBIAO PAN
By Hölder’s inequality, Fubini’s theorem and a change of variable we have
Z X Z θk+1 Z
dt
2
kXj (f )kp ≤ kΩk1
|Ω(u)| |Tk+j f (x − tu)|2 g(x)dσ(u) dx
t
Rn k∈Z θk
Sn−1
Z X Z θk+1 Z
dt
|Ω(u)| |Tk+j f (x)|2 g(x + tu)dσ(u) dx
≤ kΩk1
t
Sn−1
Rn k∈Z θk
!
Z
X
∗
≤ C kΩk1
|Tk+j f (x)|2 σΩ,1,θ
(g̃)(−x)dx,
Rn
k∈Z
∗
σΩ,h,θ
where g̃(x) = g(−x) and
is defined as in (2.2). By the last inequality, Lemma 2.2 and
using Littlewood-Paley theory ([24, p. 96]) we get
X
∗
2
2
|Tk+j f | kXj (f )kp ≤ C kΩk1 σΩ,1,θ (g̃) (p/2)0 k∈Z
Cp (log θ) kΩk21
≤
kf k2p
(p/2)
,
which in turn easily implies
kXj (f )kp ≤ Cp (q − 1)−1/2 kΩkq kf kp
(3.10)
for 2 < p < ∞.
By interpolating between (3.9) – (3.10) we get (3.2) for the case 2 ≤ p < ∞. Now, let us
estimate kXj (f )kp for the case (nq 0 γ 0 )/(γ 0 n + nq 0 − γ 0 ) < p < 2. By a change of variable we
get
2 ! 12
X Z θ Z
dt
k
Xj f (x) =
.
n−1 Ω(u)Tk+j f (x − θ tu)dσ(u) t
1
S
k∈Z
By duality, there is a function h = hk,j (x, t) satisfying khk ≤ 1 and
dt
p0
2
2
, k , dx
hk,j (x, t) ∈ L
l L [1, θ] ,
t
such that
Z
kXj (f )kp =
XZ
Rn k∈Z
Z
=
Z
Ω(u) (Tk+j f ) (x − θk tu)hk,j (x, t)dσ(u)
dt
dx
t
Ω(u) (Tk+j f ) (x)hk,j (x + θk tu, t)dσ(u)
dt
dx.
t
Sn−1
1
XZ
Rn k∈Z
θ
θZ
Sn−1
1
By Hölder’s inequality and Littlewood-Paley theory we get
! 12 X
1
2
(3.11)
|T
f
|
kXj (f )kp ≤ (L(h)) 2 k+j
p0 k∈Z
p
1
≤ Cp kf kp (L(h)) 2 ,
p0
where
L(h) =
X Z
k∈Z
1
θ
Z
dt
Ω(u)hk,j (x + θ tu, t)dσ(u)
t
Sn−1
J. Inequal. Pure and Appl. Math., 10(3) (2009), Art. 78, 15 pp.
k
2
.
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ROUGH I NTEGRAL O PERATORS AND E XTRAPOLATION
11
0
0
1/2
Since p0 > 2 and (L(h))1/2 = kL(h)kp0 /2 , there is a nonnegative function b ∈ L(p /2) (Rn )
p0
such that kbk(p0 /2)0 ≤ 1 and
Z X Z
kL(h)kp0 /2 =
Rn k∈Z
θ
1
Z
dt
Ω(u)hk,j (x + θ tu, t)dσ(u)
t
Sn−1
k
2
b(x)dx.
By the Schwarz inequality and Lemma 2.4, we obtain
Z θ Z
2
dt
k
Ω(u)hk,j (x + θ tu, t)dσ(u)
t
1
Sn−1
2
Z θ Z
dt
k
≤ C (log θ)
Ω(u)hk,j (x + θ tu, t)dσ(u)
t
1
Sn−1
Z θZ
2
dt
min{2,q}
≤ C (log θ) kΩkLq (Sn−1 )
|Ω(u)|max{0,2−q} hk,j (x + θk tu, t) dσ(u) .
t
1
Sn−1
Therefore, by the last inequality, a change of variable, Fubini’s theorem, Hölder’s inequality,
and Lemma 2.4 we get
min{2,q}
max{0,2−q}
kL(h)kp0 /2 ≤ C (log θ) kΩkLq (Sn−1 ) kΩkLq (Sn−1 )
!
0 20
Z
XZ θ
q
dt
q /2
2
MSph b̃
(−x)
dx
×
|hk,j (x, t)|
t
Rn
k∈Z 1
X Z θ
dt
≤ C (log θ) kΩk2Lq (Sn−1 ) |hk,j (x, t)|2 t
k∈Z 1
p0 /2
0
2/q q0 /2
× MSph b̃
.
0 0
(p /2)
0
0
0
By the condition on p we have (2/q )(p /2) > n/(n − 1) and hence by the Lp boundedness of
MSph and the choice of b we obtain
kL(h)kp0 /2 ≤ C(q − 1)−1 kΩk2Lq (Sn−1 ) .
Using the last inequality and (3.11) we have
(3.12)
kXj (f )kp ≤ Cp (q − 1)−1/2 kΩkq kf kp
for (nq 0 γ 0 )/(γ 0 n + nq 0 − γ 0 ) < p < 2.
Thus, by (3.9) and (3.12) and interpolation we get (3.2) for (nq 0 γ 0 )/(γ 0 n + nq 0 − γ 0 ) < p < 2.
This ends the proof of (3.1) for the case γ = 2.
Proof of (3.1) for the case 1 < γ < 2.
We will use an idea that appeared in [19]. By duality we have
(γ,∗) = kF (f )kLp (Lγ 0 (R+ , dt ),Rn ,dx) ,
SΩ (f )
Lp (Rn )
t
γ0
where F : Lp (Rn ) → Lp (L (R+ , dtt ), Rn ) is a linear operator defined by
Z
F (f )(x; t) =
f (x − tu)Ω(u)dσ(u).
Sn−1
From the inequalities (3.1) (for the case γ = 2) and (3.1) (for the case γ = 1), we interpret that
1
kF (f )kLp (L2 (R+ , dt ),Rn ) ≤ C(q − 1)− 2 kΩkLq (Sn−1 ) kf kLp (Rn )
t
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12
H USSAIN A L -Q ASSEM , L ESLIE C HENG ,
AND
Y IBIAO PAN
for (2nq 0 )/(2n + nq 0 − 2) < p < ∞ and
kF (f )kLp (L∞ (R+ , dt ),Rn ) ≤ C kΩkLq (Sn−1 ) kf kLp (Rn )
(3.13)
t
0 0
for q n ≤ p < ∞. Applying the real interpolation theorem for Lebesgue mixed normed spaces
to the above results (see [11]), we conclude that
0
kF (f )kLp (Lγ 0 (R+ , dt ),Rn ) ≤ Cp (q − 1)−1/γ kΩkLq (Sn−1 ) kf kLp (Rn )
t
0 0
holds for 1 < q ≤ 2, (nq γ )/(nγ 0 + nq 0 − γ 0 ) < p < ∞ and 1 ≤ γ ≤ 2. This completes the
proof of Theorem 1.4.
Next we shall present the proof of Theorem 1.5.
Proof. We need to consider two cases.
Case 1. 1 ≤ γ ≤ 2. By an argument appearing in [6], we may, without loss of generality,
in the definition of MΩ,h , replace |y| ≤ t with 12 t ≤ |y| ≤ t. By Minkowski’s inequality and
Fubini’s theorem we get

 12
2
Z ∞ Z
0
Ω(y )
dt

f (x − y) n−ρ h(|y|)dy 1+2σ 
1 t≤|y|≤t
t
|y|
0
2
Z
∞
Z
≤
0
Z
∞
0
Z
≤
0
0
∞
∞
Z
Sn−1
Z
Sn−1
f (x − sy)Ω(y)dσ(y) |h(s)| χ
ds
(s) 1−σ
1 t,t
[2 ] s
2
f (x − sy)Ω(y)dσ(y) |h(s)|2 χ
[
1
2 t,t
]
(s)
dt
2
dt
! 12
t1+2σ
! 12
1+2σ
t
ds
s1−σ
Z
1
Z ∞
dt 2 ds
=
n−1 f (x − sy)Ω(y)dσ(y) |h(s)|
s1−σ
t1+2σ
0
S
s
Z ∞ Z
1
|h(s)| ds .
=√
f
(x
−
sy)Ω(y)dσ(y)
s
2σ 0 Sn−1
Therefore, by duality we have
1 (γ,∗)
(γ,∗)
(3.14)
MΩ f (x) ≤ √ SΩ f (x),
2σ
and hence (1.7) holds by (1.6) for 1 ≤ γ ≤ 2.
Z
∞
Case 2. 2 < γ ≤ ∞. By a change of variable and duality we have

 12
γ 0 !2/γ 0
Z ∞ Z 1 Z
ds
dt 
(γ,∗)
MΩ f (x) ≤ 
.
n−1 f (x − stu)Ω (u) dσ(u) s
t
0
1/2
S
Using the generalized Minkowski inequality and since γ 0 < 2, we get
Z 1
1/γ 0
(γ,∗)
γ 0 ds
(3.15)
MΩ f (x) ≤
|Es f (x)|
,
s
1/2
where
Z
Es f (x) =
0
∞
2 ! 12
dt
f (x − stu)Ω (u) dσ(u)
.
t
Sn−1
Z
J. Inequal. Pure and Appl. Math., 10(3) (2009), Art. 78, 15 pp.
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ROUGH I NTEGRAL O PERATORS AND E XTRAPOLATION
(2,∗)
Since Es f (x) = SΩ
13
f (x), we obtain
(γ,∗)
MΩ
(3.16)
(2,∗)
f (x) ≤ SΩ
f (x),
and again (1.7) holds for the case 2 < γ ≤ ∞ by (1.6) in the case γ = 2.
We now give the proof of Theorem 1.6.
Proof. For part (a), we need to consider two cases.
Case 1. 1 < γ ≤ 2.
(ε)
(ε)
By definition of SΩ,h f (x) we have SΩ,h f (x) = lim SΩ,h f (x) for f ∈ S(Rn ), where SΩ,h is
ε→0
the truncated singular integral operator given by
Z
Ω(u)
(ε)
SΩ,h f (x) =
n h(|u|)f (x − u)du.
|u|>ε |u|
By Hölder’s inequality and duality we have
Z
Z ∞
(ε)
|h(t)| SΩ,h f (x) ≤
dt
f (x − tu)Ω (u) dσ(u)
t
Sn−1
ε
(∗,γ)
≤ khkLγ (R+ ,dt/t) SΩ
f (x).
Thus, by Theorem 1.4 we get
(ε)
−1/γ 0
(3.17)
khkLγ (R+ ,dt/t) kΩkLq (Sn−1 ) kf kp
SΩ,h (f ) ≤ Cp (q − 1)
p
for (nq 0 γ 0 )/(nγ 0 + nq 0 − γ 0 ) < p < ∞ and 1 < γ ≤ 2 and for some positive constant Cp
independent of ε. In particular, (3.17) holds for 2 ≤ p < ∞ and 1 < γ ≤ 2. By duality (3.17)
also holds for 1 < p ≤ 2 and 1 < γ ≤ 2. Using Fatou’s lemma and (3.17) we get (1.8) for
1 < p < ∞ and 1 < γ ≤ 2.
P
(ε)
Case 2. 2 < γ ≤ ∞. Write SΩ,h (f ) = k∈Z σΩ,h,θ,k ∗ f. By Hölder’s inequality we have
γ 0 ! γ10
Z θk+1 Z
dt
|σ̂Ω,h,θ,k (ξ)| ≤ khkLγ (R+ ,dt/t)
Ω(x)e−itξ·x dσ(x)
t
θk
Sn−1
≤ (log θ)
2−γ 0
2γ 0
khkLγ (R+ ,dt/t) Hk (ξ),
where Hk (ξ) is defined as in (3.5). Thus, by the last inequality and (3.5) we obtain
n − λ λo
0
(3.18)
|σ̂Ω,h,θ,k (ξ)| ≤ C(log θ)1/γ kΩkq min θk ξ q0 , θk+1 ξ q0 .
0
By Lemma 2.3, (3.18) and invoking Lemma 2.1 with θ = 2q , ak = θk we obtain (1.8) for
2 < γ ≤ ∞ and γ 0 < p < ∞ with Cp independent of ε. Since γ 0 < 2, we get (1.8) for
2 ≤ p < ∞ and 2 < γ ≤ ∞. By duality and Fatou’s lemma we obtain (1.8) for 1 < p < ∞ and
2 < γ ≤ ∞. This completes the proof of (1.8).
The proof of (1.9) follows immediately from Theorem 1.5 and
(γ,∗)
MΩ,h f (x) ≤ MΩ
for any h ∈ Lγ (R+ , dt/t) , 1 < γ < ∞.
f (x)
Proofs of Theorems 1.7 and 1.8. A proof of each of these theorems follows by Theorems 1.4 –
1.6 and an extrapolation argument. For more details, see [6], [21].
J. Inequal. Pure and Appl. Math., 10(3) (2009), Art. 78, 15 pp.
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14
H USSAIN A L -Q ASSEM , L ESLIE C HENG ,
AND
Y IBIAO PAN
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