ON CERTAIN ROUGH INTEGRAL OPERATORS AND
EXTRAPOLATION
HUSSAIN AL-QASSEM
LESLIE CHENG
Department of Mathematics and Physics
Qatar University,Doha-Qatar
EMail: husseink@yu.edu.jo
Department of Mathematics
Bryn Mawr College, Bryn Mawr, PA 19010, U.S.A.
EMail: lcheng@brynmawr.edu
Rough Integral Operators
and Extrapolation
Hussain Al-Qassem, Leslie Cheng
and Yibiao Pan
vol. 10, iss. 3, art. 78, 2009
YIBIAO PAN
Department of Mathematics
University of Pittsburgh, Pittsburgh, PA 15260, U.S.A.
Title Page
Contents
EMail: yibiao@pitt.edu
JJ
II
J
I
Received:
24 January, 2009
Accepted:
31 August, 2009
Communicated by:
S.S. Dragomir
2000 AMS Sub. Class.:
Primary 42B20; Secondary 42B15, 42B25.
Key words:
Maximal operator, Rough kernel, Singular integral, Orlicz spaces, Block spaces,
Extrapolation, Lp boundedness.
Full Screen
Abstract:
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.
Close
Page 1 of 29
Go Back
Contents
1
Introduction and Main Results
3
2
Definitions and Some Basic Lemmas
10
3
Proof of the Main Results
15
Rough Integral Operators
and Extrapolation
Hussain Al-Qassem, Leslie Cheng
and Yibiao Pan
vol. 10, iss. 3, art. 78, 2009
Title Page
Contents
JJ
II
J
I
Page 2 of 29
Go Back
Full Screen
Close
1.
Introduction and Main Results
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.
Rough Integral Operators
and Extrapolation
Hussain Al-Qassem, Leslie Cheng
and Yibiao Pan
vol. 10, iss. 3, art. 78, 2009
Sn−1
For 1 ≤ γ ≤ ∞, let ∆γ (R+ ) denote the collection of all measurable functions
R
1/γ
R
h : [0, ∞) −→ C satisfying sup R1 0 |h(t)|γ dt
< ∞.
Title Page
Contents
R>0
Define the singular integral operator SΩ,h , the parametric Marcinkiewicz integral
(γ,∗)
(γ,∗)
operator 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,
JJ
II
J
I
Page 3 of 29
Rn
Go Back
Z
(1.3)
MΩ,h f (x) =
0
(1.4)
∞
(γ,∗)
SΩ
2 ! 12
dt
f (x − u) |u|ρ KΩ,h (u)du
,
t
|u|≤t
Z
−ρ
t
f (x) = sup |SΩ,h f (x)| ,
h
Full Screen
Close
(γ,∗)
MΩ
(1.5)
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]).
1/γ 0
(b) Ω ∈ L(log L) (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. More(0,0)
over, the condition Ω ∈ Bq (Sn−1 ) is an optimal size condition for the Lp
boundedness of SΩ (see [4] and [5]).
Rough Integral Operators
and Extrapolation
Hussain Al-Qassem, Leslie Cheng
and Yibiao Pan
vol. 10, iss. 3, art. 78, 2009
Title Page
Contents
JJ
II
J
I
Page 4 of 29
Go Back
Full Screen
Close
(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
|Ω(y)| log (2 + |Ω(y)|)dσ(y) < ∞
kΩkL(log L)α (Sn−1 ) =
Sn−1
(0,υ)
Bq (Sn−1 ),
and
υ > −1, is a special class of block spaces whose definition will
be recalled in Section 2.
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
(0,−1/2)
Ω ∈ Bq
MΩ ([3]).
Rough Integral Operators
and Extrapolation
Hussain Al-Qassem, Leslie Cheng
and Yibiao Pan
vol. 10, iss. 3, art. 78, 2009
Title Page
Contents
(Sn−1 ) for some q > 1 and h = 1. Moreover, the condition
JJ
II
(Sn−1 ) is an optimal size condition for the L2 boundedness of
J
I
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 has attracted
the attention of many authors. See for example, [14], [1], [5], [6], [8], [9], [27].
(γ,∗)
Theorem 1.3. If Ω satisfies one of the following conditions, then SΩ
on Lp (Rn ).
is bounded
(a) Ω ∈ C(Sn−1 ), (nγ)/(nγ − 1) < p < ∞ and 1 ≤ γ ≤ 2. Moreover, the range
of p is the best possible [14].
Page 5 of 29
Go Back
Full Screen
Close
(0,−1/2)
(b) Ω ∈ Bq
(Sn−1 ) for some q > 1, γ = 2, 2 ≤ p < ∞. Moreover, the condi-
(0,−1/2)
tion Ω ∈ Bq
(Sn−1 ) is an optimal size condition for the L2 boundedness
(2,∗)
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
(2,∗)
possible 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. 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 for example, [1], [2], [6], [8], [9]) in dealing with the
(γ,∗)
(γ,∗)
operators SΩ,h , MΩ,h , SΩ and MΩ when the kernel function Ω belongs to the
α
(0,υ)
block space Bq (Sn−1 ) (for υ > −1) or Ω belongs 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.
Rough Integral Operators
and Extrapolation
Hussain Al-Qassem, Leslie Cheng
and Yibiao Pan
vol. 10, iss. 3, art. 78, 2009
Title Page
Contents
JJ
II
J
I
Page 6 of 29
Go Back
Full Screen
Close
Theorem 1.5. Let α and Ω be as in Theorem 1.4 and let 1 ≤ γ < ∞. Then
1/β
q
(γ,∗) kΩkLq (Sn−1 ) kf kLp (Rn )
(1.7)
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.
q
Theorem 1.6. Suppose that Ω∈ L (S
some 1 < γ ≤ ∞. Then
(1.8)
n−1
γ
), 1 < q ≤ 2, and h ∈ L (R+ , dt/t) for
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)
Rough Integral Operators
and Extrapolation
Hussain Al-Qassem, Leslie Cheng
and Yibiao Pan
vol. 10, iss. 3, art. 78, 2009
Title Page
kMΩ,h (f )kLp (Rn ) ≤ Cp (q − 1)−1/β khkLγ (R+ ,dt/t) kΩkLq (Sn−1 ) kf kLp (Rn )
Contents
for (nβq 0 )/(βn + nq 0 − β) < p < ∞.
JJ
II
By the conclusions from Theorems 1.4 – 1.6 and applying an extrapolation method,
we get the following results:
J
I
Theorem 1.7.
Page 7 of 29
Go Back
1/γ 0
(γ,∗)
(a) If Ω ∈ L(log L) (Sn−1 ) and 1 < γ ≤ 2, the operator SΩ
Lp (Rn ) for 2 ≤ p < ∞;
(0,1/γ 0 −1)
(γ,∗)
(b) If Ω ∈ Bq
(Sn−1 ) and 1 < γ ≤ 2, the operator SΩ
Lp (Rn ) for 2 ≤ p < ∞;
1/γ 0
is bounded on
Full Screen
Close
is bounded on
(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
Lp (Rn ) for 2 ≤ p < ∞;
(0,1/β−1)
Bq
(Sn−1 )
(b) If Ω ∈
and 1 < γ < ∞, the operator
Lp (Rn ) for 2 ≤ p < ∞.
(γ,∗)
MΩ
is bounded on
is bounded on
Rough Integral Operators
and Extrapolation
Hussain Al-Qassem, Leslie Cheng
and Yibiao Pan
vol. 10, iss. 3, art. 78, 2009
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 < ∞;
Title Page
(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
(0,υ−1)
The question regarding the relationship between Bq
Sn−1 (for υ > 0) remains open.
JJ
II
J
I
Page 8 of 29
Go Back
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
(0,υ2 )
Bq
(Sn−1 ) ⊂ Bq(0,υ1 ) (Sn−1 ) for any − 1 < υ1 <
Lγ (R+ , dt/t) ⊂ ∆γ (R+ ) for 1 ≤ γ < ∞.
Contents
Full Screen
Close
υ2 ,
υ
and L(log L) over
(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
(γ,∗)
allow us to obtain the Lp boundedness of SΩ under optimal size conditions
on Ω.
p
3. We point out that it is still an open question whether the L boundedness of
(γ,∗)
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
(γ,∗)
1 ≤ p ≤ ∞. On the other hand, 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.
Rough Integral Operators
and Extrapolation
Hussain Al-Qassem, Leslie Cheng
and Yibiao Pan
vol. 10, iss. 3, art. 78, 2009
Title Page
Contents
JJ
II
J
I
Page 9 of 29
Go Back
Full Screen
Close
2.
Definitions and Some Basic Lemmas
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 consult the book [20]. In [20], Lu introduced
(0,υ)
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
Rough Integral Operators
and Extrapolation
Hussain Al-Qassem, Leslie Cheng
and Yibiao Pan
vol. 10, iss. 3, art. 78, 2009
(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
(0,υ)
1
n−1
(0,υ)
= Ω ∈ L (S ) : Ω =
{λµ } < ∞ ,
Bq
λµ bµ , Mq
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 .
{λµ } =
Mq
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.
Title Page
Contents
JJ
II
J
I
Page 10 of 29
Go Back
Full Screen
Close
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 )
f dσΩ,h,θ,k =
h(|u|)
f (u) du;
(2.1)
|u|n
Rn
θk ≤|u|<θk+1
(2.2)
∗
σΩ,h,θ
(f ) = sup ||σΩ,h,θ,k | ∗ f | .
k∈Z
Rough Integral Operators
and Extrapolation
Hussain Al-Qassem, Leslie Cheng
and Yibiao Pan
vol. 10, iss. 3, art. 78, 2009
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.
Lemma 2.3. 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
|gk |2 .
(ii) |σk ∗ gk |2 ≤ 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
holds for all f in Lp (Rn ).
p
Title Page
Contents
JJ
II
J
I
Page 11 of 29
Go Back
Full Screen
Close
∗
Lemma 2.4. 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 )
!1/γ 0
Z θk+1 Z
0
dt
|Ω(y)| |f (x − ty)|γ dσ(y)
×
.
t
Sn−1
θk
Rough Integral Operators
and Extrapolation
Hussain Al-Qassem, Leslie Cheng
and Yibiao Pan
vol. 10, iss. 3, art. 78, 2009
Title Page
Contents
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
1/γ 0
dσ(y)
,
JJ
II
J
I
Page 12 of 29
Go Back
where
My f (x) = sup R−1
R>0
Z
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 2.4 is thus proved.
By following an argument similar to that in [17] and [2], we obtain the following:
Full Screen
Close
Lemma 2.5. Let θ and Ω be as in Lemma 2.4. 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
! 12 ! 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
holds for arbitrary measurable functions {gk } on Rn .
Rough Integral Operators
and Extrapolation
Hussain Al-Qassem, Leslie Cheng
and Yibiao Pan
vol. 10, iss. 3, art. 78, 2009
Now, we need the following simple lemma.
Lemma 2.6. 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}
(2.4) Ω(ξ)f (ξ)dσ(ξ) ≤ kΩkq
|Ω(ξ)|max{0,2−q} |f (ξ)|2 dσ(ξ)
JJ
II
and
J
I
Sn−1
Z
(2.5)
Sn−1
Title Page
Contents
Page 13 of 29
|Ω(ξ)|max{0,2−q} |f (x − tξ)| dσ(ξ)
Go Back
Sn−1
≤ C kΩkmax{0,2−q}
MSph |f |β/2
q
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
Full Screen
Close
Proof. Let us first prove (2.4). First if q ≥ 2, by Hölder’s inequality we have
Z
2
20
Z
q
0
q
2
|f (ξ)| dσ(ξ)
n−1 Ω(ξ)f (ξ)dσ(ξ) ≤ kΩkq
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
q
2−q
0
= q 0 /2. The lemma is thus proved.
Rough Integral Operators
and Extrapolation
Hussain Al-Qassem, Leslie Cheng
and Yibiao Pan
vol. 10, iss. 3, art. 78, 2009
Title Page
Contents
JJ
II
J
I
Page 14 of 29
Go Back
Full Screen
Close
3.
Proof of the Main Results
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)
≤ Cp (q − 1)−1/γ kΩkLq (Sn−1 ) kf kLp (Rn )
SΩ (f )
Lp (Rn )
holds for 1 < q ≤ 2, (nq 0 γ 0 )/(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) = n−1 f (x − tu)Ω(u)dσ(u) ∞
S
L (R+ ,dt/t)
Z
=
n−1 f (x − tu)Ω(u)dσ(u) ∞
S
L (R+ ,dt)
Z
≤ sup
|f (x − tu)| |Ω(u)| dσ(u).
t>0
vol. 10, iss. 3, art. 78, 2009
Title Page
Contents
JJ
II
J
I
Page 15 of 29
Go Back
Sn−1
Full Screen
Using Hölder’s inequality we get
(1,∗)
SΩ f (x)
Rough Integral Operators
and Extrapolation
Hussain Al-Qassem, Leslie Cheng
and Yibiao Pan
q0
≤ kΩkq MSph |f |
1/q0
(x)
.
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.
Close
Proof of (3.1) for the case γ = 2. By Hölder’s inequality we obtain
(2,∗)
SΩ f (x)
Z
∞
≤
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
−k−1
−k+1
interval [θ
,θ
]. Specifically, we require the following:
X
ϕk ∈ C ∞ , 0 ≤ ϕk ≤ 1,
ϕk (t) = 1,
0
supp ϕk ⊆ [θ−k−1 , θ−k+1 ],
k∈Z
ds ϕk (t) Cs
dts ≤ ts ,
where Cs is independent of the lacunary sequence {θk : k ∈ Z}. Define the multin
d
ˆ
plier operators Tk in RnPby (T
k f )(ξ) = ϕk (|ξ|)f (ξ). Then for any f ∈ S(R ) and
k ∈ Z we have f (x) = j∈Z (Tk+j f )(x). Thus, by Minkowski’s inequality we have

(2,∗)
SΩ f (x)
≤
k∈Z
≤
θk+1
XZ
X
θk
2  12
X Z
dt

Ω(u)Tk+j f (x − tu)dσ(u)
t
n−1
j∈Z S
vol. 10, iss. 3, art. 78, 2009
Title Page
Contents
JJ
II
J
I
Page 16 of 29
Go Back
Full Screen
Xj f (x),
Close
j∈Z
where
Xj f (x) =
Rough Integral Operators
and Extrapolation
Hussain Al-Qassem, Leslie Cheng
and Yibiao Pan
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
1
kXj (f )kp ≤ Cp (q − 1)− 2 kΩkq 2−δp |j| kf kp
(3.2)
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
(Hk (ξ))2 fˆ(ξ)
(3.3)
kXj (f )k2 =
dξ,
t
∆k+l k∈Z
where ∆k = ξ ∈ Rn : θ−k−1 ≤ |ξ| ≤ θ−k+1 and
Z
θk+1
Hk (ξ) =
(3.4)
θk
vol. 10, iss. 3, art. 78, 2009
Title Page
2 ! 12
dt
.
Ω(x)e−itξ·x dσ(x)
t
Sn−1
Z
We claim that
n − λ λo
k q0 k+1 q0
, θ ξ
,
|Hk (ξ)| ≤ C(log θ) kΩkq min θ ξ
1
2
(3.5)
Rough Integral Operators
and Extrapolation
Hussain Al-Qassem, Leslie Cheng
and Yibiao Pan
Contents
JJ
II
J
I
Page 17 of 29
Go Back
where C is a constant independent of k, ξ and q.
Let us now turn to the proof of (3.5). First, by a change of variable and since
Z
2 Z
k tξ·x
k
−iθ
Ω(x)e
dσ(x) =
Ω(x)Ω(y)e−iθ tξ·(x−y) dσ(x)dσ(y),
Sn−1
Sn−1 ×Sn−1
we obtain
(3.6)
2
Z
Z
(Hk (ξ)) ≤
Ω(x)Ω(y)
Sn−1 ×Sn−1
θ
−iθk tξ·(x−y) dt
e
1
t
dσ(x)dσ(y).
Full Screen
Close
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),
t we get
Z θ
k −α 0
−iθk tξ·(x−y) dt θ ξ |ξ · (x − y)|−α for any 0 < α ≤ 1.
e
≤
C(log
θ)
t
1
vol. 10, iss. 3, art. 78, 2009
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
Rough Integral Operators
and Extrapolation
Hussain Al-Qassem, Leslie Cheng
and Yibiao Pan
!
1
2q 0
Title Page
.
αq 0
· (x − y)|
Contents
0
0
By choosing α so that αq < 1 we obtain that the last integral is bounded in ξ ∈
Sn−1 and hence
− α
1
(3.7)
|Hk (ξ)| ≤ C(log θ) 2 kΩk θk ξ 2q0 .
JJ
II
J
I
q
Page 18 of 29
Secondly, by the cancellation conditions on Ω we obtain
Go Back
2 ! 12
−itξ·x
dt
Hk (ξ) ≤
|Ω(x)| e
− 1 dσ(x)
t
θk
Sn−1
k+1 1
≤ C(log θ) 2 kΩk1 θ ξ .
Z
θk+1
Z
Full Screen
Close
1
By combining the last estimate with the estimate |Hk (ξ)| ≤ C(log θ) 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
(3.9)
1
kXj (f )k2 ≤ C2−λ|j| (q − 1)− 2 kΩkq kf k2 .
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 first consider the case p > 2. By duality, there is a nonnegative function g in
0
L(p/2) (Rn ) with kgk(p/2)0 ≤ 1 such that
2
Z X Z θk+1 Z
dt
2
kXj (f )kp =
Ω(u)T
f
(x
−
tu)dσ(u)
k+j
t g(x)dx.
n−1
Rn k∈Z θk
S
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
≤ kΩk1
|Ω(u)| |Tk+j f (x)|2 g(x + tu)dσ(u) dx
t
Rn k∈Z θk
Sn−1
!
Z
X
∗
≤ C kΩk1
|Tk+j f (x)|2 σΩ,1,θ
(g̃)(−x)dx,
Rn
vol. 10, iss. 3, art. 78, 2009
Title Page
Contents
JJ
II
J
I
Page 19 of 29
Go Back
k∈Z
∗
where g̃(x) = g(−x) and σΩ,h,θ
is defined as in (2.2). By the last inequality, Lemma
2.4 and using Littlewood-Paley theory ([24, p. 96]) we get
X
∗
2
2
kXj (f )kp ≤ C kΩk1 σΩ,1,θ (g̃) (p/2)0 |Tk+j f | k∈Z
≤
Rough Integral Operators
and Extrapolation
Hussain Al-Qassem, Leslie Cheng
and Yibiao Pan
Cp (log θ) kΩk21
kf k2p
,
(p/2)
Full Screen
Close
which in turn easily implies
(3.10)
kXj (f )kp ≤ Cp (q − 1)−1/2 kΩkq kf kp
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
hk,j (x, t) ∈ L
l L [1, θ] ,
, k , dx
t
kXj (f )kp =
Rn k∈Z
Z
=
XZ
1
Z
Ω(u) (Tk+j f ) (x − θk tu)hk,j (x, t)dσ(u)
Sn−1
1
XZ
Rn k∈Z
θ
θ
Z
Contents
dt
dx
t
dt
Ω(u) (Tk+j f ) (x)hk,j (x + θ tu, t)dσ(u) dx.
t
Sn−1
k
By Hölder’s inequality and Littlewood-Paley theory we get
! 12 X
1
2
kXj (f )kp ≤ (L(h)) 2 |Tk+j f |
(3.11)
p0 k∈Z
p
1
≤ Cp kf kp (L(h)) 2 ,
p0
vol. 10, iss. 3, art. 78, 2009
Title Page
such that
Z
Rough Integral Operators
and Extrapolation
Hussain Al-Qassem, Leslie Cheng
and Yibiao Pan
JJ
II
J
I
Page 20 of 29
Go Back
Full Screen
Close
where
L(h) =
X Z
k∈Z
θ
Z
dt
Ω(u)hk,j (x + θ tu, t)dσ(u)
t
Sn−1
1
k
2
.
1/2
Since p0 > 2 and (L(h))1/2 = kL(h)kp0 /2 , there is a nonnegative function b ∈
L
(p0 /2)0
p0
n
(R ) such that kbk(p0 /2)0 ≤ 1 and
X Z
Z
kL(h)kp0 /2 =
Rn k∈Z
θ
Z
dt
Ω(u)hk,j (x + θ tu, t)dσ(u)
t
Sn−1
1
k
2
b(x)dx.
By the Schwarz inequality and Lemma 2.6, 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 2.6 we get
kL(h)kp0 /2 ≤ C
min{2,q}
(log θ) kΩkLq (Sn−1 )
Z
×
Rn
XZ
k∈Z
1
θ
max{0,2−q}
kΩkLq (Sn−1 )
dt
|hk,j (x, t)|
t
2
!
0 20
q
q /2
MSph b̃
(−x)
dx
Rough Integral Operators
and Extrapolation
Hussain Al-Qassem, Leslie Cheng
and Yibiao Pan
vol. 10, iss. 3, art. 78, 2009
Title Page
Contents
JJ
II
J
I
Page 21 of 29
Go Back
Full Screen
Close
X Z θ
dt
|hk,j (x, t)|2 ≤ C (log θ) kΩk2Lq (Sn−1 ) t
k∈Z 1
p0 /2
0
2/q q0 /2
× MSph b̃
.
0 0
(p /2)
By the condition on p we have (2/q 0 )(p0 /2)0 > n/(n − 1) and hence by the Lp
boundedness of MSph and the choice of b we obtain
Rough Integral Operators
and Extrapolation
Hussain Al-Qassem, Leslie Cheng
and Yibiao Pan
kL(h)kp0 /2 ≤ C(q − 1)−1 kΩk2Lq (Sn−1 ) .
vol. 10, iss. 3, art. 78, 2009
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.
Title Page
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.
Contents
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
JJ
II
J
I
Page 22 of 29
Go Back
Full Screen
Close
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
for q 0 n0 ≤ 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
holds for 1 < q ≤ 2, (nq 0 γ 0 )/(nγ 0 + nq 0 − γ 0 ) < p < ∞ and 1 ≤ γ ≤ 2. This
completes the proof of Theorem 1.4.
Rough Integral Operators
and Extrapolation
Hussain Al-Qassem, Leslie Cheng
and Yibiao Pan
vol. 10, iss. 3, art. 78, 2009
Next we shall present the proof of Theorem 1.5.
Title Page
Proof. We need to consider two cases.
Contents
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

∞
Z

0
 12
Z
2
0
Ω(y )
dt
f (x − y) n−ρ h(|y|)dy 1+2σ 
1 t≤|y|≤t
t
|y|
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
JJ
II
J
I
Page 23 of 29
Go Back
[
2
f (x − sy)Ω(y)dσ(y) |h(s)|2 χ
[
1
2 t,t
]
1 t,t
2
]
(s)
(s)
ds
s1−σ
dt
1+2σ
t
2
dt
! 12
1+2σ
t
! 12
ds
s1−σ
Full Screen
Close
Z
Z ∞
12
dt
ds
|h(s)|
=
f
(x
−
sy)Ω(y)dσ(y)
1+2σ
n−1
s1−σ
t
s
0
ZS∞ Z
1
|h(s)| ds .
=√
f
(x
−
sy)Ω(y)dσ(y)
s
2σ 0
Sn−1
Therefore, by duality we have
Z
∞
1 (γ,∗)
(3.14)
≤ √ SΩ f (x),
2σ
and hence (1.7) holds by (1.6) for 1 ≤ γ ≤ 2.
(γ,∗)
MΩ f (x)
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
0
Using the generalized Minkowski inequality and since γ < 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,∗)
Since Es f (x) = SΩ
(3.16)
∞
2 ! 12
dt
f (x − stu)Ω (u) dσ(u)
.
t
Sn−1
Z
f (x), we obtain
(γ,∗)
MΩ
(2,∗)
f (x) ≤ SΩ
f (x),
and again (1.7) holds for the case 2 < γ ≤ ∞ by (1.6) in the case γ = 2.
Rough Integral Operators
and Extrapolation
Hussain Al-Qassem, Leslie Cheng
and Yibiao Pan
vol. 10, iss. 3, art. 78, 2009
Title Page
Contents
JJ
II
J
I
Page 24 of 29
Go Back
Full Screen
Close
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)
By definition of SΩ,h f (x) we have SΩ,h f (x) = lim SΩ,h f (x) for f ∈ S(Rn ),
ε→0
(ε)
where SΩ,h is 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 ∞
dt
(ε)
|h(t)| f (x − tu)Ω (u) dσ(u)
SΩ,h f (x) ≤
t
n−1
ε
≤
Rough Integral Operators
and Extrapolation
Hussain Al-Qassem, Leslie Cheng
and Yibiao Pan
vol. 10, iss. 3, art. 78, 2009
Title Page
Contents
S
(∗,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
JJ
II
J
I
Page 25 of 29
Go Back
Full Screen
Close
we have
Z
|σ̂Ω,h,θ,k (ξ)| ≤ khkLγ (R+ ,dt/t)
≤ (log θ)
2−γ 0
2γ 0
θk+1
θk
γ 0 ! γ10
dt
Ω(x)e−itξ·x dσ(x)
t
Sn−1
Z
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
1/γ 0
k q0 k+1 q0
(3.18)
|σ̂Ω,h,θ,k (ξ)| ≤ C(log θ)
kΩkq min θ ξ
, θ ξ
.
Rough Integral Operators
and Extrapolation
Hussain Al-Qassem, Leslie Cheng
and Yibiao Pan
vol. 10, iss. 3, art. 78, 2009
0
By Lemma 2.5, (3.18) and invoking Lemma 2.3 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
f (x)
for any h ∈ Lγ (R+ , dt/t) , 1 < γ < ∞.
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].
Title Page
Contents
JJ
II
J
I
Page 26 of 29
Go Back
Full Screen
Close
References
[1] H.M. AL-QASSEM, Maximal operators related to block spaces, Kodai Math.
J., 28 (2005), 494–510.
[2] H.M. AL-QASSEM, On the boundedness of maximal operators and singular
operators with kernels in L(log L)α (Sn−1 ), J. Ineq. Appl., 2006 (2006), 1–16.
[3] H.M. AL-QASSEM AND A. AL-SALMAN, A note on Marcinkiewicz integral
operators, J. Math. Anal. Appl., 282 (2003), 698–710.
[4] H.M. AL-QASSEM, A. AL-SALMAN AND Y. PAN, Singular integrals associated to homogeneous mappings with rough kernels, Hokkaido Math. J., 33
(2004), 551–569.
Rough Integral Operators
and Extrapolation
Hussain Al-Qassem, Leslie Cheng
and Yibiao Pan
vol. 10, iss. 3, art. 78, 2009
Title Page
p
[5] H.M. AL-QASSEM AND Y. PAN, L estimates for singular integrals with kernels belonging to certain block spaces, Revista Matemática Iberoamericana,
18(3) (2002), 701–730.
[6] H.M. AL-QASSEM AND Y. PAN, On certain estimates for Marcinkiewicz integrals and extrapolation, Collect. Math., 60(2) (2009), 123–145.
Contents
JJ
II
J
I
Page 27 of 29
p
[7] A. AL-SALMAN, H.M. AL-QASSEM, L. CHENG AND Y. PAN, L bounds
for the function of Marcinkiewicz, Math. Res. Lett., 9 (2002), 697–700.
[8] A. AL-SALMAN, On maximal functions with rough
L(log L)1/2 (Sn−1 ), Collect. Math., 56(1) (2005), 47–56.
kernels
in
[9] A. AL-SALMAN, On a class of singular integral operators with rough kernels,
Canad. Math. Bull., 49(1) (2006), 3–10.
[10] A. AL-SALMAN AND Y. PAN, Singular integrals with rough kernels in
L log+ L(Sn−1 ), J. London Math. Soc., 66(2) (2002), 153–174.
Go Back
Full Screen
Close
[11] A. BENEDEK AND R. PANZONE, The spaces Lp , with mixed norm, Duke
Math. J., 28 (1961), 301–324.
[12] J. BOURGAIN, Average in the plane over convex curves and maximal operators, J. Analyse Math., 47 (1986), 69–85.
[13] A.P. CALDERÓN AND A. ZYGMUND, On singular integrals, Amer. J. Math.,
78 (1956), 289–309.
[14] L.K. CHEN AND H. LIN, A maximal operator related to a class of singular
integrals, Illi. Jour. Math., 34 (1990), 120–126.
[15] J. DUOANDIKOETXEA AND J.L. RUBIO DE FRANCIA, Maximal functions
and singular integral operators via Fourier transform estimates, Invent. Math.,
84 (1986), 541–561.
[16] D. FAN AND Y. PAN, Singular integral operators with rough kernels supported
by subvarieties, Amer. J. Math., 119 (1997), 799–839.
[17] D. FAN, Y. PAN AND D. YANG, A weighted norm inequality for rough singular integrals, Tohoku Math. J., 51 (1999), 141–161.
[18] R. FEFFERMAN, A note on singular integrals, Proc. Amer. Math. Soc., 74
(1979), 266–270.
[19] H.V. LE, Maximal operators and singular integral operators along submanifolds, J. Math. Anal. Appl., 296 (2004), 44–64.
[20] S. LU, M. TAIBLESON AND G. WEISS, Spaces Generated by Blocks, Beijing
Normal University Press, 1989, Beijing.
[21] S. SATO, Estimates for singular integrals and extrapolation, Studia Math., 192
(2009), 219–233.
Rough Integral Operators
and Extrapolation
Hussain Al-Qassem, Leslie Cheng
and Yibiao Pan
vol. 10, iss. 3, art. 78, 2009
Title Page
Contents
JJ
II
J
I
Page 28 of 29
Go Back
Full Screen
Close
[22] E.M. STEIN, On the functions of Littlewood-Paley, Lusin and Marcinkiewicz,
Trans. Amer. Math. Soc., 88 (1958), 430–466.
[23] E.M. STEIN, Maximal functions: spherical means, Proc. Nat. Acad. Sci. USA,
73 (1976), 2174–2175.
[24] E.M. STEIN, Singular integrals and Differentiability Properties of Functions,
Princeton University Press, Princeton, NJ, 1970.
[25] E.M. STEIN, Harmonic Analysis: Real-Variable Mathods, Orthogonality and
Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993.
[26] T. WALSH, On the function of Marcinkiewicz, Studia Math., 44 (1972), 203–
217.
[27] H. XU, D. FAN AND M. WANG, Some maximal operators related to families
of singular integral operators, Acta. Math. Sinica, 20(3) (2004), 441–452.
Rough Integral Operators
and Extrapolation
Hussain Al-Qassem, Leslie Cheng
and Yibiao Pan
vol. 10, iss. 3, art. 78, 2009
Title Page
Contents
JJ
II
J
I
Page 29 of 29
Go Back
Full Screen
Close