c 2005 International Press
ASIAN J. MATH. Vol. 9, No. 1, pp. 019–030, March 2005
002
ROUGH SINGULAR INTEGRALS WITH KERNELS SUPPORTED BY SUBMANIFOLDS OF FINITE TYPE∗ HUSSAIN AL-QASSEM† , AHMAD AL-SALMAN‡ , AND YIBIAO PAN§ Key words. Lp boundedness, Singular integrals, Fourier transform, block spaces AMS subject classifications. Primary 42B20; Secondary 42B15, 42B25
1. Introduction. Let n ≥ 2, Rn be the n-dimensional Euclidean space, and S denote the unit sphere in Rn equipped with the normalized Lebesgue measure dσ. For d ∈ N, let B(0, 1) be the unit ball centered at the origin in Rn and Φ : B(0, 1) → Rd be a C ∞ mapping. Define the singular integral operator TΦ and the related maximal operator MΦ by Z Ω(y) TΦ f (x) = p.v. f (x − Φ(y)) n dy, (1.1) |y| B(0,1) n−1
1 n 00 ε≤|y| 0 such that kTΦ f kLp (Rd ) ≤ Cp kf kLp (Rd )
(1.5)
kMΦ f kLp (Rd ) ≤ Cp kf kLp (Rd )
(1.6)
and
∗ Received
November 10, 2003; accepted for publication February 28, 2004. of Mathematics, Yarmouk University, Irbid-Jordan (
[email protected]). ‡ Department of Mathematics, Yarmouk University, Irbid-Jordan (
[email protected]) § Department of Mathematics, University of Pittsburgh, PA 15260, U.S.A. (
[email protected]). † Department
19
20
H. AL-QASSEM, A. AL-SALMAN AND Y. PAN
for every f ∈ Lp (Rd ). Recently, the results in Theorem 1.1 were improved by Fan, Guo, and Pan in [FGP] who showed that the Lp boundedness of TΦ and MΦ continues to hold if the condition Ω ∈ C 1 (Sn−1 ) is replaced by the weaker condition Ω ∈ Lq (Sn−1 ) for some q > 1. Also, the authors of [FGP] were able to establish the Lp boundedness of the maximal operator TΦ∗ under the condition Ω ∈ Lq (Sn−1 ) for some q > 1. The main purpose of this paper is to present further improvements of the above results in which the condition Ω ∈ Lq (Sn−1 ) is replaced by a weaker condition Ω ∈ Bq0,0 (Sn−1 ). It is worth pointing out that the authors of this paper were able in [AqAsP] to show that the condition Ω ∈ Bq0,0 (Sn−1 ) is the best possible for the Lp boundedness of the classical operator T to hold. Namely, the Lp boundedness of T may fail for any p if it is replaced by a weaker condition Ω ∈ Bq0,υ (Sn−1 ) for any −1 < υ < 0 and q > 1.The definition of the block spaces Bq0,υ (Sn−1 ) on the sphere will be recalled in Section 2. Our main results can be stated as follows. Theorem 1.2. Let TΦ and MΦ be given as in (1.1)-(1.3). Assume that: (i) Φ is of finite type at 0; (ii) Ω ∈ Bq0,0 (Sn−1 ) for some q > 1. Then kTΦ f kLp (Rd ) ≤ Cp kf kLp (Rd )
(1.7)
and kMΦ f kLp (Rd ) ≤ Cp kf kLp (Rd ) hold for all 1 < p < ∞ and f ∈ Lp Rd .
(1.8)
Theorem 1.3. Let Ω and TΦ∗ be given as in (1.3)-(1.4). Assume that: (i) Φ is of finite type at 0; (ii) Ω ∈ Bq0,0 (Sn−1 ) for some q > 1. Then for 1 < p < ∞ there exists a constant Cp > 0 such that kTΦ∗ f kLp (Rd ) ≤ Cp kf kLp (Rd )
(1.9)
for every f ∈ Lp (Rd ). 2. Preliminaries. Let us begin with the definition of block functions on Sn−1 . Definition 2.1. (1) For x′0 ∈ Sn−1 and 0 < θ0 ≤ 2, the set B(x′0 , θ0 ) = {x′ ∈ Sn−1 : |x′ − x′0 | < θ0 } is called a cap on Sn−1 . (2) For 1 < q ≤ ∞, a measurable function b is called a q−block on Sn−1 if b is a −1/q′ function supported on some cap I = B(x′0 , θ0 ) with kbkLq ≤ |I| where |I| = σ(I) and 1/q + 1/q ′ = 1. P∞ (3) Bqκ,υ (Sn−1 ) = {Ω ∈ L1 (Sn−1 ): Ω = µ=1 cµ bµ where each cµ is a complex number; each b is a q−block supported on a cap Iµ on Sn−1 ; and Mqκ,υ ({ck }, {Ik }) P∞ µ = µ=1 cµ (1 + φκ,υ ( Iµ )) < ∞}, where Z 1 υ (2.1) φκ,υ (t) = χ(0,1) (t) u−1−κ log u−1 du. t
ROUGH SINGULAR INTEGRALS
21
One observes that υ
φκ,υ (t) ∼ t−κ log (t−1 ) as t → 0 for κ > 0, υ ∈ R, φ0,υ (t) ∼ logυ+1 t−1 as t → 0 for υ > −1.
The following properties of Bqκ,υ can be found in [KS]:
(i) Bqκ,υ2 ⊂ Bqκ,υ1 if υ2 > υ1 > −1 and κ ≥ 0; (ii) Bqκ2 ,υ2 ⊂ Bqκ1 ,υ1 if υ1 , υ2 > −1 and 0 ≤ κ1 < κ2 ;
(2.2) (2.3)
(iii) Bqκ,υ ⊂ Bqκ,υ if 1 < q1 < q2 ; 2 1
(2.4)
q
n−1
(iv) L (S
)⊂
Bqκ,υ (Sn−1 )
for υ > −1 and κ ≥ 0.
(2.5)
In their investigations of block spaces, Keitoku and Sato showed in [KS] that these spaces enjoy the following properties: Lemma 2.2. (i) If 1 < p ≤ q ≤ ∞, then for κ >
1 p′
we have
Bqκ,υ (Sn−1 ) ⊆ Lp (Sn−1 ) for any υ > −1; (ii) Bqκ,υ (Sn−1 ) = Lq (Sn−1 ) if and only if κ ≥ (iii) for any υ > −1, we have [ [ 0,υ n−1 )" q>1 Bq (S
q>1 L
q
1 and υ ≥ 0; q′
(Sn−1 ).
For a q-block function b on Sn−1 supported in an interval with q > 1 and kbkq ≤ |I| , 1/q + 1/q ′ = 1, we define the function ˜b on Sn−1 by Z ˜b(x) = b(x) − b(u)dσ(u). (2.6) −1/q′
Sn−1
Then one can easily see that ˜b enjoys the following properties: Z ˜b (u) dσ (u) = 0; Sn−1
˜ −1/q′ ;
b ≤ 2 |I| q
˜
b ≤ 2. 1
(2.7) (2.8) (2.9)
To simplify matters, we shall call the function ˜b the blocklike function corresponding to the block function b. We shall need the following two lemmas from [FGP]. Lemma 2.3. Let Φ : B(0, 1) → Rd be a smooth mapping and Ω be a homogeneous function of degree 0. Suppose that Φ is of finite type at 0 and Ω ∈ Lq (Sn−1 ) for some q > 1. Then there are N ∈ N, δ ∈ (0, 1], C > 0 and j0 ∈ Z− such that Z −iξ·Φ(y) Ω(y) Nj e |ξ|)−δ (2.10) n dy ≤ C kΩkq (2 2j−1 ≤|y| 1. Then there exists a constant C = C(m, n) > 0 such that Z X 1 −iP (y) Ω(y) mj ´ e |aα |)− 2qm n dy ≤ C kΩkq (2 2j−1 ≤|y| 0, A > 1, p0 ∈ (2, ∞) we have the following:
(l) (i) σk ≤ CA; − αAl (l) (ii) ˆ σk (ξ) ≤ CA akA L (ξ) ; l l αAl (l) (l−1) ; (iii) ˆ σk (ξ) − σ ˆk (ξ) ≤ CA akA l Ll (ξ) (iv)
X 2 1
(l) 2
( σk ∗ gk )
k∈Z
p0
X
1
2 ≤ CA ( |gk | ) 2
k∈Z
holds for all functions {gk } on Rn . Then for p′0 < p < p0 there exists a positive constant Cp such that
X
(N ) σk ∗ f ≤ Cp A kf kLp (Rn )
k∈Z Lp (Rn )
X 2 12
(N )
≤ Cp A kf kLp (Rn )
σk ∗ f
k∈Z
(2.11)
p0
(2.12)
(2.13)
Lp (Rn )
hold for all f in Lp (Rn ). The constant Cp is independent of the linear transformations N {Ll }l=1 . We shall also need the following result from [DR] (see also [AqP]): Lemma 2.6. Let {λj : j ∈ Z} be a sequence of Borel measures in Rn and let
23
ROUGH SINGULAR INTEGRALS
λ∗ (f ) = sup ||λj | ∗ f | . Assume that j∈Z
kλ∗ (f )kq ≤ B kf kq for some q > 1 and B > 1. Then, for arbitrary functions {gj } on Rn and p10 − holds
12
X
1
2 2 |λk ∗ gk |
≤ (B sup kλk k)
k∈Z k∈Z
p0
1 2
=
1 2q ,
(2.14)
the following inequality
12
X
2 |gk |
k∈Z
.
(2.15)
p0
3. Lp boundedness of certain maximal functions. For given sequences {µk } k∈Z and {τk } k∈Z of nonnegative Borel measures on Rn we define the maximal functions µ∗ and τ ∗ by µ∗ (f ) = sup |µk ∗ f | and τ ∗ (f ) = sup |τk ∗ f | . k∈Z
k∈Z
We have the following lemma. Lemma 3.1. Let {µk } k∈Z and {τk } k∈Z be sequences of nonnegative Borel measures on Rn . Let L: Rn → Rm be a linear transformation. Suppose that for all k ∈ Z, ξ∈ Rn , for some a ≥ 2, α, C > 0 and for some constant B > 1 we have (i) kµk k ≤ B; kτk k ≤ B; α (ii) |ˆ µk (ξ)| ≤ CB (a kB |L(ξ)| )− B ; α (iii) |ˆ µk (ξ)−ˆ τk (ξ)| ≤ CB (a kB |L(ξ)| ) B ; (iv) kτ ∗ (f )kp ≤ B kf kp for all 1 < p ≤ ∞ and f ∈ Lp (Rn ).
(3.1)
Then the inequality kµ∗ (f )kp ≤ Cp B kf kp
(3.2)
holds for all 1 < p ≤ ∞ and f in Lp (Rn ) with a constant Cp independent of B and L. Proof. By the arguments in the proof of Lemma 6.2 in [FP], we may assume n that m ≤ n and Lξ = πm ξ = (ξ1 , . . . , ξm ) for ξ = (ξ1 , . . . , ξn ). Now, choose and fix a m ˆ ˆ θ ∈ S (R ) such that θ(ξ) = 1 for |ξ| ≤ 1 and θ(ξ) = 0 for |ξ| ≥ 2. For each k ∈ Z, kB ˆ let (θkˆ)(ξ) = θ(a ξ), and define the sequence of measures {Υk } by n ˆ k (ξ) = µ ξ)ˆ τk (ξ). Υ ˆk (ξ) − (θkˆ)(πm
(3.3)
By (i)-(iii) we get ±α ˆ n ξ|) B Υk (ξ) ≤ CB(akB |πm
for ξ ∈ Rn . Let SΥ (f ) (x) = (
X
k∈Z
(3.4)
1
|Υk ∗ f (x)|2 ) 2 and Υ∗ (f ) = sup ||Υk | ∗ f | . k∈Z
24
H. AL-QASSEM, A. AL-SALMAN AND Y. PAN
Then by using (3.3) we have µ∗ (f ) (x) ≤ SΥ (f ) (x) + C(MRm ⊗ idRn−m )(τ ∗ (f ) (x)) ∗
∗
Υ (f ) (x) ≤ SΥ (f ) (x) + 2C[(MRm ⊗ idRn−m )](τ (f ) (x))
(3.5) (3.6)
where MRd is the classical Hardy-Littlewood maximal function on Rd . By (3.4) and Plancherel’s theorem we obtain kSΥ (f )k2 ≤ CB kf k2
(3.7)
which when combined with the Lp boundedness of MRd , (3.1), and (3.6)-(3.7) gives that kΥ∗ (f )k2 ≤ CB kf k2
(3.8)
with C independent of B. By using the fact kΥk k ≤ CB together with Lemma 2.6 (for q = 2) we get
X
X
2 12 2 12 (|Υk ∗ gk | ) ≤ Cp0 B ( |gk | ) (3.9)
(
k∈Z
p0
k∈Z
p0
if 1/4 = |1/p0 − 1/2|. Now, by (3.4), (3.9) and applying Lemma 2.5 we get kSΥ (f )kp ≤ Cp B kf kp for p ∈ (
4 , 4). 3
(3.10)
Again, the Lp boundedness of MRd , (3.1), (3.6) and (3.10) imply that kΥ∗ (f )kp ≤ CB kf kp for p ∈ (
4 , 4). 3
(3.11)
Reasoning as above, (3.4), (3.11), Lemma 2.5 and Lemma 2.6 provide kSΥ (f )kp ≤ Cp B kf kp for p ∈ (
8 , 8). 7
(3.12)
By successive application of the above argument we ultimately obtain that kSΥ (f )kp ≤ Cp B kf kp for p ∈ (1, ∞).
(3.13)
Therefore, by the Lp boundedness of MRd , (3.1), (3.5) and (3.13) we conclude that kµ∗ (f )kp ≤ Cp B kf kp for p ∈ (1, ∞).
(3.14)
Finally, the inequality (3.2) holds trivially for p = ∞. This concludes the proof of our lemma. Definition 3.2. Let ˜b(·) be a blocklike function defined as in (2.2) and Γ be an arbitrary function on Rn . Define the measures {σΓ,˜b,j : j ∈ Z} and the maximal ∗ operator σΓ, on Rn by ˜ b Z
Rd
˜b (u) f (Γ (u)) n du; |u| 2j−1 ≤|u| 0 and j0 ∈ Z− such that δ − ˆ λΦ,˜b,j (ξ) ≤ C[log(|I|)](ω N j |ξ|) [log(|I|−1 )]
(3.19)
for all j ≤ j0 , ξ ∈ Rd with C independent of j and [log(|I|−1 )]. Proof. By (2.4), Lemma 2.3 and the definition of λΦ,˜b,j we get Z [log(|I|−1 )]−1 X ˜ b(y) ˆ e−iξ·Φ(y) n dy λΦ,˜b,j (ξ) ≤ |y| s=0 ω(j−1) 2s ≤|y| 1) defined as in (2.2) and R(·) be a real-valued polynomial on Rn with deg(R) ≤ m − 1. Suppose X α P (y) = aα y + R(y), |α|=m
26
H. AL-QASSEM, A. AL-SALMAN AND Y. PAN
and |I| < e−2 . Then there exists a constant C = C(m, n) > 0 such that Z X ˜ 1 − −1 −iP (y) b(y) −1 )] ´ e dy |aα |) 2qm[log(|I| ≤ C[log(|I| )](ω mj n ωj−1 ≤|u| 1 let ˜b be a q-blocklike function defined as in (2.2). Suppose that Φ is of finite type at 0. Then for 1 < p ≤ ∞ and f ∈ Lp (Rd ) there exists a positive constant Cp which is independent of ˜b such that
∗
−1 (3.20)
λΦ,˜b (f ) p d ≤ Cp [log(|I| )] kf kLp (Rd ) if |I| < e−2 ; L (R )
∗
≤ Cp kf kLp (Rd ) if |I| ≥ e−2 . (3.21)
σΦ,˜b (f ) Lp (Rd )
Proof. Assume first that |I| < e−2 . Without loss of generality we may assume that ˜b ≥ 0. By Lemma 3.4, there are N ∈ N, δ ∈ (0, 1], C > 0 and k0 ∈ Z− such that δ − ˆ −1 λΦ,˜b,k (ξ) ≤ C[log(|I| )](ω N k |ξ|) [log(|I|−1 )]
(3.22)
27
ROUGH SINGULAR INTEGRALS
for all k ≤ k0 , ξ ∈ Rd with C independent of k and [log(|I| For Φ = (Φ1 , . . . , Φd ) we let P = (P1 , . . . , Pd ) where
−1
)] where ω = 2[log(|I|
−1
)]
.
β
Pj (y) =
X
|β|≤N −1
β 1 ∂ Φj (0)y , β! ∂y β
1 ≤ j ≤ d.
Then we have ˆ ˆ ˜ (ξ) ≤ C[log(|I|−1 )]ω −N ω N k |ξ| . λΦ,˜b,k (ξ) − λ P,b,k
(3.23)
ˆ ˆ ˜ (ξ) ≤ C[log(|I|−1 )]. λΦ,˜b,k (ξ) − λ P,b,k
(3.24)
By (2.5) we have
By interpolating between this estimate and (3.23) we get
δ [log(|I|−1 )] ˆ −1 Nk ˆ . λΦ,˜b,k (ξ) − λP,˜b,k (ξ) ≤ C[log(|I| )] ω |ξ|
(3.25)
Therefore, (3.20) follows from (3.22), (3.25), Lemma 3.1 and Lemma 3.7. The proof of the inequality (3.21) will be much easier. In fact, it follows from (2.4)-(2.5), Lemma 2.3, 3.1, and 3.6. We omit the details. 4. Proofs of the theorems. By assumption, Ω can be written as Ω = where cµ ∈ C, bµ is a q-block with support on an ia cap Iµ on Sn−1 and ∞ X cµ 1 + (log |Iµ |−1 ) < ∞.
Mq0,0 ({ck }, {Ik }) =
P∞
µ=1
cµ b µ
(4.1)
µ=1
For each µ = 1, 2, ..., let ˜bµ be the blocklike function corresponding to bµ . By the vanishing condition on Ω we have Ω=
∞ X
cµ ˜bµ
(4.2)
µ=1
and hence kTΦ f kp ≤
∞ X cµ
T
µ=1
where TΦ,˜bµ f (x) = p.v.
Z
Φ,˜ bµ f p
f (x − Φ(u))
B(0,1)
,
(4.3)
˜b (u′ ) µ n du. |u|
Let δ, N, P be given as in the proof of Theorem 3.8. For 1 ≤ j ≤ d, let aj,β = β
1 ∂ Φj β! ∂y β (0).
For 0 ≤ l ≤ N − 1 we define Ql = (Ql1 , . . . , Qld ) by Qlj (y) =
X
|β|≤l
β
aj,β y ,
j = 1, . . . , d
(4.4)
28
H. AL-QASSEM, A. AL-SALMAN AND Y. PAN (l) µ ,k
when 0 ≤ l ≤ N − 1 and QN = Φ. For each 0 ≤ l ≤ N , let λ˜b
(l) µ ,k
= λQl ,˜bµ ,k and σ˜b
= σQl ,˜bµ ,k . Then by (2.3)-(2.5), Lemma 2.4 we have
(l) (4.5)
σ˜b ,k ≤ C; µ d X X − 1′ (l) ) 2q l ; σ˜b ,k (ξ) ≤ C(2lk a ξ (4.6) ˆ j,β j µ |β|=l j=l (N ) (N −1) σ˜b ,k (ξ) − σ ˆ˜b ,k (ξ) ≤ C(2N k |ξ|); (4.7) ˆ µ µ d X X (l) (l−1) lk (4.8) ˆ˜b ,k (ξ) − σ ˆ˜b ,k (ξ) ≤ C(2 aj,β ξj ) σ µ µ |β|=l j=l for Iµ ≥ e−2 , µ = 1, 2, . . . , 0 ≤ l ≤ N − 1, and k ≤ k0 . Also, by (2.3)-(2.5), Lemma 3.5, and the same argument as in the proof (3.25) we have
(l) (4.9)
λ˜b ,k ≤ CAµ ; µ d X X − 1 ′ ˆ (l) lAµ k aj,β ξj ) Aµ 2q l ; (4.10) λ˜b ,k (ξ) ≤ CAµ (2 µ |β|=l j=l d 1 X X ˆ (l) ′ (l−1) lAµ k ˆ (4.11) aj,β ξj ) Aµ 2q l λ˜b ,k (ξ) − λ˜b ,k (ξ) ≤ CAµ (2 µ µ |β|=l j=l
−1 where Aµ = [log( Iµ )], Iµ < e−2 , µ = 1, 2, . . . , k ≤ k0 , 0 ≤ l ≤ N − 1. By (3.20)-(3.22), (3.25), (4.5)-(4.11), Theorem 3.8, Lemmas 2.5-2.6, and 3.6-3.7 we get
X
(N )
≤ Cp Aµ kf k if Iµ < e−2 ; λ ∗ f (4.12)
TΦ,˜bµ f = ˜ p
b ,k p
j∈Z− µ
p
X (N )
−2 σ˜b ,k ∗ f (4.13)
TΦ,˜bµ f =
≤ Cp kf kp if Iµ ≥ e , µ p
j∈Z−
p
p
d
for every f ∈ L (R ), µ = 1, 2, ..., and for all p, 1 < p < ∞. Hence, (1.7) follows from (4.1), (4.3) and (4.12)-(4.13). On the other hand, (1.8) follows from (3.20)-(3.21), (4.2) and the following inequality ∞ X ∗ cµ σ MΦ f (x) ≤ 4 µ=1
≤4
∞ X
µ=1,|Iµ |≥e−2
∗ c σ ˜ (|f |) (x) + 8 µ Φ,b µ
Φ,˜ bµ
∞ X
(|f |) (x)
µ=1,|Iµ |
