A unifying approach for certain class of maximal functions

MAXIMAL FUNCTIONS

Received 16 January 2006; Revised 12 April 2006; Accepted 13 April 2006

We establishLp_{estimates for certain class of maximal functions with kernels inL}q_(Sn−1_). As a consequence of suchLp _{estimates, we obtain theL}p_{boundedness of our maximal} functions when their kernels are inL(logL)1/2_(Sn−1_{) or in the block spaceB}0,−1/2

q (Sn−1), q >1. Several applications of our results are also presented.

Copyright © 2006 Ahmad Al-Salman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction and statement of results

LetRn_{,n≥2, be then-dimensional Euclidean space and letS}n−1 _{be the unit sphere in} Rn_{equipped with the normalized Lebesgue measuredσ. For nonzeroy∈R}n_{, we will let} y= |y|−1_{y. LetΩbe an integrable function onS}n−1_{that is homogeneous of degree zero} onRn_{and satisfies the cancelation property}

Sn−1Ω(y

_{)dσ(y)=0. (1.1)}

Consider the maximal functionᏹΩ,

ᏹΩ(f)(x)=sup h∈U

Rn f(x−y)|y|

−n_{h|y|Ω(y)d y, (1.2)}

whereUis the class of allh∈L2_(R+,r−1_{dr) withh}

L2_(R+,r−1_dr)≤1.

The operator ᏹΩ was introduced by Chen and Lin [7]. They showed that ᏹΩ is bounded on Lp_(Rn_{) for all p >2n/(2n−1) provided that Ω∈Ꮿ(S}n−1_{). Recently, we} have been able to show that theLp_(Rn_{) boundedness ofᏹ}

Ω still holds for all p≥2 if the conditionΩ∈Ꮿ(Sn−1_{) is replaced by the more natural and weaker conditionΩ∈} L(logL)1/2_(Sn−1_{) [2]. Moreover, we showed that if the conditionΩ∈L(logL)}1/2_(Sn−1_{) is} replaced by any condition in the form Ω∈L(logL)r_(Sn−1_{) for some r <1/2, thenᏹ}

Ω might fail to be bounded onL2_.

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 56272, Pages1–17

On the other hand, whenΩlies inB0,−1/2

s (Sn−1),s >1, which is a special class of block spaces Bκ,υ

q (Sn−1) (seeSection 5 for the definition), we were able to show thatᏹΩ is bounded on Lp _{for all p≥2 [3]. Moreover, we showed that the condition Ω∈} B0,−1/2

s (Sn−1),s >1 is nearly optimal in the sense that the exponent−1/2 cannot be re-placed by any smaller number for theL2 _{boundedness ofᏹ}

Ωto hold. We remark here that block spaces have been introduced by Jiang and Lu to improve previously obtained Lp_{boundedness results for singular integrals [7]. It should be noted here that the relation} between the spacesB0,−1/2

s (Sn−1) andL(logL)1/2(Sn−1) is unknown.

However, it is known that Lq_(Sn−1_{) is properly contained in L(logL)}1/2_(Sn−1_)∩ B0,−1/2

s (Sn−1) for all q, s > 1. Moreover, it is not hard to see that every Ω in L(logL)1/2_(Sn−1_)∪B0,−1/2

s (Sn−1) can be written as an infinite sum of functions inLq(Sn−1). This gives rise to the question whether the results pertaining theLp_{boundedness ofᏹΩin} [2,3] can be obtained via certain correspondingLp_{estimates with kernels inL}q_(Sn−1_{). It} is one of our main goals in this paper to consider such problem. It should be pointed out here that a positive solution for this problem will not only make life easier when dealing with kernels inL(logL)1/2_(Sn−1_{) orB}0,−1/2

s (Sn−1), but also will pave the way for extending several results that are known when kernels are inLq_(Sn−1_).

Our work in this paper will be mainly concerned with the following general class of maximal functions:

ᏹΩ,P(f)(x)=sup h∈U

Rne

iP(y)_f(x−y)|y|−n_{h|y|Ω(y)d y, (1.3)}

whereP:Rn_{→Ris a real-valued polynomial.}

Clearly, ifP(y)=0, thenᏹΩ,P=ᏹΩ. For the significance of considering integral op-erators with oscillating kernels, we refer the readers to consult [1,4,11,16,19,22–24], among others.

Our result concerningLp_{estimates with kernels inL}q_(Sn−1_{) is the following theorem.}

Theorem1.1. Let Ω∈Lq_(Sn−1_{),q >1, be a homogeneous function of degree zero onR}n withΩ1≤1. LetP:Rn_{→Rbe a real-valued polynomial of degreed. Letᏹ}

Ω,Pbe given by (1.3). Then

_ᏹΩ,P(f)

p≤

1 + log1/2e+ΩqCp,qfp (1.4)

for allp≥2, whereCp,q=(21/q

/(21/q_−1))C

p. Here1/q=1−1/qandCpis a constant that may depend on the degree of the polynomialPbut it is independent of the functionΩ, the indexq, and the coefficients of the polynomialP.

We remark here that the constantCp,q inTheorem 1.1satisfiesCp,q→ ∞asq→1+. That is, the constantCp,q diverges when q tends to 1. This behavior ofCp,q is natural since, by [2, Theorem B(b)], the special operatorᏹΩ=ᏹΩ,0is not bounded onL2if the functionΩis assumed to satisfy only the sole condition thatΩ∈L1_(Sn−1_{) (i.e.,q=1).}

Theorem1.2. Suppose thatΩ∈L(log+L)1/2_(Sn−1_{)satisfying (1.1). LetP:R}n_{→Rbe a} real-valued polynomial. Then ᏹΩ,P is bounded onLp(Rn)for all p≥2with Lp bounds independent of the coefficients of the polynomialP.

We should point out here that an alternative proof ofTheorem 1.2can be obtained by observing thatCp,q≈Cp/(q−1), whereCp,qis the constant inTheorem 1.1, and then using a Yano-type extrapolation technique [27].

By another suitable application ofTheorem 1.1, we will prove the following extension of [3, Theorem 1.2].

Theorem1.3. Suppose thatΩ∈B0,−1/2

q (Sn−1),q >1, satisfying (1.1). LetP:Rn→Rbe a real-valued polynomial. ThenᏹΩ,P is bounded onLp(Rn)for all p≥2withLpbounds independent of the coefficients of the polynomialP.

As an immediate consequence ofTheorem 1.1and the observation that

_{TΩ,P,h(f)(x)≤ hL}2_(R+,r−1_dr)ᏹ_Ω,P(f)(x), (1.5)

we obtain the following result concerning oscillatory singular integrals.

Theorem1.4. Let Ω∈Lq_(Sn−1_{),q >1, be a homogeneous function of degree zero onR}n with Ω1≤1. Let P : Rn_{→R be a real-valued polynomial of degree d and let h∈} L2_(R+,r−1_{dr). Then the oscillatory singular integral operatorᏹ}

Ω,P;

TΩ,P,h(f)(x)=p·v

Rne

iP(y)_f(x−y)|y|−n_{h|y|Ωy)d y (1.6)}

satisfies

_TΩ,P,h(f)

p≤

1 + log1/2e+ΩqhL2_(R+,r−1_dr)C_p,qf_p (1.7)

for allp≥2, whereCp,q=(21/q

/(21/q_−1))C

p. Here1/q=1−1/qandCpis a constant that may depend on the degree of the polynomialPbut it is independent of the functionΩ, the indexq, and the coefficients of the polynomialP.

ByTheorem 1.4, we obtain the following two results.

Corollary1.5. Let Ω∈L(logL)1/2_(Sn−1_{)be a homogeneous function of degree zero on} Rn_{and satisfies (1.1). LetP:R}n_{→Rbe a real-valued polynomial of degreedand leth∈} L2_(R+,r−1_{dr). Then the oscillatory singular integral operatorᏹ}

Ω,P;

TΩ,P,h(f)(x)=p·v

Rne

iP(y)_f(x−y)|y|−n_{h|y|Ω(y)d y (1.8)}

is bounded onLp_{for allp≥2withL}p_{bounds that may depend on the degree of the} polyno-mialPbut they are independent of the coefficients of the polynomialP.

Corollary1.6. Let Ω∈B0,−1/2

h∈L2_(R+,r−1_{dr). Then the oscillatory singular integral operatorᏹ} Ω,P;

TΩ,P,h(f)(x)=p·v

Rne

iP(y)_f(x−y)|y|−n_{h|y|Ω(y)d y (1.9)}

is bounded onLp_{for allp≥2withL}p_{bounds that may depend on the degree of the} polyno-mialPbut they are independent of the coefficients of the polynomialP.

Further applications of the results stated above will be presented inSection 6.

Throughout this paper, the letterCwill stand for a constant that may vary at each occurrence, but it is independent of the essential variables.

2. Preliminary estimates

We start by recalling the following result in [10].

Lemma 2.1 (see [10]). Letᏼ=(P1,. . .,Pd)be a polynomial mapping from Rn intoRd. Suppose thatΩ∈L1_(Sn−1_)and

MΩ,ᏼf(x)=sup

j∈Z

2j_≤|y|<2j+1

_f

x−ᏼ(y)|y|−n_{Ωy)d y. (2.1)}

Then for1< p≤ ∞, there exist a constantCp>0independent ofΩand the coefficients of P1,. . .,Pdsuch that

_MΩ,ᏼf

p≤CpΩL1_(Sn−1)f_p (2.2)

for every f ∈Lp_(Rd_).

Lemma2.2 (van der Corput [26]). Supposeφis real valued and smooth in(a,b), and that |φ(k)_{(t)| ≥1for allt∈(a,b). Then the inequality}

_abe−iλφ(t)ψ(t)dt≤Ck|λ|−1/k (2.3)

holds when (i)k≥2, or

(ii)k=1andφis monotonic.

The boundCkis independent ofa,b,φ, andλ.

Lemma2.3. LetΩ∈Lq_(Sn−1_{),q >1, be a homogeneous function of degree zero onR}n_with Ω1≤1. Let P(x)= |α|≤daαxα be a real-valued polynomial of degreed >1such that |x|d _{is not one of its terms. Fork∈Z, letE}

k,Ω: [1, log(e+Ωq)]×P(Sn−1)×R→Cand letJk,Ω:Rn→Rbe given by

Ek,Ωr,P(y),s=e−i[P(2−(k+1) log(e+Ωq)r y)+2−(k+1) log(e+Ωq)sr],

Jk,Ω(ξ)=

22 log(e+Ωq)

1

Sn−1Ω(y

_)E

k,Ωr,P(y),ξ·ydσ(y) 2_d

rr.

Then,Jk,Ωsatisfies

sup ξ∈Rn

Jk,Ω(ξ)≤2(k+1)/4qloge+Ωq

|α|=d

_aα−ε/q_{C (2.5)}

for some0< ε <1, whereCis a constant that may depend on the degree of the polynomialP but it is independent of the functionΩ, the indexq, and the coefficients of the polynomialP. Proof ofLemma 2.3. First, we notice the following:

Jk,Ω(ξ)≤loge+Ωq, (2.6)

Jk,Ω(ξ)q

≤ Ω2qq

Sn−1

22 log(e+Ωq

)

1 Ek,Ω

r,P(y),ξ·y

×Ek,Ω

r,P(z_),ξ·zdr

r

qdσ(y)dσ(z). (2.7)

Next, notice that

P2−γk,Ω_{r y+ 2}−γk,Ω_{(ξ·y)r−P2}−γk,Ω_{rz+ 2}−γk,Ω_(ξ·z)r

=2−γk,Ωd_rd

|α|=d aαyα−

|α|=d aαzα

+ 2−γk,Ω_{ξ·(y−z)r+H}

k(r,y,z,ξ) (2.8)

with (dd_/drd_)H

k=0, whereγk,Ω=(k+ 1) log(e+Ωq). Thus, byLemma 2.2, we have

22 log(e+Ωq)

1 Ek,Ω

r,P(y),ξ·yEk,Ωr,P(z),ξ·zdr_r

≤2−dγk,ΩP(y)−P(z)−1/d.

(2.9)

Now, by (2.9) and the inequality

22 log(e+Ωq)

1 Ek,Ω

r,P(y),ξ·yEk,Ω

r,P(z_),ξ·zdr

r

≤Cloge+Ωq

, (2.10) we obtain

22 log(e+Ωq)

1 Ek,Ω

r,P(y),ξ·yEk,Ω

r,P(z_),ξ·zdr

r

≤2−dγk,Ω_P(y)−P(z)−1/4dq_Cloge+Ω

q

1−1/4q

.

(2.11)

Therefore, by (2.7), (2.11), and [12, (3.11)], we obtain

Jk,Ω(ξ)≤2γk,Ω/4qΩ 2q

q Cloge+Ωq

1−1/4q

Hence, by (2.6) and (2.12), we get

Jk,Ω(ξ)≤2γk,Ω/4 log(e+Ωq)q_Ω2/log(e+Ωq)

q loge+Ωq

≤2(k+1)/4q_loge+Ω qC.

(2.13)

This completes the proof.

Now, we will need the following lemma.

Lemma2.4. LetΩ∈Lq_(Sn−1_{),q >1, be a homogeneous function of degree zero onR}n_with Ω1≤1. Then

_ᏹΩ(f)

p≤log 1/2_e

+Ωq

21/q

21/q

−1

Cpfp (2.14)

for allp≥2with constantsCpindependent of the functionΩand the indexq.

We remark here that sinceLq_(Sn−1_)⊂Llog1/2_{L, it follows from [2, Theorem B(a)] that} ᏹΩp≤ ΩqCpfor allp≥2. But, clearly the constant{1 + log1/2(e+Ωq)}in (2.14) is sharper than the constantΩqthat can be deduced from [2, Theorem B(a)]. However, the former constant can be obtained by following a similar argument as in the proof of Theorem B(a) in [2] and keeping track of certain constants. For completeness, we, below, present the main ideas of the proof.

Proof ofLemma 2.4. Choose a collection of Ꮿ∞ functions {ωk}k∈Z on (0,∞) with the properties sup(ωk)⊆[2−log(e+Ωq)(k+1), 2−log(e+Ωq)(k−1)], 0≤ωk≤1, k∈Zωk(u)=1, |(ds_ω

k/dus)(u)| ≤Csu−s, whereCsis a constant independent of log(e+Ωq). Fork∈Z, letGkbe the operator defined by (Gk(f))(ξ)=ωk(|ξ|)f(ξ). Let

Ej(f)(x)=

k∈Z

22 log(e+Ωq)

1

Sn−1Ω(y

_)G

k+j(f)

x−2klog(e+Ωq)_{r ydσ(y)} 2

r−1_dr

1/2 .

(2.15)

Then

ᏹΩ(f)(x)≤

j∈Z

Ej(f)(x). (2.16)

By exactly the same argument in [2], we obtain

_Ej(f)

2≤C2−β|j|/q

log1/2e+Ωq

f2. (2.17)

On the other hand, by a duality argument; see (3.24)-(3.25) for similar argument, we get

_Ej(f)

p≤Clog 1/2

e+Ωqfp (2.18)

for all 2< p <∞. Thus, by interpolation between (2.17) and (2.18), we have

_Ej(f)

p≤C2−ε(|j|/q ₎

log1/2e+Ωq

for someε >0 and for all 2≤p <∞, andj∈Zwith constantCindependent ofΩ,k, and j. Hence, (2.14) follows by (2.16) and (2.19). This completes the proof.

3. Proof ofTheorem 1.1

Proof ofTheorem 1.1. We will argue by induction on the degree of the polynomialP. If d=deg(P)=0, then (1.4) follows easily fromLemma 2.4. In fact, ifd=0, then by duality it can be easily seen that

ᏹΩ,P(f)(x)≤CᏹΩ(f)(x). (3.1)

Thus, byLemma 2.4, we have

_ᏹΩ,P(f)

p≤

21/q

21/q

−1

log1/2e+Ωq

Cpfp

≤

₂1/q

21/q

−1

1 + log1/2e+Ωq

Cpfp

(3.2)

for allp≥2.

Now, ifd=1, that is,P(y)= −→a·yfor some−→a∈Rn_{, then by (3.2), we have}

_ᏹΩ,P(f)

p≤

21/q 21/q

−1

log1/2ΩqCpgp

=

21/q 21/q

−1

1 + log1/2e+Ωq

Cpfp,

(3.3)

whereg(y)=e−iP(y)_f(y).

Next, assume that (1.4) holds for all polynomialsQ of degree less than or equal to d >1. Let

P(x)=

|α|≤d+1

aαxα (3.4)

be a polynomial of degreed+ 1. Then by duality, we have

ᏹΩ,P(f)(x)=

_∞

0

Sn−1e iP(r y₎

Ω(y)f(x−r y)dσ(y) 2

r−1dr

1/2

. (3.5)

We may assume thatPdoes not contain|x|d+1_{as one of its terms. By dilation} invari-ance, we may also assume that

|α|=d+1

We now choose a collection{ωk}k∈ZofᏯ∞functions defined on (0,∞) that satisfy the following properties:

suppψk

⊆2−log(e+Ωq)(k+1)_{, 2}−log(e+Ωq)(k−1)_{, 0}≤_ψ k≤1,

k∈Z

ψk(u)=1. (3.7)

Set

η∞(u)=

0

k=−∞

ψk(u), η0(u)=

∞

k=1

ψk(u). (3.8)

Then,

η∞(u) +η0(u)=1,

suppη∞(u)⊂2−log(e+Ωq),∞_{, suppη0(u)}⊂_{(0, 1].} (3.9)

Define the operators᏿Ω,P,∞and᏿Ω,P,0by

᏿Ω,P,∞(f)(x)=

_∞

2−log(e+Ωq)

η∞(r)

Sn−1e iP(r y₎

Ω(y)f(x−r y)dσ(y) 2

r−1dr

1/2 ,

᏿Ω,P,0(f)(x)=

1

0

η0(r)

Sn−1e

iP(r y)_{Ω(y)f(x−r y)dσ(y)}2_r−1_dr

1/2 .

(3.10)

Thus, by (3.9), we have

᏿Ω,P(f)(x)≤᏿Ω,P,0(f)(x) +᏿Ω,P,∞(f)(x). (3.11)

Now, we estimate᏿Ω,P,0p. Let

Q(x)=

|α|≤d

aαxα. (3.12)

Assume that deg(Q)=l, where 0≤l≤d. Define the operators᏿(1)_Ω,P,0and᏿ (2) Ω,Q,0by

᏿(1)

Ω,P,0(f)(x)=

1

0

Sn−1

eiP(r y)−eiQ(r y)Ω(y)f(x−r y)dσ(y) 2

r−1dr

1/2 ,

᏿(2)

Ω,Q,0(f)(x)=

1

0

Sn−1e iQ(r y₎

Ω(y)f(x−r y)dσ(y) 2

r−1dr

1/2 .

(3.13)

Now, by the observation thatη0(r)≤1 and by Minkowski’s inequality, we obtain

᏿Ω,P,0(f)(x)≤᏿(1)Ω,P,0(f)(x) +᏿ (2)

By induction assumption, it follows that

_᏿(2)

Ω,Q,0(f)p≤

1 + log1/2(e+Ωq)

21/q

21/q

−1

Cpfp (3.15)

for allp≥2.

On the other hand, by Cauchy-Schwarz inequality, by the fact thatΩ1≤1, and the inequality

_eiP(r y)_−eiQ(r y)_≤rd+1

|α|=d+1 aαyα

≤rd+1_,

(3.16)

we get

᏿(1)

Ω,P,0(f)(x)≤

1

0

Sn−1

_eiP(r y)_−eiQ(r y)2_{Ω(y)f(x−r y)}2

dσ(y)r−1_dr

1/2

≤

1

0

Sn−1

_{Ω(y)f(x−r y)}2

dσ(y)r2d+1dr

1/2

=

₋1

j=−∞

2j+1

2j

Sn−1

_{Ω(y)f(x−r y)}2

dσ(y)r2d+1_dr

1/2

≤

₋₁

j=−∞

2(2d+2)j

2j+1

2j

Sn−1

_{Ω(y)f(x−r y)}2

dσ(y)r−1_dr

1/2

≤CMΩ|f|21/2(x),

(3.17)

whereMΩis the operator given by (2.1) withᏼ(y)=y. Thus, by (3.17), by the fact that Ω1≤1, andLemma 2.1, we obtain

_᏿(1)

Ω,P,0(f)p≤Cpfp (3.18)

for allp≥2 with constantCpindependent of the functionΩand the coefficients of the polynomialP. Therefore, by (3.14), by Minkowski’s inequality, by (3.15), and (3.18), we obtain

_᏿Ω,P,0(f)

p≤

1 + log1/2e+Ωq

21/q

21/q

−1

Cpfp (3.19)

Finally, we prove theLp_{boundedness of᏿}

Ω,P,∞. By generalized Minkowski’s

inequal-ity, we can write᏿Ω,P,∞as

᏿Ω,P,∞(f)(x)=

_∞

2−log(e+Ωq)

η∞(r)

Sn−1e iP(r y₎

Ω(y)f(x−r y)dσ(y) 2

r−1_dr

1/2

= _∞ 0 0 k=−∞

ψk(r)

Sn−1e iP(r y₎

Ω(y)f(x−r y)dσ(y)

2 1 rdr

1/2

≤ 0

k=−∞

᏿Ω,P,∞,k(f)(x),

(3.20)

where

᏿Ω,P,∞,k(f)(x)=

2−log(e+Ωq)(k−1)

2−log(e+Ωq)(k+1)

Sn−1e iP(r y₎

Ω(y)f(x−r y)dσ(y) 2

r−1_dr

1/2 .

(3.21)

By Plancherel’s theorem, Fubini’s theorem, andLemma 2.3, we have

_{᏿Ω,P,∞,k(f)}2 2=

Rn f(ξ)

2

Jk,Ω(ξ)dξ≤2(k+1)/4qloge+Ωqf22. (3.22)

Thus,

_{᏿Ω,P,∞,k(f)}

2≤2(k+1)/8q

log1/2e+Ωqf2. (3.23)

Now, forp >2, chooseg∈L(p/2)_withg(

p/2)=1 such that

_{᏿Ω,P,∞,k(f)}2 p

=

Rn

22 log(e+Ωq)

1

Sn−1Ek,Ω

r,P(y), 0Ω(y)fx−2−γk,Ω_{r ydσ(y)}

2

r−1_drg(x)dx

≤

Rn

_f(z)22

2 log(e+Ωq)

1

Sn−1

_{Ω(y)gz+ 2}−γk,Ω_{r y}dσ(y)dr

r dz

≤Cloge+Ωqf2pMΩg(z) (p/2),

(3.24)

whereMΩ is the operator given by (2.1) withᏼ(y)=y. Thus,Lemma 2.1 and (3.24) imply that

_{᏿Ω,P,∞,k(f)}

p≤log 1/2

e+Ωq

Cfp, (3.25)

which when combined with (3.23) implies

_{᏿Ω,P,∞,k(f)}

p≤2(k+1)δ/8q

log1/2e+Ωq

whereδis a constant that is independent of the essential variables. Thus, by (3.20), (3.26), and Minkowski’s inequality, we get

_{᏿Ω,P,∞(f)}

p≤Clog

1/2_e+Ω

q 2

1/q

21/q

−1

Cpfp (3.27)

for allp≥2. Hence, by Minkowski’s inequality, (3.11), (3.19), and (3.27), we obtain (1.4)

for the given polynomialP. This completes the proof.

4. Proof of results concerningL(logL)1/2_(Sn−1₎

Proof ofTheorem 1.2. Given Ω∈L(logL)1/2_(Sn−1_{), then we decomposeΩas a sum of} functions in L2_(Sn−1_{). More precisely, there exists a sequence{Ω}

m:m=0, 1, 2,. . .} of functions inL1_(Sn−1_{) with}

Ω=

∞

m=0

Ωm (4.1)

such that

Sn−1Ωm(y

_{)dσ(y)=0, Ω}

m1≤C, Ω0∈L2

Sn−1_,

_Ωm

∞≤24mC form=1, 2, 3,. . .,

(4.2)

∞

m=1 √

mΩm1≤ ΩL(logL)1/2_(Sn−1)C. (4.3)

For a detailed proof of the existence of the decomposition (4.1), one might look into [2,5].

Now, by (4.1), we have the following:

ᏹΩ,P(f)(x)≤ᏹΩ0,P(f)(x) +

∞

m=1

_Ωm

1ᏹΩm,P(f)(x). (4.4)

ByLemma 2.4, we have

_ᏹΩ0,P(f)

p≤

1 + log1/2e+Ω0₂Cpfp (4.5)

for allp≥2.

Next, by observing that

1 + log1/2e+Ωm_∞≤1 + log1/2e+ 24m≤4√m (4.6)

for allm≥1,Theorem 1.1implies that

_ᏹΩm,P(f)

p≤4 √

mCpfp (4.7)

Thus, by Minkowski’s inequality, (4.4), (4.5), (4.7), and (4.2), we obtain

_ᏹΩ,P(f)

p≤Cpfp (4.8)

for allp≥2. This completes the proof.

Proof ofCorollary 1.5. By the inequality (1.5) and the decomposition (4.1), we have

_{TΩ,P,h(f)(x)≤TΩ0,P,h(f)(x)+}∞

m=1

_Ωm

1TΩm,P,h(f)(x). (4.9)

Thus, byTheorem 1.4, (4.9), and a similar argument as in the proof ofTheorem 1.2, the

proof is complete.

5. Proof of results concerning block spaces

We start this section by recalling the definition of block spaces introduced by Jiang and Lu (see [16]).

Definition 5.1. (1) Forx0∈Sn−1and 0< θ0≤2, the setB(x0, θ0)= {x∈Sn−1:|x−x0|< θ0}is called a cap onSn−1_.

(2) For 1< q≤ ∞, a measurable functionbis called aq−block onSn−1_{ifbis a function} supported on some capI=B(x0,θ0) with bLq≤ |I|−1/q, where|I| =σ(I) and 1/q+ 1/q=1.

(3)Bκ,υ

q (Sn−1)= {Ω∈L1(Sn−1) :Ω= ∞μ=1cμbμ, where eachcμ is a complex number; eachbμis aq-block supported on a capIμonSn−1; andMqκ,υ({cμ},{Iμ})= ∞μ=1|cμ|(1 + φκ,υ(|Iμ|))<∞, whereφκ,υ(t)=

1

t u−1−κlogυ(u−1)duif 0< t <1 andφκ,υ(t)=0 ift≥1}. Notice thatφκ,υ(t)∼t−κlogυ(t−1) ast→0 forκ >0,υ∈R, and φ0,υ(t)∼logυ+1(t−1) ast→0 forυ >−1. Moreover, among many properties of block spaces [17], we cite the following which are closely related to our work:

B0,0q ⊂B0,q−1/2 (q >1),

B0,q2υ⊂B 0,υ q1

1< q1< q2,

LqSn−1⊆Bq0,υ

Sn−1 (forυ >−1),

q>1 B0,υ

q

Sn−1_⊆ p>1

LpSn−1 _{for anyυ >−1.}

(5.1)

Proof ofTheorem 1.3. Assume thatΩ∈B0,−1/2

q (Sn−1),q >1. Then

Ω=

∞

μ=1

where eachcμ is a complex number; eachbμ is aq-block supported on a capIμonSn−1; and

M0,q−1/2

cμ

,Iμ

=

∞

μ=1

_{cμ1 + log}1/2 Iμ−

1

<∞. (5.3)

Without loss of generality, we may assume that|Iμ|<1. For eachμ, let

¯

bμ(x)=bμ(x)−

Sn−1bμ(u)du. (5.4)

Then, it follows that

_b¯_μ

q≤C|I|−1/q

, b¯μ1≤C. (5.5)

By the decomposition (5.3), we have

ᏹΩ,P(f)(x)≤

∞

μ=1

_cμᏹ¯

bμ,P(f)(x). (5.6)

Thus, by Minkowski’s inequality, (5.5), andTheorem 1.1, we have

_ᏹΩ,P(f)

p≤Cp

∞

μ=1

_{cμ1 + log}1/2

e+|I|−1/q

fp

≤Cp,q

∞

μ=1

_{cμ1 + log}1/2 Iμ−1

fp

≤Cp,qfp

(5.7)

for allp≥2, where the last inequality follows by (5.3). This completes the proof. A proof ofCorollary 1.6 can be obtained by a similar argument as in the proof of Corollary 1.5. We omit the details.

6. Further applications

This section is devoted to present some results that follow by applying our results in Section 1.

Parametric Marcinkiewicz integral operators. The parametric Marcinkiewicz integral op-erator related to the opop-eratorᏹΩ,Pis defined by

μρ_Ω,Pf(x)=

_∞

−∞

2−ρt

|y|≤2te

iP(y)_f(x−y)|y|−n+ρ_{Ω(y)d y}2_dt

1/2

, (6.1)

Now, it is straightforward to see that

μρ_Ω,Pf(x)≤C(ρ)ᏹΩ,Pf(x). (6.2)

Therefore, by (6.2),Theorem 1.1, and the decompositions (4.1) and (5.2), we can easily obtain the following theorem.

Theorem 6.1. Suppose that ρ >0 and thatΩ∈L(logL)1/2_(Sn−1_{)satisfying (1.1). Then} the parametric Marcinkiewicz integral operatorμρ_Ω,Pis bounded onLp_{for allp≥2withL}p bounds that may depend on the degree of the polynomialPbut they are independent of the functionΩand the coefficients of the polynomialP.

Theorem6.2. Suppose thatρ >0and thatΩ∈B0,−1/2

q (Sn−1),q >1, satisfying (1.1). Then the parametric Marcinkiewicz integral operatorμρ_Ω,Pis bounded onLpfor allp≥2withLp bounds that may depend on the degree of the polynomialPbut they are independent of the functionΩand the coefficients of the polynomialP.

We remark here that by specializing to the caseP=0 andρ=1, the resulting operator μΩ=μ1Ω,0is the classical Marcinkiewicz integral operator introduced by Stein [25]. Thus, Theorems6.1and6.2generalize as well as improve the result in (see [25]). Furthermore, Theorems6.1and6.2generalize the corresponding results in [2,3,8]. For more back-ground information and related results about Marcinkiewicz integral operators, we refer the readers to consult [6,8,15,25], and the references therein.

Morrey spaces. In [20], Mizuhara introduced the following generalized Morrey spaces. Definition 6.3. Let φ: (0,∞)→(0,∞) be an increasing function that satisfiesφ(2r)≤ Dφ(r) for any r >0, whereDis a constant independent ofr. Let 1≤p <∞. A locally integrable functionf ∈Lp,φ_if

Br

x0

f(x)pdx≤Cpφ(r) (6.3)

for allx0∈Rn_{andr >0, whereB}

r(x0) is the ball with centerx0and radiusr.

It is worth pointing out here that Morrey spaces have been used to study several prob-lems in harmonic analysis, such as studying the local behavior of solutions to second-order elliptic partial differential equations and measuring the regularity of the solution to an elliptic second-order equation with discontinuous coefficients; see [13,21], and references therein.

ByTheorem 1.1, the decompositions (4.1) and (5.2), and following a similar argument as in the proof of Theorem 5 in [13], we obtain the following theorem.

Theorem6.4. Suppose thatΩ∈L(log+L)1/2_(Sn−1_)∪B0,−1/2

q (Sn−1),q >1, satisfying (1.1). LetP:Rn_{→Rbe a real-valued polynomial. Thenᏹ}

Ω,Pis bounded onLp,φ(Rn)for allp≥2 withLp_{bounds independent of the coefficients of the polynomialP.}

P. Moreover, by (1.5) andTheorem 6.4, it follows that the operatorTΩ,P,his bounded on Lp,φ_{for all 1< p <∞andh∈L}2_(R+,r−1_dr).

Lp _{estimates with radial weights. The results in this paper can be easily extended to the} radial weights setting introduced by Duoandikoetxea [9]. In order to state our weighted Lp_{estimates, we recall the definition of the radial weights [9,13].}

Definition 6.5. Suppose thatω(t)≥0 andω∈L1

loc(R+). For 1< p <∞,ω∈Ap(R+) if there is a constantC >0 such that for any intervalI⊆R+_,

|I|−1

Iω(t)dt

|I|−1

Iω(t)

−1/(p−1)_{dt≤C <∞. (6.4)}

If there is a constantC >0 such that

ω∗(t)≤Cω(t) for a.e.t∈R+_{, (6.5)}

whereω∗is the Hardy-Littlewood maximal function ofωonR+_{, thenω∈A1(R}+_).

We let Ap(R+) be the class of functions ω that can be written as follows:ω(x)=

ν1(|x|)ν2(|x|)1−p_{, where eitherν}

i∈A1(R+) is decreasing orνi2∈A1(R+),i=1, 2. Also, for 1< p <∞, we let

¯

ApR+=ω(x)=ω|x|:ω(t)>0,ω(t)∈L1loc

R+_,ω2_(t)∈A

pR+ (6.6)

and letAI

p(Rn) be the weighted class defined by using alln-dimensional intervals with sides parallel to coordinate axes. The weightedLp_spaceLp_(Rn_{,ω(x)dx) associated to the} weightωis defined to be the class of all measurable functions f withfLp_(ω)<∞, where

fLp_(ω)=

Rn

_f(x)p

ω(x)dx

1/ p

. (6.7)

It is known that ¯Ap(R+)⊆Ap(R+); see [13]. Moreover, ifω(t)∈A¯p(R+), thenω(|x|) is in Muckenhoupt weighted classAp(Rn) whose definition can be found in [14].

By the same argument in this paper with minor modifications, it can be easily shown that the weighted version of allLp _{estimates obtained in this paper holds. In particular,} we have the following theorem.

Theorem6.6. Suppose thatρ >0and thatΩ∈L(log+L)1/2_(Sn−1_)∪B0,−1/2

q (Sn−1),q >1, satisfying (1.1). LetP:Rn_{→Rbe a real-valued polynomial. Ifω∈A}

p/2∩AIp/2,2≤p <∞, then the operatorsᏹΩ,PandμρΩ,Pare bounded onLp(ω)withLpbounds independent of the coefficients of the polynomialP.

A special class of radial weights that have received a considerable amount of attention is the class of power weights|x|α_{. For background information and related results on} power weights, we refer the readers to consult [9,13], among others. By the observation that|x|α_∈A

Corollary6.7. Suppose thatρ >0and thatΩ∈L(log+L)1/2_(Sn−1_)∪B0,−1/2

q (Sn−1),q >1, satisfying (1.1). LetP:Rn_{→Rbe a real-valued polynomial. Then the operatorsᏹ}

Ω,Pand μρ_Ω,Pare bounded onLp(|x|α)ifα∈(−1,p/2−1)withLp(|x|α)bounds independent of the coefficients of the polynomialP.

Acknowledgment

The author wishs to thank the referee of this paper for helpful comments.

References

[1] A. Al-Salman, Rough oscillatory singular integral operators of nonconvolution type, Journal of Mathematical Analysis and Applications299(2004), no. 1, 72–88.

[2] ,On maximal functions with rough kernels inL(logL)1/2_(Sn−1_{), Collectanea Mathematica}

56(2005), no. 1, 47–56.

[3] ,On a class of singular integral operators with rough kernels, Canadian Mathematical Bul-letin49(2006), no. 1, 3–10.

[4] A. Al-Salman and A. Al-Jarrah,Rough oscillatory singular integral operators. II, Turkish Journal of Mathematics27(2003), no. 4, 565–579.

[5] A. Al-Salman and Y. Pan,Singular integrals with rough kernels inLlog+L(Sn−1_{), Journal of the}

London Mathematical Society. Second Series66(2002), no. 1, 153–174.

[6] J. Chen, D. S. Fan, and Y. Pan, A note on a Marcinkiewicz integral operator, Mathematische Nachrichten227(2001), no. 1, 33–42.

[7] L.-K. Chen and H. Lin,A maximal operator related to a class of singular integrals, Illinois Journal of Mathematics34(1990), no. 1, 120–126.

[8] Y. Ding, S. Lu, and K. Yabuta,A problem on rough parametric Marcinkiewicz functions, Journal of the Australian Mathematical Society72(2002), no. 1, 13–21.

[9] J. Duoandikoetxea,Weighted norm inequalities for homogeneous singular integrals, Transactions of the American Mathematical Society336(1993), no. 2, 869–880.

[10] D. S. Fan, K. Guo, and Y. Pan,Lp_{estimates for singular integrals associated to homogeneous} sur-faces, Journal f¨ur die reine und angewandte Mathematik542(2002), 1–22.

[11] D. S. Fan and Y. Pan,Boundedness of certain oscillatory singular integrals, Studia Mathematica 114(1995), no. 2, 105–116.

[12] ,Singular integral operators with rough kernels supported by subvarieties, American Jour-nal of Mathematics119(1997), no. 4, 799–839.

[13] D. S. Fan, Y. Pan, and D. Yang,A weighted norm inequality for rough singular integrals, The To-hoku Mathematical Journal. Second Series51(1999), no. 2, 141–161.

[14] J. Garc´ıa-Cuerva and J. L. Rubio de Francia,Weighted Norm Inequalities and Related Topics, North-Holland Mathematics Studies, vol. 116, North-Holland, Amsterdam, 1985.

[15] L. H¨ormander,Estimates for translation invariant operators inLp_{spaces, Acta Mathematica104} (1960), 93–140.

[16] Y. S. Jiang and S. Z. Lu,Oscillatory singular integrals with rough kernel, Harmonic Analysis in China (M. D. Cheng, D. G. Deng, S. Gong, and C.-C. Yang, eds.), Math. Appl., vol. 327, Kluwer Academic, Dordrecht, 1995, pp. 135–145.

[18] S. Lu, M. Taibleson, and G. Weiss,Spaces Generated by Blocks, Beijing Normal University Press, Beijing, 1989.

[19] S. Z. Lu and Y. Zhang,Criterion onLp_{-boundedness for a class of oscillatory singular integrals with} rough kernels, Revista Matem´atica Iberoamericana8(1992), no. 2, 201–219.

[20] T. Mizuhara,Boundedness of some classical operators on generalized Morrey spaces, Harmonic Analysis (Sendai, 1990) (S. Igari, ed.), ICM-90 Satell. Conference Proceedings, Springer, Tokyo, 1991, pp. 183–189.

[21] C. B. Morrey Jr.,On the solutions of quasi-linear elliptic partial differential equations, Transactions of the American Mathematical Society43(1938), no. 1, 126–166.

[22] Y. Pan,L2_{estimates for convolution operators with oscillating kernels, Mathematical Proceedings}

of the Cambridge Philosophical Society113(1993), no. 1, 179–193.

[23] F. Ricci and E. M. Stein,Harmonic analysis on nilpotent groups and singular integrals. I. Oscillatory integrals, Journal of Functional Analysis73(1987), no. 1, 179–194.

[24] P. Sj¨olin,Convolution with oscillating kernels, Indiana University Mathematics Journal30(1981), no. 1, 47–55.

[25] E. M. Stein,On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz, Transactions of the American Mathematical Society88(1958), no. 2, 430–466.

[26] , Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, New Jersey, 1993.

[27] S. Yano,Notes on Fourier analysis. XXIX. An extrapolation theorem, Journal of the Mathematical Society of Japan3(1951), 296–305.

Ahmad Al-Salman: Department of Mathematics, Faculty of Science, Yarmouk University, Irbid, Jordan