Rough Marcinkiewicz integral operators

PII. S0161171201006548 http://ijmms.hindawi.com © Hindawi Publishing Corp.

ROUGH MARCINKIEWICZ INTEGRAL OPERATORS

HUSSAIN AL-QASSEM and AHMAD AL-SALMAN

(Received 16 January 2001)

Abstract.We study the Marcinkiewicz integral operator M_ᏼf (x)=(_−∞∞ |_|y|≤2tf (x− ᏼ(y))(Ω(y)/|y|n−1_{) dy|}2_dt/22t₎1/2_{, whereᏼis a polynomial mapping fromR}n_intoRd

andΩis a homogeneous function of degree zero onRn_{with mean value zero over the}

unit sphereSn−1. We prove anLpboundedness result ofM_ᏼfor roughΩ. 2000 Mathematics Subject Classification. 42B20, 42B15, 42B25.

1. Introduction. LetRn_{,n≥2 be then-dimensional Euclidean space andS}n−1_be the unit sphere inRn _{equipped with the induced Lebesgue measure. Consider the} Marcinkiewics integral operator

µf (x)=

∞

−∞

Ft(x) 2dt

22t

1/2

, (1.1)

where

Ft(x)=

|x−y|≤2tf (y)

Ω(x−y)

|x−y|n−1dy, (1.2)

andΩis a homogeneous function of degree zero which has the following properties:

Ω∈L1Sn−1,

Sn−1Ω

ydσy=0. (1.3)

WhenΩ∈Lipα(Sn−1),(0< α≤1), Stein proved theLpboundedness ofµ(f )for all 1< p≤2. Subsequently, Benedek, Calderón, and Panzone proved theLp_boundedness ofµ(f )for all 1< p <∞under the conditionΩ∈C1_(Sn−1_{)(see [2]).}

The authors of [3] were able to prove the following result for the more general class of operators

µPf (x)=

∞

−∞

FP ,t(x) 2dt

22t

1/2

, (1.4)

where

FP ,t(x)=

|y|≤2tf

x−P (|y|)y Ω(y)

|y|n−1dy (1.5)

andP is a real-valued polynomial onRand satisfiesP (0)=0.

Theorem_{1.1(see [3])}. _{Letα >0, andΩ}∈_{Vα(n). Then the operatorµP is bounded}

In [1], Al-Salman and Pan studied the singular integral operator

TΩ,ᏼf (x)=p.v.

Rnf

x−ᏼ(y)Ω

y

|y|n dy, (1.6)

whereᏼ=(P1, . . . , Pd):Rn→Rd is a polynomial mapping,d≥1,n≥2. The authors of [1] proved thatTΩ,ᏼis bounded inLp(Rd)whenever(2+2α)/(1+2α) < p <2+2α

andΩ∈Wα(n). HereWα(n)is a subspace ofL1(Sn−1)and its definition as well as the definition ofVα(n)will be reviewed inSection 2. It was shown in [1] thatWα(n)= Vα(n),if n=2 and it is a proper subspace ofVα(n)ifn≥3.

Our purpose in this paper is to study theLp_{boundedness of the class of operators}

Mᏼf (x)=

∞

−∞

_Fᏼ,t(x)2dt 22t

1/2

, (1.7)

where

Fᏼ,t(x)=

|y|≤2tf

x−ᏼ(y) Ω(y)

|y|n−1dy. (1.8) Our main result in this paper is the following theorem.

Theorem _1.2. _{Letα >0, and Ω}∈_{Wα(n). Then the operator Mᏼ is bounded in} Lp_(Rd_{)for(2α+2)/(2α+1) < p <2+2α. The bound ofM}

ᏼf is independent of the coefficients of{Pj}.

By [1, Theorem 3.1] andTheorem 1.2we have the following corollary.

Corollary 1.3. Letα >0, Ω∈V_α(2) andᏼ:R2→Rd. Then M_ᏼ is bounded in Lp_(Rd_{)for(2α+2)/(2α+1) < p <2+2α. The bound ofM}

ᏼ is independent of the coefficients of{Pj}.

2. Preparation. We start this section by recalling the following definition from [1].

Definition2.1. Forα >0,N≥1, letᐂ(n, N) = N_m=1ᐂ(n, m)and letW_α(N, n)be the subspace ofL1_(Sn−1_{)defined by}

Wα(N, n)=

Ω∈L1_Sn−1_:

Sn−1Ω

ydσy=0, Mα(N, n) <∞

, (2.1)

where

Mα(N, n)

=max

Sn−1 _Ω

y

log 1 |P (y_)|

1+α

dσy:P∈ᐂ(n, N) withP =1

.

(2.2)

Forα >0, we defineWα(n)to be

Wα(n)= ∞

N=1

Wα(N, n). (2.3)

Hereᐂ(n, m)is the space of all real-valued homogeneous polynomials onRn_with degree equal tomand with norm·defined by

|α|=m aαyα

=

|α|=m

aα. (2.4)

Now we need to recall the following results.

Lemma_{2.2(see van der Corput [7])}. _{Supposeφandψare 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

|ψ(b)|+

b

a

ψ(t)dt

, (2.5)

holds when

(i) k≥2, or

(ii) k=1andφis monotonic.

The boundCkis independent ofa,b,φ, andλ.

Lemma_{2.3(see [7])}. _Letᏼ=_{(P1, . . . , Pd)be a polynomial mapping from}Rn_intoRd_.

Letdeg(ᏼ)=max1_≤j≤ddeg(Pj). SupposeΩ∈L1(Sn−1)and

µΩ,ᏼf (x)=sup

h>0

1

hn

|y|<hf

x−ᏼ(y)Ωydy. (2.6)

Then for every1< p≤ ∞, there exists a constantCp>0which is independent ofΩand

the coefficients of{Pj}such that

µΩ,ᏼfp≤CpΩL1(Sn−1)fp (2.7)

for everyf∈Lp_(Rd_).

To each polynomial mappingᏼ=(P1, . . . , Pd):Rn→Rd with

degᏼ=max

1≤j≤ddegPj=N, d≥1, n≥2, (2.8)

we define a family of measures

ϑtl, λlt:l=0,1, . . . , N, t∈R

(2.9)

as follows.

For 1≤j≤d, 0≤l≤N letPj=

|α|≤NCjαyα and let Ql=(Ql1, . . . , Qld) where Ql

j=

|α|≤lCjαyα.

Now for 0≤l≤Nandt∈R, letϑl

t andλlt be the measures defined in the Fourier transform side by

ϑltˆ

(ξ)=

|y|≤2te

−2π iξ·Ql_(y)Ωy |y|n−1

dy 2t ,

λltˆ

(ξ)=

|y|≤2te

−2π iξ·Ql_(y)Ωy |y|n−1

dy 2t .

The maximal functions(ϑl₎∗_{defined by}

ϑl∗_{(f )(x)=sup} t∈R

λl

t∗f (x), (2.11)

forl=0,1, . . . , N.

For later purposes, we need the following definition.

Definition 2.4. For each 1≤l≤N, letN_l= |{α∈Nn:|α| =l}|and let {α∈

Nn _{:|α| =l} = {α}

1, . . . , αNl}. For each 1≤l≤N, define the linear transformations Lα_lj:Rd_→RandL

l:Rd→RNl by

Lα_lj(ξ)= d

i=1

C_i,α

jyαj

ξi, j=1, . . . , Nl,

Ll(ξ)=

Lα1

l (ξ), . . . , L α_Nl l (ξ)

.

(2.12)

To simplify the proof of our result we need the following lemma.

Lemma_2.5. _Let{_σtl_:l=_{0,1, . . . , N, t}∈R}_{be a family of measures such thatσt}0=₀

for allt∈R. LetDl:Rn→Rd,l=0,1, . . . , N be linear transformations. Suppose that

for allt∈Randl=0,1, . . . , N,then _σl

t≤C(l),

σl t

_ˆ

(ξ)≤C Mα

logc2lt_Dl(ξ)1+α,

_σl t

_ˆ

(ξ)−σtl−1

_ˆ

(ξ)≤C2ltDl(ξ).

(2.13)

Then there exists a family of measures{νtl:l=1, . . . , N}t∈Rsuch that

νl

t≤C(l),

νtlˆ

(ξ)≤C Mα logc2lt_D

l(ξ) 1+α,

_νl tˆ

(ξ)≤C2ltDl(ξ),

σN t =

N

l=1 νl

t.

(2.14)

Proof. _{By [5, Lemma 6.1], for eachl}=_{1, . . . , Nchoose two nonsingular linear}

trans-formations

Al:Rr (l) →Rd, Bl:Rd →Rd, (2.15)

such that

Alπr (l)d Bl(ξ)≤Dl(ξ)≤NAlπr (l)d Bl(ξ), ξ∈Rd, (2.16)

Now chooseη∈C0∞(R)such thatη(t)=1 for|t| ≤1/2 andη(t)=0 for|t| ≥1. Let ϕ(t)=φ(t2_{)and let}

νl

tˆ

(ξ)=σl t

_ˆ (ξ)

l<j≤N

ϕ2tj_A

jπr (j)d Bj(ξ)

−σtl−1

_ˆ

(ξ)

l−1<j≤N

ϕ2tj_A

jπr (j)d Bj(ξ)

(2.17)

with the conventionj∈∅aj=1, 1≤l≤N.

Hence, one can easily see that {σtl:l=1, . . . , N, t∈R} is the desired family of measures.

Now for the boundedness of the maximal functions(ϑl₎∗_{,l=0,1, . . . , N, we have the} following lemma whose proof is an easy consequence ofLemma 2.3, polar coordinates and Hölder’s inequality:

Lemma_2.6. _Forl=_{1, . . . , Nandp}∈_(1,∞_{), there exists a constantCp,lwhich is}

inde-pendent of the coefficients of the polynomial components of the mappingQl_{such that}

ϑl∗_f

p≤Cp,lfp. (2.18)

3. Boundedness of some square functions. For a nonnegativeC∞radial function

ΦonRn_with

supp(Φ)⊂

x∈Rn:1

2≤ |x| ≤2

,

∞

0

Φ(t)

t dt=1, (3.1)

and for a linear transformationL:Rn_→Rd_{, define the functionsψt,t∈Rby ˆψ} t(y)=

Φ(2t_L(y)).

For a family of measures{σt}t∈R, real numberuandl∈N, letJlu(f )be the square function defined by

Jlu(f )(x)=

∞

−∞

σt∗ψl(t+u)∗f (x) 2

dt 1/2

. (3.2)

For such a square function we have the following theorem.

Theorem_3.1. _If{_σt}t_{∈Ris a family of measures such that the corresponding}

maxi-mal function

σ∗(f )(x)=sup t∈R

_{σt∗f (x) (3.3)}

is bounded onLp_(Rd_{)for every1< p <∞, then}

Jl

u(f )Lp_Rd≤Cp,l

σ∗

(p/2)sup t∈R

σtf_Lp_Rd (3.4)

for every1< p <∞. HereCp,lis a constant that depends only onpand the dimension

Proof. _{If supt∈Rσ}t = ∞_{, then the inequality holds trivially. Thus we may assume}

that supt_∈Rσt<∞. In this case we follow a similar argument as in [4]. Letp >2 and q=(p/2). Choose a nonnegative functionv∈Lq+withvq=1 such that

Jl u(f )

2 p=

Rd

∞

−∞

σt∗ψl(t+u)∗f (x) 2

dt

v(x)dx. (3.5)

Thus it is easy to see that

_Jl u(f )

2 p≤sup

t∈R

_σt∞ −∞

Rd

_{ψl(t+u)∗f (z)}2

σ∗(v)(−z)dz dt

≤sup t∈R

_σt

Rd[g(f )]

2_(z)σ∗_{(v)(−z) dz,}

(3.6)

where

g(f )(x)=

∞

−∞

_{ψl(t+u)∗f (x)}2 dt

1/2

. (3.7)

Now since_Rdψt(x) dx=0, it is well known that

g(f )p≤Cpfp ∀1< p <∞ (3.8)

with constantCpthat depends only onpand the dimension of the underlying space. Thus by (3.6) and Hölder’s inequality we have

Jl u(f )

2 p≤sup

t∈R

σtg(f )2pσ∗(u)q

≤Cp2sup t∈R

_σtσ∗

(p/2)f2p.

(3.9)

Hence our result follows by taking the square root on both sides. The casep <2 follows by duality.

4. Proof of the main theorem. Let α >0, Ω∈Wα(n). Let ᏼ=(P1, . . . , Pd) be a polynomial mapping fromRn_intoRd_{with degᏼ=max1}

≤j≤ddegPj=N, whered≥1 andn≥2. For 0≤l≤NletNl,Ql_,νl

t,λlt, andLlbe as inSection 3. The first step in our proof is to show that eachϑl

t,l=1, . . . , Nsatisfies the hypothe-ses ofLemma 2.5, that is,

_ϑl

t≤C(l), (4.1)

ϑl tˆ

(ξ)≤C Mα logc2lt_L

l(ξ)1+α

, (4.2)

_ϑl tˆ

(ξ)−ϑlt−1ˆ

(ξ)≤C2ltLl(ξ). (4.3)

One can easily see that (4.1) holds trivially. Using the cancellation property ofΩ, it is easy to see that (4.3) holds. Thus, we need only to verify (4.2). To see that, we notice that

_ϑl tˆ

(ξ)≤

Sn−1 _Ω

y 1

0e

Now the quantityξ·Ql₍₂tl_{r y)can be written in the form}

ξ·Ql2tlr y=2tlrlλGly+ξ·R2tr y, (4.5)

whereQl_{is a homogeneous polynomial of degreelwithG}l _{=1,Ris a polynomial} of degree at mostl−1 in the variabler,

λ= Nl

j=1

_Lα_j

l (ξ)≥NlLl(ξ) (4.6)

andα1, . . . , α_Nl are the constants that appeared inSection 2. Thus by van der Corput lemma, we have

₀1e−2π iξ·Ql₍₂t_{r y)}

dr≤Cmin1,2tl_L

l(ξ)Gly−1/l

(4.7)

and hence

₀1e−2π iξ·Ql₍₂t_{r y)}

dr≤C

logcGl_y−11+α

logc2tl_Ll(ξ)1+α , (4.8)

whereC is a constant independent oft andξ. SinceΩ∈Wα(n), the estimate (4.2) follows.

ByLemma 2.5, there exists a family of measures{νtl:l=1, . . . , N, t∈R}such that

_νl

t≤C(l), (4.9)

_νl tˆ

(ξ)≤C Mα logc2lt_L

l(ξ)

1+α, (4.10)

_νl tˆ

(ξ)≤C2ltLl(ξ), (4.11)

ϑtN= N

l=1

νtl. (4.12)

Also byLemma 2.6and the definition ofνtl(see the proof ofLemma 2.5), we have

νl∗_f

p≤Cp,lfp ∀1< p <∞. (4.13)

Now one can easily see that

2−tFᏼ,t(x)=ϑNt ∗f (x)= N

l=1

νtl∗f (x). (4.14)

Therefore,

M_ᏼf_p≤ N

l=1

Ml

ᏼfp, (4.15)

where

M_ᏼlf (x)=

∞

−∞ _νl

t∗f (x) 2

dt 1/2

Thus to show the boundedness ofMᏼf, it suffices to show that _Ml

ᏼfp≤Cp,lfp (4.17)

forp∈((2+2α)/(1+2α),2+2α), and for alll=1, . . . , N.

To show (4.17), we proceed as follows: letΦandψtbe as inSection 3. Then

Ml

ᏼf (x)=log 2l

∞

−∞ _−∞∞ νl

t∗ψl(t+u)∗f (x) du

2dt

1/2

≤log 2l

∞

−∞S l

uf (x) du,

(4.18)

where

Sulf (x)=

∞

−∞ _νl

t∗ψl(t+u)∗f (x) 2

dt 1/2

. (4.19)

Now by (4.13) andTheorem 3.1, we have

Sl

ufp≤Cpfp (4.20)

for allp∈(1,∞)and forl=1, . . . , N which in turn implies that

1

−1

_Sl

ufpdu≤2Cpfp ∀p∈(1,∞). (4.21)

On the other hand, ifu≥1, by the estimate (4.11) we have

Sl uf

2 2=

Rd ∞

−∞ νl

t∗ψl(t+u)∗f (x)2dt dx

=

∞

−∞

Rd

Φ2lt+lu_L l(ξ)

2 νl

tˆ

(ξ)2f (ξ)ˆ 2dξ dt

≤22l−2lu

Rd

_{f (ξ)}ˆ 2log(2 l_/|L

l(ξ)|)−u

log(1/2l_|L l(ξ)|)−u

dt

dξ

=2 log 2l22l−2luf2 2.

(4.22)

Thus

Sl uf2≤

2 log 2l₂l−lu_{f2. (4.23)}

By interpolating between (4.20) and (4.23) we get

_Sl

ufp≤Cp,l2θl−θlufp (4.24)

for all 1< p <∞and for someθ=θ(p) >0. Hence we have

_∞

1

_Sl

Finally, if u <−1, by the estimate (4.10) and similar argument as in the case of u≤1, we get

Sl

uf2≤Cl(|u|)−1−αf2. (4.26)

By interpolating between (4.26) and anyp ∈(1,∞) in (4.20), we get that, ifp∈ ((2+2α)/(1+2α),2+2α)there existsβ >0 such that

Sl

ufp≤Cp(|u|)−βfp, (4.27)

which implies that

₋1

−∞ _Sl

ufpdu≤Cpfp (4.28)

forp∈((2+2α)/(1+2α),2+2α).

Hence by combining (4.18), (4.21), (4.25), and (4.28) we get (4.17).

Acknowledgement. _{The publication of this paper was supported by Yarmouk}

University Research Council.

References

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Hussain Al-Qassem: Department of Mathematics, Yarmouk University, Irbid, Jordan E-mail address:husseink@yu.edu.jo