R E S E A R C H
Open Access
A note on certain integrals along polynomial
compound curves
Feng Liu
1*and Pu Zhang
2*Correspondence:
1College of Mathematics and
Systems Sciences, Shandong University of Science and Technology, Qingdao, P.R. China Full list of author information is available at the end of the article
Abstract
In this paper we consider a singular integral operator and a parametric Marcinkiewicz integral operator with rough kernel. These operators have singularity along sets of the form curves{x=P(
ϕ
(|y|))y}, wherePis a real polynomial satisfyingP(0) = 0 andϕ
satisfies certain smooth conditions. Under the conditions thatΩ
∈H1(Sn–1) and h∈γ(R+) for some
γ
> 1, we prove that the above operators are bounded on theLebesgue spaceL2(Rn). Moreover, theL2-bounds of the maximal functions related to
the above integrals are also established. Particularly, the bounds are independent of the coefficients of the polynomialP. In addition, we also present certain Hardy type inequalities related to these operators.
MSC: Primary 42B20; secondary 42B15; 42B25; 42B99
Keywords: Singular integrals; Marcinkiewicz integrals; Maximal operators; Polynomial compound curves; Hardy spaces
1 Introduction
Let Rn(n≥2) be then-dimensional Euclidean space and Sn–1denote the unit sphere in
Rnequipped with the induced Lebesgue measuredσ. LetK(·) be a kernel of Calderón–
Zygmund type on Rngiven by
K(y) =Ω(y)h(|y|)
|y|n ,
wherehis a suitable function defined on R+:= (0,∞) andΩ is homogeneous of degree
zero, withΩ∈L1(Sn–1) and
Sn–1
Ω(u)dσ(u) = 0. (1.1)
Suppose thatPis a real polynomial on R of degreeNand satisfiesP(0) = 0. For a suitable functionϕ defined on R+, we consider that the singular integral operatorTh,Ω,P,ϕ along
the “polynomial compound curve”P(ϕ(|y|))yon Rnis defined by
Th,Ω,P,ϕf(x) := p.v.
Rnf
x–Pϕ|y|yK(y)dy, (1.2)
wherey=|yy| for any nonzero pointy∈Rn.
For the sake of simplicity, we denoteTh,Ω,P,ϕ=Th,ΩifP(t)≡tandϕ(t)≡tandTh,Ω=TΩ
ifh(t)≡1. In their fundamental work on singular integrals, Calderón and Zygmund [1] proved thatTΩ is bounded onLp(Rn) for 1 <p<∞, provided thatΩ∈LlogL(Sn–1). The
same conclusion under the less restrictive condition thatΩ∈H1(Sn–1) was later obtained
independently by Coifman and Weiss [2] and Connett [3]. It should be pointed out that the condition thatΩ∈H1(Sn–1) turns out to be the most desirable size condition for the
Lpboundedness ofTΩ. For the singular integral operator with radial kernel, Fefferman [4]
in 1978 first introduced the singular integral operatorTh,Ωand established theLp(1 <p<
∞) boundedness ofTh,Ω, provided thatΩ∈Lipα(Sn–1) for someα> 0 andh∈L∞(R+).
Later on, Namazi [5] improved Fefferman’s result by assumingΩ∈Lq(Sn–1) for someq> 1
instead ofΩ∈Lipα(Sn–1). In 1986, Duoandikoetxea and Rubio de Francia [6] used the Littlewood–Paley theory to improve the above results to the caseΩ∈Lq(Sn–1) for any
q> 1 andh∈2(R+). Hereγ(R+) (γ > 0) is the set of all measurable functionsh: R+→R
satisfying
hγ(R+):=sup
R>0
R–1
R
0
h(t)γdt
1/γ
<∞.
It is clear that
L∞(R+) =∞(R+)γ2(R+)γ1(R+), ∀1≤γ1<γ2<∞ (1.3)
and
LipαSn–1LqSn–1LlogLSn–1H1Sn–1L1Sn–1, ∀α> 0 andq> 1. In bothTΩandTh,Ω, the singularity is along the diagonal{x=y}. However, many
prob-lems in analysis have led one to consider singular integral operators with singularity along more general sets. One of the principal motivations for the study of such operators is the requirements of several complex variables and large classes of “subelliptic” equations. We refer the readers to Stein’s survey articles [7,8] for more background information. During the last several years theLpmapping properties for singular integral operators with singu-larity along various sets and with rough kernel inH1(Sn–1) have been actively studied by
many authors. For example, see [9] for polynomial mappings, [10] for real-analytic sub-manifolds, [11,12] for homogeneous mappings, [13] for polynomial curves. For further re-sults on the singular integral operators with singularity along the above curves or surfaces, we refer the readers to consult [14–17], among others. Particularly, Fan and Pan [9] estab-lished theLpboundedness of the singular integral operator along polynomial mappings for
pwith|1/p– 1/2|<max{1/2, 1/γ}under the conditions thatΩ∈H1(Sn–1) andh∈ γ(R+)
for someγ > 1. In this paper we focus on theLp-boundedness of the singular integral
operator along the polynomial compound curves with rough kernelsΩ∈H1(Sn–1) and
h∈γ(R+) for someγ> 1. Recently, Fan and Pan [13] proved the following result. Theorem A([13]) Letϕ(t)≡t.IfΩ∈H1(Sn–1)satisfies(1.1)and h∈L∞(R+),then the
operator Th,Ω,P,ϕis bounded on L2(Rn).More precisely,we have
Th,Ω,P,ϕfL2(Rn)≤ChL∞(R+)ΩH1(Sn–1)fL2(Rn).
Before stating our main results, let us introduce two classes of functions.
Definition 1.1(Gclass) LetGbe the set of all nonnegative (or non-positive) and mono-tonicC1(R
+) functionsϕsuch thatΥϕ(t) :=tϕ(ϕ(tt)) with|Υϕ(t)| ≤Cϕ, whereCϕis a positive
constant that depends only onϕ.
Definition 1.2(Fclass) We denote byFthe set of all functionsφ satisfying one of the following conditions:
(i) ϕis a positive increasingC1(R+)function such thattϕ(t)≥Cϕϕ(t)and
ϕ(2t)≤cϕϕ(t)for allt> 0, whereCϕandcϕare independent oft.
(ii) ϕis a positive decreasingC1(R
+)function such thattϕ(t)≤–Cϕϕ(t)and
ϕ(t)≤cϕϕ(2t)for allt> 0, whereCϕandcϕare independent oft.
Remark1.1 It is clear thatFG. There are some model examples for the classF, such as
tα(α= 0),tα(ln(1 +t))β(α,β> 0),tln ln(e +t),t–1ln(1 +t–1), real-valued polynomialsPon Rwith positive coefficients andP(0) = 0, and so on. Note that the following facts are valid (see [18]): ifϕ∈G, then
(a) limt→0ϕ(t) = 0andlimt→∞|ϕ(t)|=∞ifϕis nonnegative and increasing, or non-positive and decreasing;
(b) limt→0|ϕ(t)|=∞andlimt→∞ϕ(t) = 0ifϕis nonnegative and decreasing, or non-positive and increasing.
In this paper we shall establish the following.
Theorem 1.1 Let Th,Ω,P,ϕbe the singular integral operator defined by(1.2)andϕ∈G.If
Ω∈H1(Sn–1)satisfies(1.1)and h∈γ(R+)for someγ∈(1,∞],then the operator Th,Ω,P,ϕ
is bounded on L2(Rn).More precisely,we have
Th,Ω,P,ϕfL2(Rn)≤Chγ(R+)ΩH1(Sn–1)fL2(Rn),
where C> 0is independent of h,γ,Ω,f and the coefficients of P,but depends onϕand
deg(P).
Remark1.2 By (1.3), Theorem1.1generalizes and improves TheoremA.
The second type of our operators we consider is the parametric Marcinkiewicz integral operator along polynomial compound curves. More precisely, leth,Ω,P,ϕ be given as in (1.2). For a complex numberρ=τ +iϑ (τ,ϑ∈Rwithτ > 0), we consider the para-metric Marcinkiewicz integral operatorMh,Ω,P,ϕ,ρalong “polynomial compound curve”
P(ϕ(|y|))yon Rnby
Mh,Ω,P,ϕ,ρf(x) =
∞
0
t1ρ
|y|≤t
fx–Pϕ|y|yh(|y|)Ω(y )
|y|n–ρ dy
2dtt 1/2. (1.4)
Ifρ= 1,P(t)≡t, andϕ(t)≡t,Mh,Ω,P,ϕ,ρis just the classical Marcinkiewicz integral
oper-ator, which is denoted byMh,Ω. TheLpmapping properties ofMh,Ω and other extensions
Ding et al. [21] proved that the parametric Marcinkiewicz integral operator along polyno-mial mappings is of type (p,p) for 1 <p<∞ifΩ∈H1(Sn–1) andh∈L∞(R
+). In this paper
we focus on the parametric Marcinkiewicz integral operator along polynomial compound curves. More precisely, we shall establish the following result.
Theorem 1.2 LetMh,Ω,P,ϕ,ρbe the Marcinkiewicz integral operator defined by(1.4)and
ϕ∈F.IfΩ∈H1(Sn–1)satisfies(1.1)and h∈
γ(R+)for someγ ∈(1,∞],then the operator Mh,Ω,P,ϕ,ρis bounded on L2(Rn).More precisely,we have
Mh,Ω,P,ϕ,ρfL2(Rn)≤C
γ
γ – 1hγ(R+)ΩH1(Sn–1)fL2(Rn),
where C> 0is independent of h,γ,Ω,f and the coefficients of P,but depends onϕ,ρand
deg(P).
Remark1.3
(i) It was shown in [22] that the operatorMh,Ω,P,ϕ,ρwithρ> 0andP(t)≡tis bounded
onLp(Rn)forpwith|1/p– 1/2|<min{1/2, 1/γ}under the same conditions of Theorem1.2(see also [27] for more generalization results).
(ii) Theorem1.2is new even in the special caseρ= 1,h(t)≡1, andϕ(t)≡t.
In 2009, Sato [29] introduced a class of functionsNα(R+), which may be the most
desir-able size condition on the radial kernel of rough singular integrals so far. HereNα(R+) for
α> 0 is the set of all measurable functionsh: R+→Rsatisfying
Nα(h) =
m=1
mα2msup k∈Z2
–kE(k,m)<∞,
whereE(k, 1) ={t∈(2k, 2k+1];|h(t)| ≤2}and
E(k,m) = t∈(2k, 2k+1]; 2m–1<h(t)≤2m form≥2.
It is easy to check that
γ(R+)Nα(R+), ∀γ> 1 andα> 0.
Applying Theorems1.1–1.2and the extrapolation arguments following from [29], we have the following result.
Corollary 1.1 Let Th,Ω,P,ϕandMh,Ω,P,ϕ,ρbe defined by(1.2)and(1.4),respectively.LetΩ
satisfy(1.1)andΩ∈H1(Sn–1).Then
(i) Ifh∈Nα(R+)for someα> 0andϕ∈G,then the operatorTh,Ω,P,ϕis bounded on
L2(Rn).More precisely,we have Th,Ω,P,ϕfL2(Rn)≤C
1 +Nα(h)
ΩH1(Sn–1)fL2(Rn),
whereC> 0is independent ofh,α,Ω,f and the coefficients ofP,but depends onϕ
(ii) Ifh∈N1(R+)andϕ∈F,then the operatorMh,Ω,P,ϕ,ρis bounded onL2(Rn).More precisely,we have
Mh,Ω,P,ϕ,ρfL2(Rn)≤C1 +N1(h)ΩH1(Sn–1)fL2(Rn),
whereC> 0is independent ofh,Ω,f and the coefficients ofP,but depends onϕ,ρ
anddeg(P).
The third type of our operators is the maximal functions related to the singular inte-grals and Marcinkiewicz inteinte-grals along polynomial compound curves. More precisely, letTh,Ω,P,ϕandMh,Ω,P,ϕ,be defined as in (1.2) and (1.4), respectively. We define the
maxi-mal operatorsSΩ,P,ϕandMΩ,P,ϕ,ρalong the “polynomial compound curve”P(ϕ(|y|))yon Rnby
SΩ,P,ϕf(x) = sup
h∈K2
Th,Ω,P,ϕf(x), (1.5)
MΩ,P,ϕ,ρf(x) = sup
h∈K2
Mh,Ω,P,ϕ,ρf(x), (1.6)
whereK2is the set of all measurable functionsh: R+→RwithhL2(R+,r–1dr)≤1. Here
Lγ(R
+,r–1dr) (γ > 0) is the set of all measurable functionsh: R+→Rthat satisfies
hLγ(R+,r–1dr):= ∞
0
h(r)γr–1dr
1/γ
<∞.
Clearly,Lγ(R
+,r–1dr)γ(R+) forγ > 0.
The rest of the main results can be formulated as follows.
Theorem 1.3 LetSΩ,P,ϕandMΩ,P,ϕ,ρbe the maximal operators defined by(1.5)and(1.6),
respectively.IfΩ∈H1(Sn–1)satisfies(1.1)andϕ∈G,then both the operatorsS
Ω,P,ϕand
MΩ,P,ϕ,ρare bounded on L2(Rn).More precisely,we have
SΩ,P,ϕfL2(Rn)≤CΩH1(Sn–1)fL2(Rn),
MΩ,P,ϕ,ρfL2(Rn)≤CΩH1(Sn–1)fL2(Rn),
where C> 0is independent ofΩ,f and the coefficients of P,but depends onϕanddeg(P).
Remark1.4 The maximal operator related to singular integrals, which is denoted bySΩ
and corresponds to the special case ofSΩ,P,ϕwithP(t) =ϕ(t) =t, was first introduced by
Chen and Lin [30]. Chen and Lin proved that if Ω∈C(Sn–1), thenS
Ω is of type (p,p)
for anyp> 2n/(2n– 1) and the range ofpis best possible. Subsequently, theLpmapping
properties ofSΩ have been discussed extensively by many authors. Particularly, Xu et al.
[31] established theLp(Rn) bounds forS
Ω with 2≤p<∞, provided thatΩ∈H1(Sn–1)
satisfying (1.1). It should be pointed out that Theorem1.3is new even in the special case
ϕ(t)≡t.
Hardy type inequalities related to the parametric Marcinkiewicz integral operators and maximal operators related to singular integrals along polynomial compound curves in Sect.4. We would like to remark that our main results and proofs are inspired by the work in [13], but our main results and proofs are more delicate and complex than those of [13]. Some ideas in our proofs are taken from [18,29,32,33]. Throughout the paper, we denote bypthe conjugate index ofp, which satisfies 1/p+ 1/p= 1. The letterCorc, sometimes with certain parameters, will stand for positive constants that are not necessarily the same ones at each occurrence, but are independent of the essential variables. In what follows, we denote e =limx→∞(1 +1x)x.
2 Preliminary definitions and lemmas
Recall that the Hardy spaceH1(Sn–1) is the set of all functionsΩ∈L1(Sn–1) satisfying the
condition
ΩH1(Sn–1):= sup
0≤r<1
Sn–1
Ω(θ)Pr(·)(θ)dσ(θ)
L1(Sn–1)
<∞,
wherePrw(θ) = 1–r 2
|rw–θ|n for 0≤r< 1 andθ,w∈Sn–1denotes the Poisson kernel on Sn–1. Definition 2.1(H1(Sn–1) atoms) A functiona: Sn–1→Cis a (1,∞) atom if there exist
ϑ∈Sn–1and∈(0, 2] such that
supp(a)⊂Sn–1∩B(ϑ,), whereB(ϑ,) = y∈Rn:|y–ϑ|<; (2.1)
aL∞(Sn–1)≤–n+1; (2.2)
Sn–1
a(y)dσ(y) = 0. (2.3)
An important property ofH1(Sn–1) is the atomic decomposition, which is listed as fol-lows:
Lemma 2.1([34,35]) IfΩ∈H1(Sn–1)satisfies the cancelation condition(1.1),then there
exist a sequence of complex numbers{cj}and a sequence of(1,∞)atoms{Ωj}such that
Ω=jcjΩjandΩH1(Sn–1)≈j|cj|.
The following results are known (see [36]).
Lemma 2.2([36]) Suppose that n≥3and b(·)is a(1,∞)atom onSn–1supported inSn–1∩
B(ζ,),whereζ∈Sn–1and∈(0, 2].Let
Fb(s) =
1 –s2(n–3)/2χ(–1,1)(s)
Sn–2b
s,1 –s21/2y˜dσ(˜y).
Then there exists a constant C,independent of b,such that
supp(Fb)⊂
ζ1– 2r(ζ),ζ1+ 2r(ζ)
; (2.4)
FbL∞(R)≤ C
R
Fb(s)ds= 0, (2.6)
whereζ = (ζ1, . . . ,ζn),r(ζ) =|L(ζ)|,and L(ζ) = (2ζ1,ζ2, . . . ,ζn).
Lemma 2.3([36]) Suppose that n= 2and b(·)is a(1,∞)atom onS1supported inS1∩ B(ζ,),whereζ∈S1and∈(0, 2].Let
Fb(s) =
1 –s2–1/2χ(–1,1)(s)
bs,1 –s21/2+bs, –1 –s21/2.
Then Fb(·)satisfies(2.4), (2.6)and FbLq(R)≤CL(ζ)–1+1/q
for some q∈(1, 2),whereζ= (ζ1,ζ2),r(ζ) =|L(ζ)|,and L(ζ) = (2ζ1,ζ2).
The following lemmas will play key roles in the proof of Theorem1.1.
Lemma 2.4([18]) Letϕ∈Gand h∈γ(R+)for someγ > 1,then
hϕ–1Υϕ
ϕ–1
γ(R+)≤Chγ(R+),
where the constant C> 0depends only onϕ.
Lemma 2.5 Letϕ∈GandΥϕ(t) =tϕϕ(t()t).Then
(i) ifϕis nonnegative and increasing,Th,Ω,P,ϕf =Th(ϕ–1)Υϕ(ϕ–1),Ω,Pf; (ii) ifϕis nonnegative and decreasing,Th,Ω,P,ϕf = –Th(ϕ–1)Υ
ϕ(ϕ–1),Ω,Pf;
(iii) ifϕis non-positive and decreasing,Th,Ω,P,ϕf =Th(ϕ–1)Υ
ϕ(ϕ–1),Ω,˜Pf;
(iv) ifϕis non-positive and increasing,Th,Ω,P,ϕf= –Th(ϕ–1)Υ
ϕ(ϕ–1),Ω,˜Pf, whereΩ˜(y) =Ω(–y).
Lemma2.5can be proved by similar arguments as in the proof of [18, Lemma 2.3], we omit the details.
3 Proofs of the main results
In this section we shall prove Theorems1.1–1.3. In what follows, we letdeg(P) =Nand
P(t) =Nk=1aktkand assume thatΩ∈H1(Sn–1) satisfies (1.1).
Proof of Theorem1.1 By Lemmas2.4and2.5, to prove Theorem1.1, it suffices to show that
Th,Ω,PfL2(Rn)≤Chγ(R+)ΩH1(Sn–1)fL2(Rn), (3.1)
whereC> 0 is independent ofh,Ω,f and the coefficients ofP, but depends ondeg(P). It is clear thatTh,Ω,Pf(x) =Kh,Ω,P∗f(x), where the functionKh,Ω,Pis defined by
Kh,Ω,P(ξ) =
Rn
e–2πiP(|y|)y·ξh(|y|)Ω(y )
By Plancherel’s theorem, (3.1) reduces to the following:
Kh,Ω,PL∞(Rn)≤Chγ(R+)ΩH1(Sn–1), (3.2)
whereC> 0 is independent ofh,Ωand the coefficients ofP. Invoking Lemma2.1, we can writeΩ=jλjajsatisfying
j|λj| ≤CΩH1(Sn–1), where eachajis a (1,∞) atom and
eachλjis a complex number. Therefore, to prove (3.2), it is enough to prove that
Kh,Ω,PL∞(Rn)≤Ch
γ(R+), (3.3)
whereΩis a (1,∞) atom andC> 0 is independent ofh,Ωand the coefficients ofP. We now prove (3.3). LetΩbe a (1,∞) atom satisfying (2.1)–(2.3). Without loss of gener-ality, we may assume thatϑ= (1, 0, . . . , 0). We only prove the casen≥3, since the proof for
n= 2 is essentially the same (using Lemma2.3instead of Lemma2.2). Fixξ= (0, 0, . . . , 0) and writeξ=ξ/|ξ|= (ξ1, . . . ,ξn). We can choose a rotationOsuch thatO(ξ) =ϑ. By the change of variables, we write
Kh,Ω,P(ξ) = ∞
0
h(t)
Sn–1
e–2πiP(t)|ξ|y·ιΩO–1ydσydt t .
Letb(y) =Ω(O–1y). It is easy to see thatbis a regular (1,∞) atom satisfying (2.2)–(2.3)
andsupp(b)⊂B(ξ,)∩Sn–1. By the change of variables, we have
Kh,Ω,P(ξ) = ∞
0
h(t)
R
Fb(s)e–2πiP(t)|ξ|sds
dt t ,
whereFb is the function defined in Lemma2.2. We know by Lemma2.2thatsupp(Fb)⊂
(ξ1–2r(ξ),ξ1+2r(ξ)) andFbsatisfies (2.5)–(2.6), wherer(ξ) =|(2ξ1,ξ2, . . . ,ξn)|. We set
Nb(s) =r(ξ)Fb(r(ξ)s+ξ1). One can easily check thatsupp(Nb)⊂(–2, 2) andNbL∞(R)≤C
andRNb(s)ds= 0. By the change of variables again,
Kh,Ω,P(ξ) = ∞
0
h(t)
R
Nb(s)e–2πiP(t)|ξ|r(ξ
)s dsdt
t .
For convenience, we set
bk= 2πak|ξ|r
ξ, |βκ|1/κ= max 1≤k≤N|bk|
1/k, and β=|β
κ|–1/κ.
We can write
Kh,Ω,P(ξ) = β
0
h(t)
R
Nb(s)e–2πiP(t)|ξ|r(ξ
)s dsdt
t
+
∞
β
h(t)
R
Nb(s)e–2πiP(t)|ξ|r(ξ
)s dsdt
t
ForI1, let us choose an integerK0such that 2K0≤β< 2K0+1. By the cancelation condition
ofNband Hölder’s inequality, we have
|I1|=
β
0
h(t)
R
Nb(s)
e–2πiP(t)|ξ|r(ξ)s– 1dsdt t ≤ K0
j=–∞
2j+1
2j
h(t)
R Nb(s)
e–2πiP(t)|ξ|r(ξ)s– 1dsdt
t
≤Chγ(R+)
K0
j=–∞
2j+1
2j
N
k=1
|bk|tk
γ
dt t
1/γ
≤Chγ(R+) K0
j=–∞
N
k=1
|bk|
2j+1
2j t
kγ–1dt
1/γ
=Chγ(R+) K0
j=–∞
N
k=1
|bk|kγ–1/γ2kγ– 11/γ2kj
≤Chγ(R+), (3.5)
where in the last inequality of (3.5) we have used the fact thatαα≤1 for allα∈(0, 1] and
max1≤k≤N|bk|2K0k≤1. HereC> 0 is independent off,ζ and the coefficients ofP.
ForI2, letΦ∈C0∞(R) such thatΦ(t)≡1 if|t| ≤1 andΦ(t)≡0 if|t| ≥2. For anyj∈Z,
we setRj= [2j, 2j+1) and define the operatorTjby
Tjf(t) =χRj(t)
R
Φ(s)f(s)e–2πiP(t)|ξ|r(ξ)sds.
From the estimate on page 60 in [32], there exists a large integerΛ> 0 independent ofj
such that
TjfL2(R)≤C2j/2|βk|–1/(2Λ)2–jκ/(2Λ)fL2(R), (3.6)
whereC> 0 is independent off,ξand the coefficients ofP. By (3.6) and Hölder’s inequality,
|I2| ≤ ∞
j=K0 2j+1
2j
h(t)
R
Nb(s)e–2πiP(t)|ξ|r(ξ
)s dsdt
t
≤2hγ(R+)
∞
j=K0 2j+1
2j
RNb(s)e–2πiP(t)|ξ|r(ξ
)s ds γ dt t 1/γ
≤Chγ(R+)
∞
j=K0 2j+1
2j
RNb(s)e–2πiP(t)|ξ|r(ξ
)s ds
2
dt t
1/γ˜
≤Chγ(R+) ∞
j=K0
2–j/γ˜
R χRj(t)
R
Nb(s)e–2πiP(t)|ξ|r(ξ
)s ds
2
dt
1/γ˜
≤Chγ(R+) ∞
j=K0
2–j/γ˜2j|β
κ|–1/Λ2–jκ/ΛNb2L2(R) 1/γ˜
where in the last inequality of (3.7) we have used the fact that 2K0κ ≥2–κβ–1
κ . Hereγ˜= max{2,γ}and the constantC> 0 is independent ofh,b,ξ and the coefficients ofP. We get from (3.4)–(3.5) and (3.7) that
Kh,Ω,P(ξ)≤Chγ(R+),
whereC> 0 is independent ofh,Ω,ξand the coefficients ofP. This yields (3.3) and com-pletes the proof of Theorem1.1.
Proof of Theorem1.2 Leth,Ω,ϕbe given as in Theorem1.2. We only prove Theorem1.2
for the caseϕ∈Fsatisfying condition (a), and another case is discussed similarly. Fort> 0, we define the measureσh,Ω,P,ϕ,tby
σh,Ω,P,ϕ,t(x) =
1
tρ
t/2<|y|≤t
e–2πiP(ϕ(|y|))x·yh(|y|)Ω(y )
|y|n–ρ dy. (3.8)
By Minkowski’s inequality and the change of variables, one can easily verify that
Mh,Ω,P,ϕ,ρf(x)≤
1 1 – 2–τ
∞
0
σh,Ω,P,ϕ,t∗f(x)
2dt
t
1/2
. (3.9)
By Plancherel’s theorem and Fubini’s theorem, to prove Theorem 1.2, we only need to show that
0∞σh,Ω,P,ϕ,t(·)2
dt t
1/2
L∞(Rn)≤
Chγ(R+)ΩH1(Sn–1), (3.10)
whereC> 0 is independent ofh,Ωand the coefficients ofP, but depends onϕandN. Invoking Lemma2.1, (3.10) reduces to the following:
∞
0
σh,Ω,P,ϕ,t(·)
2dt
t
1/2
L∞(Rn)≤
Chγ(R+), (3.11)
whereΩis a (1,∞) atom andC> 0 is independent ofh,Ω and the coefficients ofP, but depends onϕandN.
Givenξ= (0, 0, . . . , 0). To prove (3.11), we want to show that
∞
0
σh,Ω,P,ϕ,t(ξ)2
dt
t ≤Ch
2
γ(R+), (3.12)
whereΩis a (1,∞) atom andC> 0 is independent ofh,Ω,ξ and the coefficients ofP. We now prove (3.12). Without loss of generality we may assume thatΩis a (1,∞) atom satisfying (2.1)–(2.3) withϑ= (1, 0, . . . , 0) and only prove the casen≥3. LetbandNbbe
given as in the proof of Theorem1.1. By some change of variables, we write
σh,Ω,P,ϕ,t(ξ) =
1
tρ
t
t/2
h(r)
R
Nb(s)e–2πiP(ϕ(r))|ξ|r(ξ
)s ds dr
By (3.13) and Hölder’s inequality, we obtain that
σh,Ω,P,ϕ,t(ξ)≤Chγ(R+)
t
t/2
RNb(s)e–2πiP(ϕ(r))|ξ|r(ξ
)s ds γ dr r 1/γ .
Hence, to prove (3.12), it suffices to show that
∞
0
t
t/2
RNb(s)e–2πiP(ϕ(r))|ξ|r(ξ
)s ds γ dr r 2/γ dt
t ≤C, (3.14)
whereC> 0 is independent ofh,Ω,ξ and the coefficients ofP.
Next we shall prove (3.14). By a change of variable and the properties ofϕ,
∞
0
t
t/2
RNb(s)e–2πiP(ϕ(r))|ξ|r(ξ
)s ds γ dr r 2/γ dt t
≤Cϕ
∞
0
ϕ(t)
ϕ(t/2)
RNb(s)e–2πiP(r)|ξ|r(ξ
)s ds γ dr r 2/γ dt
t . (3.15)
Letβbe given as in the proof of Theorem1.1andδ=ϕ–1(β). We write
∞
0
ϕ(t)
ϕ(t/2)
RNb(s)e–2πiP(r)|ξ|r(ξ
)s ds γ dr r 2/γ dt t = δ 0
ϕ(t)
ϕ(t/2)
RNb(s)e–2πiP(r)|ξ|r(ξ
)s ds γ dr r 2/γ dt t + ∞ δ
ϕ(t)
ϕ(t/2)
RNb(s)e–2πiP(r)|ξ|r(ξ
)s ds γ dr r 2/γ dt t
=:J1+J2. (3.16)
ForJ1, by the fact that
RNb(s)ds= 0 and the change of variables, we have
J1 =
δ
0
ϕ(t)
ϕ(t/2)
RNb(s)
e–2πiP(r)|ξ|r(ξ)s– 1ds
γ dr r 2/γ dt t ≤C N k=1
|bk|2
δ
0
ϕ(t)
ϕ(t/2)
rkγ–1dr
2/γ dt t ≤C N k=1
|bk|2
δ
0
ϕ(t)2kdt
t
≤C
N
k=1
|bk|2
ϕ(δ)
0
t2k–1dt
≤C
N
k=1
|bk|2β2k≤C, (3.17)
ForJ2. Fixt≥δ, there exists an integerj0such that 2j0≤ϕ(t/2) < 2j0+1. By the fact that
ϕ(2t)≤cϕϕ(t), there exists an integerk0such that 2k0≤cϕ< 2k0+1and thenϕ(t)≤2k0+j0+2.
From (3.6) we have
ϕ(t)
ϕ(t/2)
RNb(s)e–2πiP(r)|ξ|r(ξ
)s ds
2
dr r
≤ k0+j0+1
j=j0 2j+1
2j
RNb(s)e–2πiP(r)|ξ|r(ξ
)s ds
2
dr r
≤ k0+j0+1
j=j0
2–j2j|βκ|–1/Λ2–jκ/ΛNb2L2(R)
≤C
k0+j0+1
j=j0
|βκ|–1/Λ2–jκ/Λ
≤C|βκ|–1/Λϕ(t)–κ/Λ, (3.18)
whereC> 0 is independent ofΩ,ξ and the coefficients ofP, but depends onϕ. (3.18) together with Hölder’s inequality and the change of variable shows that
J2≤C(ϕ)
∞
δ
ϕ(t)
ϕ(t/2)
RNb(s)e–2πiP(r)|ξ|r(ξ
)s ds
2
dr r
2/γ˜
dt t
≤C
∞
δ
|βκ|–1/Λϕ(t)–κ/Λ
2/γ˜dt
t
≤C|βκ|–2/(Λγ˜)
∞
δ
ϕ(t)–2κ/(Λγ˜)dt
t
≤C|βκ|–2/(Λγ˜)
∞
β
t–2κ/(Λγ)–1˜ dt
≤CΛγ˜
2κ . (3.19)
HereC> 0 is independent ofh,Ω,ξ and the coefficients ofP, but depends onϕ. Then (3.14) follows from (3.15)–(3.17) and (3.19). This proves Theorem1.2.
Proof of Theorem1.3 LetΩ,ϕbe given as in Theorem1.3. By arguments similar to those used in deriving (3.14) and (3.16) in [37], one can easily get that
MΩ,P,ϕ,ρf(x)≤C(ρ)SΩ,P,ϕf(x).
Thus, we only prove Theorem1.3for the operatorSΩ,P,ϕ. Define the measureσΩ,P,ϕ,tby
σΩ,P,ϕ,t(ξ) =
Sn–1
e–2πiP(ϕ(t))y·ξΩydσy. (3.20)
By duality we can write
SΩ,P,ϕf(x) =
∞
0
σΩ,P,ϕ,t∗f(x)
2dt
t
1/2
By (3.21) and the same arguments as in the proof of Theorem1.2, to prove Theorem1.3
for the operatorSΩ,P,ϕ, it suffices to show that
∞
0
σΩ,P,ϕ,t(·)
2dt
t
1/2
L∞(Rn)≤
CΩH1(Sn–1), (3.22)
whereC> 0 is independent ofΩand the coefficients ofP. By Lemma2.1and Minkowski’s inequality, (3.22) reduces to the following:
0∞σΩ,P,ϕ,t(·)2
dt t
1/2
L∞(Rn)
≤C, (3.23)
whereΩis a (1,∞) atom andC> 0 is independent ofΩand the coefficients ofP. We now prove (3.23). We only assume thatΩis a (1,∞) atom satisfying (2.1)–(2.3) with
ϑ= (1, 0, . . . , 0) and consider the casen≥3. Letξ,b, andNbbe given as in the proof of
Theorem1.1. By the change of variables, we have
∞
0
σΩ,P,ϕ,t(ξ)
2dt
t =
∞
0
RNb(s)e–2πiP(ϕ(t))|ξ|r(ξ
)s ds
2
dt t .
By a change of variable and the properties ofϕ, we obtain that
∞
0
σΩ,P,ϕ,t(ξ)
2dt
t ≤C(ϕ)
∞
0
RNb(s)e–2πiP(t)|ξ|r(ξ
)s ds
2
dt
t . (3.24)
Letbk,βbe given as in the proof Theorem1.1. We write ∞
0
RNb(s)e–2πiP(t)|ξ|r(ξ
)s ds 2 dt t = β 0
RNb(s)e–2πiP(t)|ξ|r(ξ
)s ds 2 dt t + ∞ β
RNb(s)e–2πiP(t)|ξ|r(ξ
)s ds
2
dt t
=:L1+L2. (3.25)
ForL1, by the cancelation condition ofNb, we have
L1=
β
0
RNb(s)
e–2πiP(t)|ξ|r(ξ)s– 1ds
2
dt t ≤C
N
k=1
|bk|2
β
0
t–1+2kdt≤C, (3.26)
whereC> 0 is independent ofb,ξand the coefficients ofP.
ForL2, letK0be given as in the proof of Theorem1.1. Applying (3.6), we get
L2≤ ∞
j=K0 2j+1
2j
RNb(s)e–2πiP(t)|ξ|r(ξ
)s ds 2 dt t ≤ ∞
j=K0
2–j
R χRj(t)
R
Nb(s)e–2πiP(t)|ξ|r(ξ
)s ds 2 dt ≤C ∞
j=K0
where in the last inequality of (3.27) we have used the fact that 2K0κ≥2–κβ–1
κ andC> 0 is
independent ofb,ξ and the coefficients ofP. We get from (3.24)–(3.27) that
∞
0
σΩ,P,ϕ,t(ξ)
2dt
t ≤C,
whereC> 0 is independent ofΩ,ξ and the coefficients ofP. This yields (3.23) and com-pletes the proof of Theorem1.3.
4 Hardy-type inequalities
In this section we shall establish the following Hardy-type inequalities.
Theorem 4.1 Let P be a real polynomial onRsatisfying P(0) = 0andΩ∈H1(Sn–1)
satis-fying(1.1).Then we have
(i) Ifϕ∈F,then
Rn
|x1|ρ
|x|/2≤|y|<|x|
e–2πiP(ϕ(|y|))x·y Ω(y)
|y|n–ρdy
2|dxx|n
1/2
≤CΩH1(Sn–1), (4.1)
whereCis a positive constant independent ofΩand the coefficients ofP,but depends onρ,ϕ,anddeg(P).
(ii) Ifϕ∈G,then
Rn
Sn–1
e–2πiP(ϕ(|x|))x·yΩydσy
2
dx
|x|n 1/2
≤CΩH1(Sn–1), (4.2)
whereCis a positive constant independent ofΩand the coefficients ofP,but depends onϕanddeg(P).
Proof of Theorem4.1 We first prove (i). Using Lemma2.1, (4.1) reduces to the following:
Rn
|x1|ρ
|x|/2≤|y|<|x|
e–2πiP(ϕ(|y|))x·y Ω(y)
|y|n–ρdy
2|dxx|n
1/2
≤C, (4.3)
whereΩis a (1,∞) atom andCis a positive constant independent ofΩand the coefficients ofP. By the polar coordinates,
Rn
|x1|ρ
|x|/2≤|y|<|x|
e–2πiP(ϕ(|y|))x·y Ω(y)
|y|n–ρdy
2|dxx|n
=
Sn–1 ∞
0
t1ρ
t
t/2
Sn–1
e–2πiP(ϕ(r))x·yΩydσy dr r1–ρ
2dtt dσx
=
Sn–1 ∞
0
σb,Ω,P,ϕ,t
x2dt t dσ
x
≤ωn–1
∞
0
σb,Ω,P,ϕ,t(·)
2dt
t
1/2
2 L∞(Sn–1)
. (4.4)
Hereωn–1is the surface area of the unit Sn–1andσb,Ω,P,ϕ,tis defined as in (3.8) withb(·)≡1.
It remains to prove (ii). To prove (4.2), by Lemma2.1it suffices to show that
Rn
Sn–1
e–2πiP(ϕ(|x|))x·yΩydσy
2
dx
|x|n 1/2
≤C, (4.5)
whereΩis a (1,∞) atom andCis a positive constant independent ofΩand the coefficients ofP. Using the polar coordinates, we can obtain
Rn
Sn–1
e–2πiP(ϕ(|x|))x·yΩydσy
2
dx
|x|n
=
Sn–1 ∞
0
Sn–1
e–2πiP(ϕ(t))x·yΩydσy
2
dt t dσ
x
=
Sn–1 ∞
0
σΩ,P,ϕ,t
x2dt t dσ
x
≤ωn–1
0∞σΩ,P,ϕ,t
x2dt t
1/2
2 L∞(Sn–1)
, (4.6)
whereσΩ,P,ϕ,tis defined as in (3.20). (4.6) together with (3.22) yields (4.5). Theorem4.1is
proved.
Acknowledgements
The authors would like to express their sincere thanks to the referees for their careful reading and invaluable comments.
Funding
The first author was supported partially by the NNSF of China (No. 11701333) and Support Program for Outstanding Young Scientific and Technological Top-notch Talents of College of Mathematics and Systems Science (No. Sxy2016K01). The second author was supported partially by the NNSF of China (No. 11571160).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors worked jointly in drafting and approving the final manuscript.
Author details
1College of Mathematics and Systems Sciences, Shandong University of Science and Technology, Qingdao, P.R. China. 2Department of Mathematics, Mudanjiang Normal University, Mudanjiang, P.R. China.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Received: 20 November 2018 Accepted: 13 March 2019
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