# Boundedness for Parametrized Littlewood-Paley Operators with Rough Kernels on Weighted Weak Hardy Spaces

Authors: Ximei Wei; Shuangping Tao
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The authors prove that the parametrized area integralμΩ,Sρ and gλ* function μλ*,ρ are bounded from the weighted weak Hardy space WHw1(ℝn) to the weighted weak Lebesgue space WLw1(ℝn) as Ω satisfies a class of the integral Dini condition, respectively.


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## Body

## 1. Introduction and Main Results

Suppose thatΩ∈L1(Sn-1) is homogeneous of degree zero on ℝn and satisfies
(1)∫Sn-1‍Ω(x′)dσ(x′)=0,
where Sn-1 denotes the unit sphere of ℝn  (n⩾2) equipped with normalized Lebesgue measure dσ=dσ(x′), x′=x/|x|, x≠0. The parametrized area integral μΩ,Sρ and gλ* function μλ*,ρ are defined by
(2)μΩ,Sρ(f)(x)=(∫‍∫Γ(x)‍|1tρ∫|y-z|<t‍Ω(y-z)|y-z|n-ρf(z)dz|2dydttn+1)1/2,μλ*,ρ(f)(x)=(|1tρ∫|y-z|<t‍Ω(y-z)|y-z|n-ρf(z)dz|2∫‍∫ℝ+n+1‍(tt+|x-y|)λn×|1tρ∫|y-z|<t‍Ω(y-z)|y-z|n-ρf(z)dz|2dydttn+1)1/2,
respectively, where
(3)Γ(x)={(y,t)∈ℝ+n+1:|x-y|<t},ℝ+n+1=ℝn×(0,∞),ρ>0,λ>1.It is well known that Littlewood-Paley functions are very important tools in harmonic analysis and PDE (see [1–3]). Some well-known results related to the classical Littlewood-Paley operators can be seen in [4–8]. In 1999, inspired by Hörmander’s work [9], when Ω satisfies the Lipschitz condition of α,  Sakamoto and Yabuta  [10] established the Lp  (1<p<∞) boundedness of the parametrized area integral μΩ,Sρ and the parametrized gλ* function μλ*,ρ and gave the boundedness on BMO spaces and Campanato spaces. For any 0<α⩽1, 1<q⩽∞, it is easy to see that the inclusion relationship
(4)Lipα(Sn-1)Lq(Sn-1)Llog+L(Sn-1)Lipα(Sn-1)H1(Sn-1)L1(Sn-1)
holds. In 2002,  Ding et al.  [11] extended the previous Lp-boundedness to the case as Ω belongs to Llog+L(Sn-1). In 2007,  Ding et al.  [12, 13] gave the boundedness of the parametrized area integral μΩ,Sρ and gλ* function μλ*,ρ on the Hardy space and weak Hardy space when Ω satisfies a class of the integral Dini conditions. Recently,  Wang and Liu  [14] obtained the boundedness on the weighted Hardy space for the parametrized Littlewood-Paley operators with Ω satisfying the logarithmic type Lipschitz conditions. On the other hand, the boundedness properties of the intrinsic square functions on weighted weak Hardy spaces were studied by Wang in [15]. Inspired by the results mentioned previously, in this paper, we will study the boundedness of the parametrized area integral μΩ,Sρ and gλ* function μλ*,ρ on the weighted weak Hardy spaces.Before stating our main results, let us recall some definitions. Firstly, letΩ(x′)∈Lq(Sn-1), q⩾1. Then, the integral modulus ωq(δ) of continuity of order q of Ω is defined by
(5)ωq(δ)=sup∥γ∥⩽δ(∫Sn-1‍|Ω(γx′)-Ω(x′)|qdσ(x′))1/q,
where, γ denotes a rotation on Sn-1 and ∥γ∥=supx′∈Sn-1|γx′-x′|. The function Ω is said to satisfy the Lq-Dini condition, if
(6)∫01‍ωq(δ)δdδ<∞.
Secondly, given a weight function w on ℝn, for 1⩽p<∞, the weighted Lebesgue spaces is defined by
(7)Lwp(ℝn)={f:∥f∥p,w=(∫ℝn‍|f(x)|pw(x)dx)1/p<∞}.
And also, the weighted weak Lebesgue spaces is defined by
(8)WLwp(ℝn)={f:∥f∥WLwp=supλ>0λ·w({x∈ℝn:|f(x)|>λ})1/p<∞∥f∥WLwp=supλ>0}.
Let us now turn to recall the definition of the weighted weak Hardy spaces. The weak Hardy spaces were first introduced in [16]. The atomic decomposition theory of weak H1 spaces on ℝn was given by Fefferman and Soria in [17]. Later, Liu established the weak Hp spaces on homogeneous groups in [18]. In 2000, Quek and Yang introduced the weighted weak Hardy spaces WHwp(ℝn) in [19] and established their atomic decompositions. Moreover, by using the atomic decomposition theory of WHwp(ℝn), Quek and Yang also obtained the boundedness of C-Z operators on these weighted spaces in [19]. Let w∈A∞, 0<p⩽1, and N=[n(qw/p-1)]. Define
(9)𝒜N,w={supx∈ℝnsup|α|⩽N+1φ∈𝒮(ℝn):supx∈ℝnsup|α|⩽N+1(1+|x|)N+n+1|Dαφ(x)|⩽1supx∈ℝnsup|α|⩽N+1},
where, α=(α1,…,αn)∈(ℕ∪{0})n, |α|=α1+⋯+αn, Dαφ=∂|α|φ/(∂x1α1⋯∂xnαn).Forf∈𝒮′(ℝn), the grand maximal function of f is defined by
(10)Gwf(x)=supφ∈𝒜N,wsup|y-x|<t|(φt*f)(y)|.
Then, weighted weak Hardy space is defined by WHwp(ℝn)={f∈𝒮′(ℝn):Gwf∈WLwp(ℝn)}. Moreover, we set ∥f∥WHwp=∥Gwf∥WLwp.Our main results are stated as follows.Theorem 1.
LetΩ∈L2(Sn-1) satisfying (1) and the following condition
(11)∫01‍ω2(δ)δ1+αdδ<∞,0<α⩽1.
Then, for ρ>n/2, w∈A1, there exists a constant C>0 such that
(12)∥μΩ,Sρ(f)∥WLw1⩽C∥f∥WHw1.The relationship between condition (11) and Lipα(Sn-1) condition is not clear up to now. We point that the conclusion of Theorem 1 still holds if we replace the condition (11) by the Lipα(Sn-1)(0<α⩽1) condition. In other words, we have the following result.Theorem 2.
LetΩ∈Lipα(Sn-1), 0<α⩽1, satisfying (1). Then, for ρ>n/2, w∈A1, there exists a constant C>0 such that
(13)∥μΩ,Sρ(f)∥WLw1⩽C∥f∥WHw1.Theorem 3.
LetΩ∈L2(Sn-1) satisfying (1) and the following condition
(14)∫01‍ω2(δ)δ1+αdδ<∞,0<α⩽1.
Then, for ρ>n/2, w∈A1, λ>2, there exists a constant C>0 such that
(15)∥μλ*,ρ(f)∥WLw1⩽C∥f∥WHw1.


## 2. Notations and Preliminaries

In this section, we will introduce some notations and preliminary lemmas used in the proofs of our main theorems in the next section.The classicalAp weighted theory was first introduced by Muckenhoupt in the study of weighted Lp boundedness of Hardy-Littlewood maximal functions in [20]. A weight w is a locally integrable function on ℝn which takes values in (0,∞) at almost everywhere. Given a ball B and λ>0, λB denotes the ball with the same center as B whose radius is λ times that of B. We also denote the weighted measure of E by w(E); that is, w(E)=∫E‍w(x)dx. We say that w∈Ap with 1<p<∞ if there exists a constant C>0, such that for every ball B⊂ℝn,
(16)(1|B|∫B‍w(x)dx)(1|B|∫B‍w(x)-1/(p-1)dx)p-1⩽C.
We say that w∈A1 if
(17)1|B|∫B‍w(x)dx⩽C·essinfx∈Bw(x)foreveryballB⊂ℝn.
A weight function w∈A∞ if it satisfies the Ap condition for some 1<p<∞. It is well known that if w∈Ap, 1<p<∞, then w∈Ar for all r>p, and w∈Aq for some 1<q<p. We thus write qw≡inf{q>1:w∈Aq} to denote the critical index of w.Lemma 4 (see [21]).
Let1⩽p<∞, w∈Ap. Then, for any ball B, there exists an absolute constant C>0, such that
(18)w(2B)⩽Cw(B).
In general, for any λ>0, we have
(19)w(λB)⩽Cλnpw(B),
where C does not depend on B nor on λ.Lemma 5 (see [19]).
Let0<p⩽1, w∈A∞. For every f(x) belongs to WHwp(ℝn), there exists a sequence of bounded measurable functions {fk(x)}k=-∞∞ such that(i)
f(x)=∑k=-∞∞‍fk(x), in 𝒮′,(ii)
eachfk can be further decomposed into fk=∑i‍bik, where bik satisfies the following conditions.
(a)
supp(bik)⊂Qik, where Qik denotes the ball with center xik and radius rik. Moreover,
(20)∑i‍w(Qik)⩽C12-kp,∑i‍χQik⩽C1,
where χE denotes the characteristic function of the set E and C1⩽∥f∥WHwpp.(b)
∥bik∥L∞⩽C2k, where C>0 is independent of i, k.(c)
∫ℝn‍bikxαdx=0 for every multi-index α with |α|⩽[n(qw/p-1)].
Conversely, iff∈𝒮′(ℝn) have a decomposition satisfying (i) and (ii), then f∈WHwp(ℝn). Moreover, we have ∥f∥WHwpp~C.In the end of this section, we need the following lemmas used in the next section.Lemma 6 (see [22]).
Suppose thatΩ∈L2(Sn-1) satisfies (1) and the following condition
(21)∫01‍ω2(δ)δ(1+|logδ|)σdδ<∞,σ>1,w∈Ap. Then, for ρ>n/2, λ>2, and f∈Lp(ℝn)(1<p<∞), there is a constant C independent of f, such that
(22)∥μΩ,Sρ(f)∥Lwp⩽C∥f∥Lwp,∥μλ*,ρ(f)∥Lp⩽C∥f∥Lwp.Lemma 7 (see [23]).
Suppose thatρ>0, Ω is homogeneous of degree zero and satisfies the L2-Dini condition. If there exists a constant 0<θ<1/2 such that |x|<θR, then we have
(23)(∫R<|y|<2R‍|Ω(y-x)|y-x|n-ρ-Ω(y)|y|n-ρ|2dy)1/2⩽CRn/2-(n-ρ){|x|R+∫|x|/2R<δ<|x|/R‍ω2(δ)δdδ},
where the constant C>0 is independent of R, x.


## 3. Proof of Main Results

Proof of Theorem1.
In order to prove Theorem1, it suffices to show that there exists a constant C>0, for any f∈WHw1(ℝn) and β>0, such that
(24)βw({x∈ℝn:|μΩ,Sρ(f)(x)|>β})⩽C∥f∥WHw1.
Take k0∈ℤ such that 2k0⩽β<2k0+1; then by Lemma 5 we can write
(25)f=∑k=-∞∞‍fk=∑k=-∞k0‍fk+∑k=k0+1∞‍fk=:F1+F2,
where F1=∑k=-∞k0∑ibik, F2=∑k=k0+1∞∑ibik, and {bik} satisfies (a)–(c) in Lemma 5. Then, we have
(26)βw({x∈ℝn:|μΩ,Sρ(f)(x)|>β})⩽βw({x∈ℝn:|μΩ,Sρ(F1)(x)|>β2})+βw({x∈ℝn:|μΩ,Sρ(F2)(x)|>β2})=:I1+I2.
First, we claim that the following inequality holds:
(27)∥F1∥Lw2⩽Cβ1/2∥f∥WHw11/2.
In fact, since supp(bik)⊂Qik=Q(xik,rik), ∥bik∥L∞⩽C2k, then it follows from Minkowski's integral inequality that
(28)∥F1∥Lw2⩽C∑k=-∞k0‍∑i‍∥bik∥L∞w(Qik)1/2∥F1∥Lw2⩽C∑k=-∞k0‍2k(∑i‍w(Qik))1/2∥F1∥Lw2⩽C∑k=-∞k0‍2k/2∥f∥WHw11/2∥F1∥Lw2⩽C∑k=-∞k0‍2(k-k0)/2β1/2∥f∥WHw11/2⩽Cβ1/2∥f∥WHw11/2.
Let us estimate I1. By Chebyshev's inequality, Lemma 6 and (27), we have
(29)I1⩽β4β2∫ℝn‍|μΩ,Sρ(F1)(x)|2w(x)dxI1=C1β∥μΩ,Sρ(F1)∥Lw22I1⩽C1β∥F1∥Lw22I1⩽C∥f∥WHw1.
Now we turn our attention to the estimate of I2. If we set
(30)Ak0=⋃k=k0+1∞‍⋃i‍Q~ik,
where Q~ik=Q(xik,8τ(k-k0)/nrik) and τ is a fixed positive number such that 1<τ<2, therefore,
(31)I2=βw({x∈2Ak0:|μΩ,Sρ(F2)(x)|>β2})I2=+βw({x∈(2Ak0)c:|μΩ,Sρ(F2)(x)|>β2})I2=:I2′+I2′′.
Since w∈A1, then by Lemma 4 we can get
(32)I2′⩽Cβ∑k=k0+1∞‍∑i‍w(Q~ik)I2′⩽Cβ∑k=k0+1∞‍τk-k0∑i‍w(Qik)I2′⩽C∥f∥WHw1∑k=k0+1∞‍(τ2)k-k0I2′⩽C∥f∥WHw1.
An application of Chebyshev’s inequality and Minkowski integral inequality gives us that(33)I2′′⩽β2β∫(2Ak0)c‍μΩ,Sρ(F2)(x)w(x)dxI2′′=C∫(2Ak0)c‍(∫‍∫|y-x|<t‍|∑k=k0+1∞‍∑i‍∫|y-z|<t‍Ω(y-z)|y-z|n-ρbik(z)dz|2dtdytn+2ρ+1)1/2w(x)dxI2′′⩽C∑k=k0+1∞‍∑i‍∫(2Ak0)c‍(∫‍∫|y-x|<t‍|∫|y-z|<t‍Ω(y-z)|y-z|n-ρbik(z)dz|2dtdytn+2ρ+1)1/2w(x)dxI2′′⩽C∑k=k0+1∞‍∑i‍[∫(2Ak0)c‍(∫‍∫y∈4Qik|y-x|<t‍|∫|y-z|<t‍Ω(y-z)|y-z|n-ρbik(z)dz|2dtdytn+2ρ+1)1/2w(x)dxI2′′⩽I2′′⩽I2′′⩽⩽+∫(2Ak0)c‍(∫‍∫y∈(4Qik)c|y-x|<t‍|∫|y-z|<t‍Ω(y-z)|y-z|n-ρbik(z)dz|2dtdytn+2ρ+1)1/2w(x)dx]I2′′=:C∑k=k0+1∞‍∑i‍[I3+I4].Firstly, let us estimate I3. As y∈4Qik, x∈(2Ak0)c, z∈Qik, it is easy to see that(34)|y-x|⩾|x-xik|-|y-xik|⩾|x-xik|2,|y-z|<8rik,I3⩽C2k∫(2Ak0)c‍(∫‍∫y∈4Qik|x-xik|/2⩽|y-x|<t‍|∫|y-z|<t‍Ω(y-z)|y-z|n-ρdz|2dtdytn+2ρ+1)1/2w(x)dxI3⩽C2k∫(2Ak0)c‍(∫y∈4Qik‍∫|x-xik|/2∞‍|∫Sn-1‍∫08rik‍|Ω(z′)|sn-ρsn-1dsdσ(z′)|2dtdytn+2ρ+1)1/2w(x)dxI3⩽C2k∫(2Ak0)c‍(∫y∈4Qik‍∫|x-xik|/2∞‍(8rik)2ρdtdytn+2ρ+1)1/2w(x)dxI3⩽C2k(8rik)ρ∫(2Ak0)c‍(∫y∈4Qik‍dy|x-xik|n+2ρ)1/2w(x)dxI3⩽C2k(8rik)ρ|4Qik|1/2∫(2Ak0)c‍w(x)|x-xik|n/2+ρdx.Since x∈(2Ak0)c, Q~ik=Q(xik,8τ(k-k0)/nrik), then |x-xik|>8τ(k-k0)/nrik>8rik. By Lemma 4, we obtain that
(35)I3⩽C2k(rik)ρ+n/2I3⩽×∑j=3∞‍∫2jτ(k-k0)/nrik<|x-xik|<2j+1τ(k-k0)/nrikw(x)|x-xik|n/2+ρdxI3⩽C2k(rik)ρ+n/2I3⩽×∑j=3∞‍(2j+1τ(k-k0)/nrik)-n/2-ρI3⩽×∫|x-xik|<2j+1τ(k-k0)/nrik‍w(x)dxI3⩽C2k∑j=3∞‍2-(ρ-n/2)j12(j+1)n(2j+1τ(k-k0)/n)nw(Qik)I3⩽C2k∑j=3∞‍2-(ρ-n/2)j(τ(k-k0)/n)n/2-ρw(Qik)I3⩽C2k(τ(k-k0)/n)n/2-ρw(Qik).
Noticing that ρ>n/2, 1<τ<2, we have
(36)∑k=k0+1∞‍∑i‍I3⩽C∑k=k0+1∞‍2k(τ(k-k0)/n)n/2-ρ∑i‍w(Qik)∑k=k0+1∞‍∑i‍I3⩽C∑k=k0+1∞‍(τ(k-k0)/n)n/2-ρ∥f∥WHw1∑k=k0+1∞‍∑i‍I3⩽C∥f∥WHw1.
Now we consider I4. Write(37)I4⩽C∫(2Ak0)c‍(∫‍∫y∈(4Qik)c|y-x|<tt⩽|y-xik|+2rik‍|∫|y-z|<t‍Ω(y-z)|y-z|n-ρbik(z)dz|2dtdytn+2ρ+1)1/2w(x)dxI4⩽+C∫(2Ak0)c‍(∫‍∫y∈(4Qik)c|y-x|<t|y-xik|+2rik<t‍|∫|y-z|<t‍Ω(y-z)|y-z|n-ρbik(z)dz|2dtdytn+2ρ+1)1/2w(x)dxI4=:C(I41+I42). Take 0<ɛ<min{1/2,α,ρ-n/2}. First we deal with I41. As |y-z|<t, we have
(38)t>|y-z|⩾|y-xik|-|xik-z|t⩾|y-xik|-2rik,|x-xik|⩽|x-y|+|y-xik|⩽t+|y-xik||x-xik|⩽2|y-xik|+2rik⩽3|y-xik|,|1(|y-xik|-2rik)n+2ρ-1(|y-xik|+2rik)n+2ρ|⩽Crik|y-xik|n+2ρ+1.
Using the Minkowski inequality, we get that(39)I41⩽∫(2Ak0)c‍∫ℝn‍|bik(z)|(∫‍∫y∈(4Qik)c|y-x|<tt⩽|y-xik|+2rik|y-z|<t‍|Ω(y-z)|2|y-z|2n-2ρdtdytn+2ρ+1)1/2dzw(x)dx⩽∫(2Ak0)c‍∫ℝn‍|bik(z)|(∫y∈(4Qik)c|x-xik|⩽3|y-xik|‍|Ω(y-z)|2|y-z|2n-2ρ(∫|y-xik|-2rik|y-xik|+2rik‍dttn+2ρ+1)dy)1/2dzw(x)dx⩽C∫(2Ak0)c‍∫ℝn‍|bik(z)|×(∫y∈(4Qik)c|x-xik|⩽3|y-xik|‍|Ω(y-z)|2|y-z|2n-2ρ|1(|y-xik|-2rik)n+2ρ-1(|y-xik|+2rik)n+2ρ|dy∫y∈(4Qik)c|x-xik|⩽3|y-xik|‍)1/2dzw(x)dx⩽C∫(2Ak0)c‍∫ℝn‍|bik(z)|(∫y∈(4Qik)c|x-xik|⩽3|y-xik|‍|Ω(y-z)|2|y-z|2n-2ρrik|y-xik|n+2ρ+1dy∫y∈(4Qik)c|x-xik|⩽3|y-xik|‍)1/2dzw(x)dx⩽C∫(2Ak0)c‍∫ℝn‍|bik(z)|(∫y∈(4Qik)c‍|Ω(y-z)|2|y-z|n-2ɛ+1rik|x-xik|2n+2ɛdy)1/2dzw(x)dx⩽C(rik)1/2∫(2Ak0)c‍∫ℝn‍|bik(z)||x-xik|n+ɛ(∫y∈(4Qik)c‍|Ω(y-z)|2|y-z|n-2ɛ+1dy)1/2dzw(x)dx⩽C(rik)1/2∫(2Ak0)c‍∫ℝn‍|bik(z)||x-xik|n+ɛ(∫Sn-1‍∫2rik∞‍|Ω(z′)|2s2-2ɛdsdσ(z′))1/2dzw(x)dx⩽C(rik)ɛ∫(2Ak0)c‍w(x)|x-xik|n+ɛdx∫Qik‍|bik(z)|dz⩽C2k(rik)ɛ+n∑j=3∞‍∫2jτ(k-k0)/nrik<|x-xik|<2j+1τ(k-k0)/nrikw(x)|x-xik|n+ɛ‍dx⩽C2k(rik)ɛ+n∑j=3∞‍(2j+1rik)-n-ɛτ(k-k0)(-n-ɛ)/n∫|x-xik|<2j+1τ(k-k0)/nrik‍w(x)dx⩽C2k∑j=3∞‍(2j+1)-n-ɛτ(k-k0)(-n-ɛ)/n2(j+1)nτk-k0w(Qik)⩽C2k[τ(k-k0)/n]-ɛ∑j=3∞‍2-ɛ(j+1)w(Qik).By Lemma 5, we have
(40)∑k=k0+1∞‍∑i‍I41⩽C∑k=k0+1∞‍2k(τ(k-k0)/n)-ɛ∑i‍w(Qik)⩽C∑k=k0+1∞‍(τ(k-k0)/n)-ɛ∥f∥WHw1⩽C∥f∥WHw1.
Now let us considerI42. It is easy to check that Qik⊂{z:|y-z|<t} as y∈(4Qik)c, t>|y-xik|+2rik. Thus, we can obtain by the condition (c) of bik in Lemma 5(41)I42=∫(2Ak0)c(∫‍∫y∈(4Qik)c|y-x|<t|y-xik|+2rik<t‍|∫|y-z|<t‍Ω(y-z)|y-z|n-ρ-Ω(y-xik)|y-xik|n-ρbik(z)dz|2dtdytn+2ρ+1)1/2w(x)dxI42⩽∫(2Ak0)c∫ℝn‍|bik(z)|(∫‍∫y∈(4Qik)cmax{|y-x|,|y-xik|+2rik,|y-z|}<t|x-xik|⩽2|y-xik|‍|Ω(y-z)|y-z|n-ρ-Ω(y-xik)|y-xik|n-ρ|2dtdytn+2ρ+1)1/2dzw(x)dxI42=+∫(2Ak0)c∫ℝn‍|bik(z)|(∫‍∫y∈(4Qik)cmax{|y-x|,|y-xik|+2rik,|y-z|}<t|x-xik|>2|y-xik|‍|Ω(y-z)|y-z|n-ρ-Ω(y-xik)|y-xik|n-ρ|2dtdytn+2ρ+1)1/2dzw(x)dxI42=I42′+I42′′.I42′⩽∫(2Ak0)c∫ℝn‍|bik(z)|(∫y∈(4Qik)c|x-xik|⩽2|y-xik|‍|Ω(y-z)|y-z|n-ρ-Ω(y-xik)|y-xik|n-ρ|2(∫|y-xik|+2rik∞‍dttn+2ρ+1)dy)1/2dzw(x)dxI42′⩽C∫(2Ak0)c∫ℝn‍|bik(z)|(∫y∈(4Qik)c|x-xik|⩽2|y-xik|‍|Ω(y-z)|y-z|n-ρ-Ω(y-xik)|y-xik|n-ρ|21(|y-xik|+2rik)n+2ρdy)1/2dzw(x)dxI42′⩽C∫(2Ak0)c∫ℝn‍|bik(z)|(|x-xik|+2rik)n+ɛ(∫y∈(4Qik)c‍|Ω(y-z)|y-z|n-ρ-Ω(y-xik)|y-xik|n-ρ|21(|y-xik|+2rik)2ρ-n-2ɛdy)1/2dzw(x)dx.By using Lemma 7, we can get(42)I42′⩽C∫(2Ak0)c‍∫ℝn‍|bik(z)|(|x-xik|+2rik)n+ɛI42′⩽×∑j=1∞‍(1(|y-xik|+2rik)2ρ-n-2ɛ∫2jrik<|y-xik|<2j+1rik‍|Ω(y-z)|y-z|n-ρ-Ω(y-xik)|y-xik|n-ρ|21(|y-xik|+2rik)2ρ-n-2ɛdy)1/2dzw(x)dx⩽C∫(2Ak0)c‍∫ℝn‍|bik(z)|(|x-xik|+2rik)n+ɛ∑j=1∞‍1(2jrik+2rik)ρ-n/2-ɛ×(∫2jrik<|y-xik|<2j+1rik‍|Ω(y-z)|y-z|n-ρ-Ω(y-xik)|y-xik|n-ρ|2dy)1/2dzw(x)dx⩽C∫(2Ak0)c‍∫ℝn‍|bik(z)|(|x-xik|+2rik)n+ɛ∑j=1∞‍1(2jrik+2rik)ρ-n/2-ɛ(2jrik)n/2-(n-ρ){|z-xik|2jrik+∫|z-xik|/2j+1rik|z-xik|/2jrik‍w2(δ)δdδ}dzw(x)dx⩽C∫(2Ak0)c‍∫ℝn‍|bik(z)|(|x-xik|+2rik)n+ɛ∑j=1∞‍(2jrik)ɛ{12j+∫|z-xik|/2j+1rik|z-xik|/2jrik‍w2(δ)δdδ}dzw(x)dx⩽C∫(2Ak0)c‍∫ℝn‍|bik(z)|(|x-xik|+2rik)n+ɛ∑j=1∞‍(2jrik)ɛ12jdzw(x)dx+C∫(2Ak0)c‍∫ℝn‍|bik(z)|(|x-xik|+2rik)n+ɛ∑j=1∞‍(2jrik)ɛ∫|z-xik|/2j+1rik|z-xik|/2jrik‍w2(δ)δdδdzw(x)dx=:C(U1+U2).It is easy to see that
(43)U1⩽C2k(rik)ɛ+nU1⩽×∫(Ak0)c‍w(x)(|x-xik|+2rik)n+ɛdx∑j=1∞‍2j(ɛ-1)U1⩽C2k(rik)ɛ+n∫(Ak0)c‍w(x)(|x-xik|+2rik)n+ɛdx.
Using the same method as what used to deal with the inequality (39), we can obtain that
(44)U1⩽C2k[τ(k-k0)/n]-ɛw(Qik).
For U2, we have
(45)U2⩽C∫(2Ak0)c‍∫ℝn‍|bik(z)|(|x-xik|+2rik)n+ɛ×∑j=1∞‍(2jrik)ɛ2jα∫|z-xik|/2j+1rik|z-xik|/2jrik‍w2(δ)δ1+αdδdzw(x)dxU2⩽C∫(2Ak0)c‍∫ℝn‍(rik)ɛ|bik(z)|(|x-xik|+2rik)n+ɛdzw(x)dx×∑j=1∞‍2j(ɛ-α)∫01‍w2(δ)δ1+αdδU2⩽C2k(rik)ɛ+n∫(2Ak0)c‍w(x)(|x-xik|+2rik)n+ɛdxU2⩽C2k[τ(k-k0)/n]-ɛw(Qik).
Hence, by the inequalities (44) and (45), we have
(46)I42′⩽C(U1+U2)⩽C2k[τ(k-k0)/n]-ɛw(Qik).
Now we give the estimate for I42′′. Since |x-xik|>2|y-xik|, t>|y-xik|+2rik, then
(47)t>|x-y|⩾|x-xik|-|y-xik|⩾|x-xik|2.
Thus,(48)I42′′⩽∫(2Ak0)c‍∫ℝn‍|bik(z)|I42′′⩽×(∫‍∫y∈(4Qik)cmax{|y-x|,|y-xik|+2rik,|y-z|}<t‍1(|y-xik|+2rik)2ρ-n-2ɛ|Ω(y-z)|y-z|n-ρ-Ω(y-xik)|y-xik|n-ρ|2dtdyt2n+2ɛ+1)1/2dzw(x)dxI42′′⩽∫(2Ak0)c‍∫ℝn‍|bik(z)|(∫y∈(4Qik)c‍1(|y-xik|+2rik)2ρ-n-2ɛ|Ω(y-z)|y-z|n-ρ-Ω(y-xik)|y-xik|n-ρ|2(∫|x-xik|/2∞‍dtt2n+2ɛ+1)dy)1/2dzw(x)dxI42′′⩽∫(2Ak0)c‍∫ℝn‍|bik(z)|(1|x-xik|2n+2ɛdy∫y∈(4Qik)c‍1(|y-xik|+2rik)2ρ-n-2ɛ|Ω(y-z)|y-z|n-ρ-Ω(y-xik)|y-xik|n-ρ|21|x-xik|2n+2ɛdy)1/2dzw(x)dxI42′′⩽∫(2Ak0)c‍∫ℝn‍|bik(z)||x-xik|n+ɛ(∫y∈(4Qik)c‍1(|y-xik|+2rik)2ρ-n-2ɛ|Ω(y-z)|y-z|n-ρ-Ω(y-xik)|y-xik|n-ρ|2dy)1/2dzw(x)dx.Repeating this process which is similar to the one of estimating I42′ (from (42) to (46)), we may have
(49)I42′′⩽C2k[τ(k-k0)/n]-ɛw(Qik).
Thus by (36) and (40), we get
(50)I2=I2′+I2′′I2⩽I2′+C∑k=k0+1∞‍∑i‍(I3+I4)I2⩽I2′+C∑k=k0+1∞‍∑i‍(I3+I41+I42)I2⩽I2′+C∑k=k0+1∞‍∑i‍(I3+I41+I42′+I42′′)I2⩽C∥f∥WHw1.
This completes the proof of Theorem 1.Proof of Theorem2.
Combining the idea of proving Theorem1 with the similar steps as in [12] and the following inequalities
(51)|Ω(y-z)|y-z|n-ρ-Ω(y-z)|y-xik|n-ρ|⩽C|1|y-z|n-ρ-1|y-xik|n-ρ||Ω(y-z)|y-z|n-ρ-Ω(y-z)|y-xik|n-ρ|⩽Crik|y-xik|n-ρ+1,|Ω(y-z)-Ω(y-xik)|⩽C|y-z|y-z|-y-xik|y-xik||α|Ω(y-z)-Ω(y-xik)|⩽C(rik)α|y-xik|α,
it is not difficult to get the proof of Theorem 2. We omit the details here.Proof of Theorem3.
We follow the strategy of the proof of Theorem1. It suffices to show that there exists a constant C>0, such that, for any f∈WHw1(ℝn), β>0,
(52)βw({x∈ℝn:|μλ*,ρ(f)(x)|>β})⩽C∥f∥WHw1.
Take k0∈ℤ such that 2k0⩽β<2k0+1, we have
(53)βw({x∈ℝn:|μλ*,ρ(f)(x)|>β})⩽βw({x∈ℝn:|μλ*,ρ(F1)(x)|>β2})+βw({x∈ℝn:|μλ*,ρ(F2)(x)|>β2})=:J1+J2,
where the notations F1, F2 are the same as in the proof of Theorem 1. Using the same method of the proof of Theorem 1, we can get
(54)J1⩽4β-1∫ℝn‍|μλ*,ρ(F1)(x)|w(x)dxJ1⩽Cβ-1∥μλ*,ρ(F1)∥Lw22J1⩽Cβ-1∥F1∥Lw22J1⩽C∥f∥WHw1.Below, we will give the estimate of J2. If we set
(55)Ak0=⋃k=k0+1∞‍⋃i‍Q~ik,
where Q~ik=Q(xik,8τ(k-k0)/nrik), τ is a fixed positive number such that 1<τ<2; thus,
(56)J2=βw({x∈2Ak0:|μλ*,ρ(F2)(x)|>β2})J2=+βw({x∈(2Ak0)c:|μλ*,ρ(F2)(x)|>β2})J2=:J2′+J2′′.
Noting that w∈A2, then by Lemmas 4 and 5, we have(57)J2′⩽2β∑k=k0+1∞‍∑i‍w(Q~ik)⩽Cβ∑k=k0+1∞‍τk-k0∑i‍w(Qik)⩽C∑k=k0+1∞‍(τ2)k-k0∥f∥WHw1⩽C∥f∥WHw1.J2′′⩽β2β∫(2Ak0)c‍μλ*,ρ(F2)(x)w(x)dxJ2′′=C∫(2Ak0)c‍(∫‍∫ℝ+n+1‍(tt+|x-y|)λn|∑k=k0+1∞‍∑i‍∫|y-z|<t‍Ω(y-z)|y-z|n-ρbik(z)dz|2dtdytn+2ρ+1)1/2w(x)dxJ2′′⩽C∑k=k0+1∞‍∑i‍∫(2Ak0)c‍(∫‍∫ℝ+n+1‍(tt+|x-y|)λn|∫|y-z|<t‍Ω(y-z)|y-z|n-ρbik(z)dz|2dtdytn+2ρ+1)1/2w(x)dxJ2′′⩽C∑k=k0+1∞‍∑i‍∫(2Ak0)c‍(J3+J4)w(x)dx,where(58)J3=(∫‍∫|y-x|<t‍(tt+|x-y|)λn|∫|y-z|<t‍Ω(y-z)|y-z|n-ρbik(z)dz|2dtdytn+2ρ+1)1/2,J4=(∫‍∫|y-x|⩾t‍(tt+|x-y|)λn|∫|y-z|<t‍Ω(y-z)|y-z|n-ρbik(z)dz|2dtdytn+2ρ+1)1/2.Similarly as in the proof of Theorem 1, for x∈(2Ak0)c, if the integration domains of J4 is y∈4Qik, |y-x|⩾t, then we denote it by J41. If y∈(4Qik)c, |y-x|⩾t, we denote J4 by J42. Moreover, if t⩽|y-xik|+2rik in the integration domains of J42, we denote it by J5, otherwise denote it by J6. Further, we divide J5 again by the integration domains; namely, if |x-xik|⩽2|y-xik|, we denote it by J51, otherwise denote it by J52. Now, we are in a position to give the estimates of J3, J41, J51, J52, J6, respectively. First, we take 0<ɛ<min{1/2,ρ-n/2,α,(λ-2)n/2} in the whole proof of Theorem 2. Obviously,
(59)∫(2Ak0)c‍J3w(x)dx⩽∫(2Ak0)c‍(∫‍∫|y-x|<t‍|∫|y-z|<t‍Ω(y-z)|y-z|n-ρ×bik(z)dz∫|y-z|<t‍|2dtdytn+2ρ+1∫‍∫|y-x|<t‍|∫|y-z|<t‍Ω(y-z)|y-z|n-ρ)1/2w(x)dx.
By the proof of Theorem 1, we have
(60)∑k=k0+1∞‍∑i‍∫(2Ak0)c‍J3w(x)dx⩽C∥f∥WHw1.
Notice that if y∈4Qik, x∈(2Ak0)c, |y-x|⩾t, z∈Qik, it is easy to check that(a′)
|y-x|⩾|x-xik|-|y-xik|⩾|x-xik|/2, |y-z|<8rik.Ify∈(4Qik)c, x∈(2Ak0)c, |y-x|⩾t, z∈Qik, then(b′)
|y-z|~|y-xik|, |y-xik|-2rik⩽t⩽|y-xik|+2rik, for t⩽|y-xik|+2rik;(c′)
|1/(|y-xik|-2rik)n+2ρ-1/(|y-xik|+2rik)n+2ρ|⩽C(rik/|y-xik|n+2ρ+1);(d′)
|y-x|⩾|x-xik|-|y-xik|⩾|x-xik|/2, for |x-xik|⩾2|y-xik|;(e′)
t+|x-y|⩾t+|x-xik|-|y-xik|⩾|x-xik|+2rik, for t>|y-xik|+2rik.
Now, let us estimateJ41. By the fact (a′) and the Minkowski inequality, we have(61)J41⩽(∫‍∫y∈4Qik|y-x|>|x-xik|/2|y-x|⩾t‍(tt+|x-y|)2n+2ɛ|∫|y-z|<8rik|y-z|<t‍Ω(y-z)|y-z|n-ρbik(z)dz|2dtdytn+2ρ+1)1/2J41⩽C2k∫Qik‍(∫‍∫y∈4Qik|y-x|>|x-xik|/2|y-z|<8rik,|y-z|<t|y-x|⩾t‍(tt+|x-xik|/2)2n+2ɛ|Ω(y-z)|2|y-z|2n-2ρdtdytn+2ρ+1)1/2dzJ41⩽C2k(1|x-xik|/2)n+ɛ/2∫Qik‍(∫y∈4Qik|y-x|>|x-xik|/2|y-z|<8rik‍|Ω(y-z)|2|y-z|2n-2ρ∫0|y-x|‍t2n+2ɛ|x-xik|ɛ|y-z|2ρ-n-ɛt2n+ɛ+1dtdy)1/2dzJ41⩽C2k1|x-xik|n+ɛ/2∫Qik‍(∫y∈4Qik|y-x|>|x-xik|/2|y-z|<8rik‍|Ω(y-z)|2|y-x|ɛ|y-z|n-ɛ|x-xik|ɛdy)1/2dzJ41⩽C2k1|x-xik|n+ɛ/2∫Qik‍(∫|y-z|<8rik‍|Ω(y-z)|2|y-z|n-ɛdy)1/2dzJ41⩽C2k1|x-xik|n+ɛ/2∫Qik‍(∫Sn-1‍∫08rik‍|Ω(y′)|2|s|n-ɛsn-1dsdσ(y′))1/2dzJ41⩽C2k1|x-xik|n+ɛ/2∫Qik‍(8rik)ɛ/2dzJ41⩽C2k(rik)n+ɛ/2|x-xik|n+ɛ/2.As for J51, notice that y∈(4Qik)c, x∈(2Ak0)c, |y-x|⩾t, z∈Qik, using the Minkowski inequality and the previous facts (b′) and (c′), we have
(62)J51⩽∫Qik‍|bik(z)|×[∫y∈(4Qik)c|x-xik|⩽2|y-xik|‍(∫|y-xik|-2rik|y-xik|+2rik‍dttn+2ρ+1)×|Ω(y-z)|2|y-z|2n-2ρdy∫y∈(4Qik)c|x-xik|⩽2|y-xik|‍]1/2dzJ51⩽C2k∫Qik‍(∫y∈(4Qik)c|x-xik|⩽2|y-xik|‍|Ω(y-z)|2|y-z|2n-2ρ×rik|y-xik|n+2ρ+1dy∫y∈(4Qik)c|x-xik|⩽2|y-xik|‍)1/2dzJ51⩽C2k×∫Qik‍(∫y∈(4Qik)c|x-xik|⩽2|y-xik|‍|Ω(y-z)|2|y-z|2n-2ρ×rik|y-xik|2ρ-n+1-2ɛ|y-xik|2n+2ɛdy∫y∈(4Qik)c|x-xik|⩽2|y-xik|‍)1/2dzJ51⩽C2k(rik)1/2|x-xik|n+ɛ×∫Qik‍(∫y∈(4Qik)c‍|Ω(y-z)|2|y-z|n+1-2ɛdy)1/2dzJ51⩽C2k(rik)n+1/2|x-xik|n+ɛ×(∫Sn-1‍∫4rik∞‍|Ω(y′)|2s2-2ɛdsdσ(y′))1/2J51⩽C2k(rik)n+ɛ|x-xik|n+ɛ.
Now we consider J52. By the fact (d′) and t/(t+|x-y|)<t, we have(63)J52⩽∫Qik‍|bik(z)|[∫y∈(4Qik)c|x-xik|⩾2|y-xik|‍(∫|y-xik|-2rik|y-xik|+2rik‍(tt+|x-y|)2n+2ɛdttn+2ρ+1)|Ω(y-z)|2|y-z|2n-2ρdy]1/2dzJ52⩽C2k∫Qik‍(∫y∈(4Qik)c|x-xik|⩾2|y-xik|‍(∫|y-xik|-2rik|y-xik|+2rik‍tn+2ɛ-2ρ-1|x-y|2n+2ɛdt)|Ω(y-z)|2|y-z|2n-2ρdy)1/2dzJ52⩽C2k∫Qik‍(∫y∈(4Qik)c|x-xik|⩾2|y-xik|‍|Ω(y-z)|2|x-y|2n+2ɛ|y-z|2n-2ρrik|y-xik|2ρ-n-2ɛ+1dy)1/2dzJ52⩽C2k∫Qik‍(∫y∈(4Qik)c‍|Ω(y-z)|2|y-z|n-2ɛ+1rik|x-xik|2n+2ɛdy)1/2dzJ52⩽C2k(rik)n+ɛ|x-xik|n+ɛ.For y∈(4Qik)c, since t>|y-xik|+2rik, then we have Qik⊂{z:|y-z|<t}. Thus, by the cancellation of bik, we have ∫|y-z|<t‍bik(z)dz=0. By using the fact (e′), we get(64)J6=(∫‍∫y∈(4Qik)c|y-xik|+2rik<t|y-x|⩾t‍(tt+|x-y|)λn|∫|y-z|<t‍(|Ω(y-z)||y-z|n-ρ-|Ω(y-xik)||y-xik|n-ρ)bik(z)dz|2|dydttn+2ρ+1)1/2J6⩽∫ℝn‍|bik(z)|(∫‍∫y∈(4Qik)c|y-xik|+2rik<t|y-x|⩾t|y-z|<t‍tλn(t+|x-y|)2n+2ɛ1(t+|x-y|)λn-2n-2ɛ||Ω(y-z)||y-z|n-ρ-|Ω(y-xi)k||y-xik|n-ρ|2dydttn+2ρ+1)1/2dzJ6⩽C∫Qik‍|bik(z)|(|x-xik|+2rik)n+ɛ(∫‍∫y∈(4Qik)c|y-xik|+2rik<t|y-x|⩾t|y-z|<t‍tλn(t+|x-y|)λn-2n-2ɛ||Ω(y-z)||y-z|n-ρ-|Ω(y-xik)||y-xik|n-ρ|2dydttn+2ρ+1)1/2dzJ6⩽C∫Qik‍|bik(z)|(|x-xik|+2rik)n+ɛ×[∫y∈(4Qik)c|y-x|>|y-xik|+2rik‍||Ω(y-z)||y-z|n-ρ-|Ω(y-xik)||y-xik|n-ρ|2(∫|y-xik|+2rik|y-x|‍tλn-2n-2ɛ(t+|x-y|)λn-2n-2ɛt2ρ-n+1-2ɛdt)dy]1/2dz.Noting that 0<ɛ<ρ-n/2, we have 2ρ-n-2ɛ>0. Hence
(65)∫|y-xik|+2rik|y-x|‍1t2ρ-n+1-2ɛdt⩽∫|y-xik|+2rik∞‍1t2ρ-n+1-2ɛdt⩽C1(|y-xik|+2rik)2ρ-n-2ɛ.
Thus,
(66)J6⩽C∫Qik‍|bik(z)|(|x-xik|+2rik)n+ɛJ6⩽×(∫y∈(4Qik)c‍||Ω(y-z)||y-z|n-ρ-|Ω(y-xik)||y-xik|n-ρ|2×1|y-xik|2ρ-n-2ɛdy∫y∈(4Qik)c‍||Ω(y-z)||y-z|n-ρ-|Ω(y-xik)||y-xik|n-ρ|2)1/2dzJ6⩽C∫Qik‍|bik(z)|(|x-xik|+2rik)n+ɛJ6⩽×(∑j=2∞‍∫2jrik⩽|y-xik|<2j+1rik‍||Ω(y-z)||y-z|n-ρ-|Ω(y-xik)||y-xik|n-ρ|2×1|y-xik|2ρ-n-2ɛdy∑j=2∞‍∫2jrik⩽|y-xik|<2j+1rik‍||Ω(y-z)||y-z|n-ρ-|Ω(y-xik)||y-xik|n-ρ|2)1/2dzJ6⩽C∫Qik‍|bik(z)|(|x-xik|+2rik)n+ɛJ6⩽×∑j=2∞‍(2jrik)ρ-n/2-ɛJ6⩽×(∫2jrik⩽|y-xik|<2j+1rik‍||Ω(y-z)||y-z|n-ρ-|Ω(y-xik)||y-xik|n-ρ|2dy)1/2dzJ6⩽C∫Qik‍|bik(z)|(|x-xik|+2rik)n+ɛ∑j=2∞‍(2jrik)ɛJ6⩽×{12j+12jα∫|z-xik|/2j+1rik|z-xik|/2jrik‍w2(δ)δ1+αdδ}dzJ6⩽C∫Qik‍|bik(z)|(rik)ɛ(|x-xik|+2rik)n+ɛdzJ6⩽C2k(rik)n+ɛ|x-xik|n+ɛ.
From (61) to (66), we can obtain
(67)∑k=k0+1∞‍∑i‍∫(2Ak0)c‍J4w(x)dx⩽C∑k=k0+1∞‍∑i‍∫(2Ak0)c‍(J41+J42)w(x)dx⩽C∑k=k0+1∞‍∑i‍∫(2Ak0)c‍(J41+J5+J6)w(x)dx⩽C∑k=k0+1∞‍∑i‍∫(2Ak0)c‍(J41+J51+J52+J6)w(x)dx⩽C∑k=k0+1∞‍∑i‍∫(2Ak0)c‍2k×((rik)n+ɛ/2|x-xik|n+ɛ/2+(rik)n+ɛ|x-xik|n+ɛ)w(x)dx⩽C∑k=k0+1∞‍2k∑i‍(rik)ɛ/2+n×∑j=3∞‍∫2jτ(k-k0)/nrik<|x-xik|<2j+1τ(k-k0)/nrik‍w(x)|x-xik|n+ɛ/2dx+C∑k=k0+1∞‍2k∑i‍(rik)ɛ+n×∑j=3∞‍∫2jτ(k-k0)/nrik<|x-xik|<2j+1τ(k-k0)/nrik‍w(x)|x-xik|n+ɛdx⩽C∑k=k0+1∞‍2k∑i‍(rik)ɛ/2+n×∑j=3∞‍1(2jrik)ɛ/2+n∫|x-xik|<2j+1τ(k-k0)/nrik‍w(x)dx+∑k=k0+1∞‍2k∑i‍(rik)ɛ+n×∑j=3∞‍1(2jrik)ɛ+n∫|x-xik|<2j+1τ(k-k0)/nrik‍w(x)dx⩽C∑k=k0+1∞‍2k∑i‍∑j=3∞‍2-(j+1)(ɛ/2)τ-ɛ(k-k0)/2nw(Qik)+C∑k=k0+1∞‍2k∑i‍∑j=3∞‍2-(j+1)ɛτ-ɛ(k-k0)/nw(Qik)⩽C∑k=k0+1∞‍τ-ɛ(k-k0)/2n∥f∥WHw1+C∑k=k0+1∞‍τ-ɛ(k-k0)/n∥f∥WHw1⩽C∥f∥WHw1.
We conclude the proof of Theorem 3.



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*Source: 651941-2013-08-01.xml*