KYUNGPOOK Math. J. 2019; 59(2): 259-275  
On the Boundedness of Marcinkiewicz Integrals on Variable Exponent Herz-type Hardy Spaces
Rabah Heraiz
Department of Mathematics, Laboratory of Functional Analysis and Geometry of Spaces, M’sila University, P. O. Box 166, M’sila 28000, Algeria
e-mail : heraizrabeh@yahoo.fr and rabah.heraiz@univ-msila.dz
* Corresponding Author.
Received: March 22, 2018; Revised: November 2, 2018; Accepted: December 6, 2018; Published online: June 23, 2019.
© Kyungpook Mathematical Journal. All rights reserved.

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Abstract

The aim of this paper is to prove that Marcinkiewicz integral operators are bounded from K˙p(·)α(·),q(·)(n) to K˙p(·)α(·),q(·)(n) when the parameters α(·), p(·) and q(·) satisfies some conditions. Also, we prove the boundedness of μ on variable Herz-type Hardy spaces HK˙p(·)α(·),q(·)(n).

Keywords: Herz spaces, Herz-type Hardy spaces, variable exponent, HardyLittlewood maximal operator, Marcinkiewicz integral operators.
1. Introduction and Preliminaries

Function spaces with variable exponent are being actively studied not only in the field of real analysis but also in partial differential equations and in applied mathematics. The theory of function spaces with variable exponents has rapidly made progress in the last three decades.

For 0 < β ≤ 1, the Lipschitz space Lipβ(ℝn) is defined as

Lipβ(n):={f:fLipβ(n)=supx,yn;xyf(x)-f(y)x-yβ<}.

Given Ω ∈ Lipβ(ℝn) be a homogeneous function of degree zero and

Sn-1Ω(x)   dσ(x)=0

where x′ = x/|x| for any x ≠ 0 and Sn−1 denotes the unit sphere in ℝn (n ≥ 2) equipped with the normalized Lebesgue measure.

The Marcinkiewicz integral μ is defined by

μ(f)(x):=(0FΩf(x)2dtt3)12

where

FΩf(x):=x-ytΩ(x-y)x-yn-1f(y)dy.

It is well known that the operator μ was first defined by Stein [13] and under the conditions above, Stein proved that μ is of type (p, p) for 1 < p ≤ 2 and of weak type (1, 1). Benedek et al. [2] showed that μ of type (p, p) with 1 < p < ∞.

Recently, the boundedness of Marcinkiewicz integral operators μ on variable function spaces have attracted great attention (see [14, 15, 18] and their references).

The purpose of this paper is to generalize some results concerning Marcinkiewicz integral operators μ on variable Herz spaces K˙p(·)α(·),q(·)(n) and variable Herz-type Hardy spaces HK˙p(·)α(·),q(·)(n). We define the set of variable exponents by

P0(n):={pmeasurable:p(·):n[c,]   for some c>0}.

The subset of variable exponents with range [1,∞) is denoted by 𝒫(ℝn). For p ∈ 𝒫0(ℝn), we use the notation

p-=essinfxnp(x),         p+esssupxnp(x).

Definition 1.1

Let p ∈ ℘0(ℝn). The variable exponent Lebesgue space Lp(·)(ℝn) is the class of all measurable functions f on ℝn such that the modular

ϱp(·)(f):=nf(x)p(x)dx

is finite. This space is a quasi-Banach function space equipped with the norm

fp(·):=inf{μ>0:ϱp(·)(1μf)1}.

If p(x) ≡ p is constant, then Lp(·)(ℝn) = Lp(ℝn) is the classical Lebesgue space.

Definition 1.2

We say that a function g : ℝn → ℝ is locally log-Hölder continuous, if there exists a constant clog > 0 such that

g(x)-g(y)clogln(e+1/x-y)

for all x, y ∈ ℝn. If

g(x)-g(0)clogln(e+1/x)

for all x ∈ ℝn, then we say that g is log-Hölder continuous at the origin (or has a log decay at the origin). If, for some g ∈ ℝ and clog > 0, there holds

g(x)-gclogln(e+x)

for all x ∈ ℝn, then we say that g is log-Hölder continuous at infinity (or has a log decay at infinity).

As an example of a function locally log-Hölder continuous, see E. Nakai and Y. Sawano [12, Example 1.3].

The set P0log(n) and Plog(n) consist of all exponents p ∈ ℘(ℝn) which have a log decay at the origin and at infinity, respectively. The set ℘log(ℝn) is used for all those exponents p ∈ ℘(ℝn) which are locally log-Hölder continuous and have a log decay at infinity, with p := lim|x|→∞p(x).

It is well known that if p ∈ ℘log(ℝn) then p′ ∈ ℘log(ℝn), where p′ denotes the conjugate exponent of p given by 1/p(·) + 1/p′ (·) = 1.

Definition 1.3

Let p, q ∈ ℘0(ℝn). The mixed Lebesgue-sequence space ℓq(·)(Lp(·)) is defined on sequences of Lp(·)-functions by the modular

ϱq(·)(Lp(·))((fv)v)=vinf{λv>0:ϱp(·)(fvλv1/q(·))1}.

The (quasi)-norm is defined from this as usual:

(fv)vq(·)(Lp(·))=inf{γ>0:ϱq(·)(Lp(·))(1γ(fv)v)1}.

Since q+< ∞, then we can replace by the simpler expression ϱq(·)(Lp(·))((fv)v)=vfvq(·)p(·)q(·).

If E ⊂ ℝn is a measurable set, then |E| stands for the (Lebesgue) measure of E and χE denotes its characteristic function. Before giving the definition of variable Herz spaces, let us introduce the following notations

Bk:=B(0,2k),Rk:=Bk/Bk-1   and   χRk,k.

Definition 1.4

Let p, q ∈ ℘0(ℝn) and α : ℝn → ℝ with αL(ℝn). The inhomogeneous Herz space Kp(·),q(·)α(·)(n) consists of all fLLocp(·)(n) such that

fKp(·),q(·)α(·):=fχB0p(·)+(2kα(·)fχk)k1q(·)(Lp(·))<.

Similarly, the homogeneous Herz space K˙p(·)α(·),q(·)(n) is defined as the set of all fLLocp(·)(n/{0}) such that

fK˙p(·)α(·),q(·)(n):=(2kα(·)fχk)kq(·)(Lp(·))<.

The variable Herz spaces Kp(·)α(·),q(·)(n) and K˙p(·)α(·),q(·)(n), were first introduced by Izuki and Noi in [8]. In [6] the authors obtained a new equivalent norm of these function spaces. We refer the reader to the paper [5, 16] for further results for these function spaces. If α(·), p(·) and q(·) are constants, then K˙q(·)α(·),p(·)(n) is the classical Herz space K˙qα,p(n).

The following proposition is very important for the proof of the main results in this paper; it is from D. Drihem and F. Seghiri in [6].

Proposition 1.5

Let αL(ℝn), p, q ∈ ℘0(ℝn). If α and q are log-Hölder continuous at infinity, then

Kp(·)α(·),q(·)(n)=Kp(·)α,q(n).

Additionally, if α(·) and q(·) have a log decay at the origin, then

fK˙p(·)α(·),q(·)(n)(k=--12kα(0)fχkp(·)q(0))1/q(0)+(k=02kαfχkp(·)q)1/q.

The Hardy-Littlewood maximal operator ℳ is defined on Lloc1 by

M(f)(x):=supr>01B(x,r)B(x,r)f(y)dy,

where B(x, r) is the open ball in ℝn centered at x ∈ ℝn and radius r > 0. It was shown in [4, Theorem 4.3.8] that ℳ : Lp(·)Lp(·) is bounded if p ∈ ℘log and p > 1, see also [3, Theorem 1.2].

Let ϕC0(n) with supp ϕB0, ∫nϕ(x)dx ≠ 0 and ϕt(·)=t-nϕ(·t) for any t > 0. Let ℳϕ(f) be the grand maximal function of f defined by

Mϕ(f)(x):=supt>0ϕt*f(x).

Here we give the definition of the homogeneous Herz-type Hardy spaces HK˙p(·)α(·),q(·).

Definition 1.6

Let p, q ∈ ℘0(ℝn) and α : ℝn → ℝ with αL(ℝn). The homogeneous Herz-type Hardy space HK˙p(·)α(·),q(·)(n) is defined as the set of all such that Mϕ(f)K˙p(·)α(·),q(·)(n) and we define

fHK˙p(·)α(·),q(·):=Mϕ(f)K˙p(·)α(·),q(·).

It can be shown that if α(·), p(·) and q(·) satisfy the conditions of Definition 1.6, then the quasi-norm fHK˙p(·)α(·),q(·) does not depend, up to the equivalence of quasi-norms, on the choice of the function ϕ and, hence, the space HK˙p(·)α(·),q(·)(n) is defined independently of the choice of ϕ. If pP0log(n)Plog(n) with -np+<α-α+<n-np- and q ∈ ℘0(ℝn) then HK˙p(·)α(·),q(·)(n)=K˙p(·)α(·),q(·)(n). If α (·) = 0, p (·) = q (·) then HK˙p(·)α(·),q(·)(n) and K˙p(·)α(·),q(·)(n) coincide with Lp(·)(ℝn).

One recognizes immediately that if α(·), p(·) and q(·) are constants, then the spaces HK˙pα,q are just the usual Herz-type Hardy spaces were recently studied in [10] and [11].

We refer the reader to the recent monograph [4, section 4.5] for further details, historical remarks and more references on variable exponent spaces.

2. Some Technical Lemmas

In this section, we present six lemmas used to prove our main theorems in Section 3. Recall that the expression fg means that fcg for some independent constant c (and non-negative functions f and g), and fg means fgf.

Lemma 2.1 plays an important role in the proof of main results; Lemma 2.2 is a Hardy-type inequality which is easy to prove; Lemma 2.3 presents the Hölder inequality in Lp(·)(ℝn); Lemma 2.4 presents the Lp(·)-boundedness of μ; Lemma 2.5 treats the boundedness of fractional integral on variable Lebesgue space; and the last Lemma presents the boundedness of homogeneous function of degree zero.

Lemma 2.1.([1])

Let pPlog(n)and let R=B(0,r)/B(0,r2). If |R| ≥ 2n, then

χRp(·)R1p(x)R1p

with the implicit constants independent of r and xR.

The left-hand side equivalence remains true for every |R| > 0 if we assume, additionally, pP0log(n)Plog(n).

Lemma 2.2.([5])

Let 0 < a < 1 and 0 < q ≤ ∞. Let {ɛk}k∈ℤbe a sequence of positive real numbers, such that

{ɛk}kq=I<.

Then the sequences {δk : δk = ∑jk akjɛj}k∈ℤand {ηk : ηk = ∑jk ajkɛj}k∈ℤbelong to ℓq, and

{δk}kq+{ηk}kqcI,

with c > 0 only depending on a and q.

Lemma 2.3.([4])

Let p ∈ ℘(ℝn). Then for all fLp(·)(ℝn) and gLp′(·)(ℝn), fgL1(ℝn) and

fg12fp(·)gp(·).

Lemma 2.4.([9])

Let p ∈ ℘log(ℝn), then there exists a constant C such that for any fLp(·)(ℝn)

μ(f)p(·)Cfp(·).

Lemma 2.5.([17])

Suppose that p1, p2 ∈ ℘log(ℝn) with p1+<nσand 1p1(·)-1p2(·)=σn. Then for all fLp1(·) (ℝn), we have

nf(y)·-yn-σdyp2(·)cfp1(·).

Lemma 2.6.([14])

If a > 0, 1 ≤ s ≤ ∞, 0 < ds andn + (n − 1)d/s < τ < ∞, then

(yaxyτΩ(x-y)ddy)1dcx(τ+n/d)ΩLs(Sn-1).
3. Variable Herz Estimate of Marcinkiewicz Integral Operators

In this section, we present two results concerning the Marcinkiewicz integral operator μ. In the first, we show that μ is bounded from K˙p(·)α(·),q(·)(n) to K˙p(·)α(·),q(·)(n) for α(·), p(·) and q(·) satisfies some conditions. Next, we present the boundedness of μ on variable Herz-type Hardy spaces HK˙p(·)α(·),q(·)(n).

Motivated by [9] and [14], we generalize the boundedness for Marcinkiewicz integral operators μ to the case of variable Herz spaces ( all exponents are variables). One of our main results can be stated as follows.

Theorem 3.1

Suppose that 0 < τ ≤ 1, p ∈ ℘log(ℝn) with p+< ∞, Ω ∈ Ls(Sn−1), s > (p′)and αL(ℝn), q ∈ ℘0(ℝn). If α and q have a log decay at the origin such that

-np(0)-ns-τ<α(0)<n-np(0)-ns-τ   and   -np-ns-τ<α<n-np-ns-τ

then μ is bounded from K˙p(·)α(·),q(·)(n)(or Kp(·)α(·),q(·)(n)) to K˙p(·)α(·),q(·)(n)(or Kp(·)α(·),q(·)(n)).

Remark 3.2

We would like to mention if α(·) and q(·) are constants, then the statements corresponding to Theorem 3.1 can be found in Theorem 2.1 of [14].

In the following, we use c as a generic positive constant, i.e. a constant whose value may change from appearance to appearance.

Proof of Theorem 3.1

We show that

μ(f)K˙p(·)α(·),q(·)(n)cfK˙p(·)α(·),q(·)(n)

for all fK˙p(·)α(·),q(·)(n). Using Proposition 1.5, we have μ(f) in K˙p(·)α(·),q(·)(n)-norm is equivalent to

{k=--12kα(0)q(0)μ(f)χkp(·)q(0)}1/q(0)+{k=0+2kα(q)μ(f)χkp(·)(q)}1/(q),

we write

f=jfχj=jfj,

then

μ(f)K˙p(·)α(·),q(·)(n){k=--12kα(0)q(0)(j=-k-2μ(fj)χkp(·))q(0)}1/q(0)+{k=--12kα(0)q(0)(j=k-2k+1μ(fj)χkp(·))q(0)}1/q(0)+{k=--12kα(0)q(0)(j=k+2μ(fj)χkp(·))q(0)}1/q(0)+{k=02kαq(j=-k-2μ(fj)χkp(·))q}1/q+{k=02kαq(j=k-2μ(fj)χkp(·))q}1/q+{k=02kαq(j=k+2μ(fj)χkp(·))q}1/q=:H1+H2+H3+H4+H5+H6.

Let us estimate H1 and H4. We consider

μfj(x)(0x|x-ytΩ(x-y)x-yn-1fj(y)dy|2dtt3)12+(x|x-ytΩ(x-y)x-yn-1fj(y)dy|2dtt3)12:=I1+I2.

Observe that in this case xRk, yRj and jk − 2. So we know that |xy| ≈ |x| ≈ 2k, and by mean value theorem, we have

|1x-y2-1x2|cyx-y3.

By (3.1), the Minkowski inequality and the generalized Hölder inequality, we have

I1nΩ(x-y)x-yn-1fj(y)|x-yxdtt3|12dynΩ(x-y)x-yn-1fj(y)y12x-y32dy2j/2xn+1/2nΩ(x-y)fj(y)dy2-nkfjp(·)Ω(x-·)χjp(·).

The estimation of I2 is the same as before since we never use |xy| ≈ |x|, then we obtain

I22-nkfjp(·)Ω(x-·)χjp(·).

We can obtain that each term (I1 and I2) is no more than

c2-nkfjp(·)Ω(x-·)χjp(·),

in the other hand, by Hölder inequality and Lemma 2.6, we obtain

Ω(x-·)χjp(·)Ω(x-·)χjsχjθ(·)2-jτ(RjysτΩ(x-y)sdy)1sχjθ(·)2-jτ2k(τ+ns)ΩLs(Sn-1)χjθ(·)

where 1p(·)=1s+1θ(·). Since θPlog(n), we have for any j

χjθ(·)χjp(·)Rj-1s,

which gives

μ(fj)χkp(·)2-nk2-(ns+τ)(j-k)fjp(·)χjp(·)χkp(·)ΩLs(Sn-1).

Estimation of H1. In this case, since k and j are negative integers, by Lemma 2.1 and since Ω ∈ Ls(Sn−1), we have

μ(fj)χkp(·)2(n-np(0)-τ-ns)(j-k)fjp(·)   ΩLs(Sn-1)2(n-np(0)-τ-ns)(j-k)fjp(·),

which gives

H1{k=--12kα(0)q(0)(j=-k-22(j-k)(n-np(0)-τ-ns)q(0)fjp(·)q(0))}1/q(0)=c{k=--1(j=-k-22(j-k)(n-α(0)-np(0)-τ-ns)q(0)(2jα(0)q(0)fjp(·)q(0)))}1/q(0)

since α(0)-n+np(0)-ns-τ<0, then by Lemma 2.2, we have

H1c{j=--12jα(0)q(0)fjp(·)q(0)}1/q(0)cfK˙p(·)α(·),q(·)(n).

Estimation of H4. We split

j=-k-2μ(fj)χkp(·)=j=--1+j=0k-2

then H4 can be estimated by

H41+H42,

where

H41:={k=02kαq(j=--1μ(fj)χkp(·))q}1/q

and

H42:={k=02kαq(j=0k-2μ(fj)χkp(·))q}1/q

Let estimate H41. We have in this case j < 0 ≤ k. By Lemma 2.1, we obtain

μ(fj)χkp(·)c2k(np-n+ns+τ)2j(n-ns-τ-np(0))fjp(·)

therefore, H41 is bounded by

csupj02α(0)fjp(·){k=02k(α+np-n+ns+τ)q(j=--12j(n-ns-τ-np(0)))q}1/q,

by embedding q(0) and since α+np-n+ns+τ<0<n-ns-τ-np(0), we have

H41c(j=--12α(0)q(0)jfjp(·)q(0))1/q(0)cfK˙p(·)α(·),q(·)(n).

We can estimate H42 by the same argument used in the estimation of H1 if we replace α(0), p(0) and q(0) by α, p and q respectively, then H42 is bounded by

{k=0(j=0k-22(j-k)(n-np-τ-ns-α)(2jαfjp(·)))q}1/qc(k=02jαqfjp(·)q)1/qcfK˙p(·)α(·),q(·)(n)

Let us estimate H2+H5. By the (Lp(·) (ℝn), Lp(·) (ℝn))-boundedness of μ, we have

H2+H5(k=--12kα(0)fχkp(·)q(0))1/q(0)+(k=02kαfχkp(·)q)1/qfK˙p(·)α(·),q(·)(n).

Let us estimate H3. It is possible to prove the following estimation (similar to the estimate for I1 and I2)

μ(fj)c2-njfjp(·)Ω(x-·)χjp(·),

for the detailed proof of this estimation, see [14, p.259–260]. By Hölder inequality and Lemma 2.6, the right-hand of (3.4) is bounded by

c2-jτ(RjysτΩ(x-y)sdy)1sχjθ(·)2-nj2k(τ+ns)χjp(·)ΩLs(Sn-1),

which gives

μ(fj)χkp(·)2-nj2(k-j)(τ+ns)fjp(·)χjp(·)χkp(·).

We split

j=k+2μ(fj)χkp(·)=j=k+2-1+j=0

Then H3 is bounded by

{k=--12kα(0)q(0)(j=k+2-1μ(fj)χkp(·))q(0)}1/q(0)+{k=--12kα(0)q(0)(j=0μ(fj)χkp(·))q(0)}1/q(0)=:H31+H32

For H31, since j and k are negative integers, we have

H31{k=--1(j=k+2-12(α(0)+np(0)+τ+ns)(k-j)2jα(0)fjp(·))q(0)}1/q(0)

Since α(0)+np(0)+τ+ns>0, by Lemma 2.2, we obtain

H31(k=--12kα(0)q(0)fχkp(·)q(0))1/q(0)fK˙p(·)α(·),q(·)(n).

For H32, since k < 0 ≤ j, then we have

μ(fj)χkp(·)c2(k-j)(τ+ns)fjp(·)2-njp2nkp(0)c2k(τ+ns+np(0))2jαfjp(·)2-j(np+α+τ+ns),

by Hölder inequality in 1 and since γ=np+α+τ+ns>0, we have

H32{k=--12k(τ+ns+np(0))(j=02jαfjp(·)2-jγ)q(0)}1/q(0)(k=--12kηq(0))1/q(0)(j=02-jγq)1/q(j=02jαqfjp(·)q)1/q(j=02jαqfjp(·)q)1/qfK˙p(·)α(·),q(·)(n),

where η=τ+ns+np(0)>0.

Let us estimate H6. In this case, since k and j are non-negative integers, by (3.3) and Lemma 2.1, we have

μ(fj)χkp(·)c2(-n+np+τ+ns)(k-j)fjp(·),

which gives

H6{k=0(j=k+22(k-j)(n-γ)q(2jαqfjp(·)q))}1/q,

since nγ > 0, by Lemma 2.2, we obtain

H6{j=02jαqfjp(·)q}1/qcfK˙p(·)α(·),q(·)(n).

Remark 3.3

A non-homogeneous counterpart of Theorem 3.1 is available. Since Kp(·)α(·),q(·)(n)=Kp(·)α,q(n), their proof is an immediate consequence of [14, Theorem 2.1].

To prove the Theorem 3.6, we need the notation of atomic decomposition.

Definition 3.4

Let αL(ℝn), p ∈ ℘(ℝn), q ∈ ℘0(ℝn) and m ∈ ℕ0. A function a is said to be a central (α(·), p(·))-atom, if

  • suppaB(0,r)¯={xn:xr}, r > 0.

  • ap(·)B(0,r)¯-α(0)/n, 0 < r < 1.

  • ap(·)B(0,r)¯-α/n, r ≥ 1.

  • nxβa(x)dx = 0, |β| ≤ m.

A function a on ℝn is said to be a central (α(·), p(·))-atom of restricted type, if it satisfies the conditions (iii), (vi) above and suppaB(0, r), r ≥ 1.

If r = 2k for some k ∈ ℤ in Definition 3.4, then the corresponding central (α(·), p(·))-atom is called a dyadic central (α(·), p(·))-atom.

The following theorem presents the atomic decomposition characterization of variable Herz-type Hardy spaces, see [6].

Theorem 3.5

Let α and q are be log-Hölder continuous, both at the origin and at infinity and p ∈ ℘log(ℝn) with 1 < pp+<. For any fHK˙p(·)α(·),q(·)(n), we have

f=k=-λkak,

where the series converges in the sense of distributions, λk ≥ 0, each ak is a central (α(·), p(·))-atom with suppaBk and

(k=--1λkq(0))1/q(0)+(k=0λkq)1/qcfHK˙p(·)α(·),q(·).

Conversely, if α(·)n(1-1p-)and m[α++n(1p--1)], and if holds, then fHK˙p(·)α(·),q(·)(n), and

fHK˙p(·)α(·),q(·)inf{(k=--1λkq(0))1/q(0)+(k=0λkq)1/q},

where the infimum is taken over all the decompositions of f as above.

In the next result we treat the boundedness of Marcinkiewicz integral operators with homogeneous kernel on variable Herz-type Hardy spaces.

Theorem 3.6

Suppose that p1, p2 ∈ ℘log(ℝn) with p1+<2nand 1p1(·)-1p2(·)=12n, αL(ℝn), q1, q2 ∈ ℘0(ℝn),Ω ∈ Ls(Sn−1) with s>(p1)-. If α, q1and q2are log-Hölder continuous, both at the origin and at infinity such that

α(·)n(1-1p1-),q1(0)q2(0)   and   (q1)(q2).

Then μ is bounded from HK˙p1(·)α(·),q1(·)(n)to K˙p2(·)α(·),q2(·)(n).

Proof

We must show that

μ(f)K˙p2(·)α(·),q2(·)(n)cfHK˙p1(·)α(·),q1(·)(n)

for all fHK˙p1(·)α(·),q1(·)(n). Using Theorem 3.5, we may assume that

f=i=-+λiai

where λi ≥ 0 and ai’s are (α (·), p1 (·))-atom with suppaiBi. Using Proposition 1.5, we have

μ(f)K˙p2(·)α(·),q2(·)(n){k=--12kα(0)q2(0)μ(f)χkp2(·)q2(0)}1/q2(0)+{k=--12kα(q2)μ(f)χkp2(·)(q2)}1/(q2){k=--12kα(0)q2(0)(i=-k-3λiμ(ai)χkp2(·))q2(0)}1/q2(0)+{k=--12kα(0)q2(0)(i=k-2λiμ(ai)χkp2(·))q2(0)}1/q2(0)+{k=0+2kα(q2)(i=-k-3λiμ(ai)χkp2(·))(q2)}1/(q2)+{k=0+2kα(q2)(i=k-2+λiμ(ai)χkp2(·))(q2)}1/(q2)=:F1+F2+F3+F4.

Let us estimate F1. We consider

μ(ai)(x)(0x|x-ytΩ(x-y)x-yn-1ai(y)dy|2dtt3)12+(x|x-ytΩ(x-y)x-yn-1ai(y)dy|2dtt3)12:=Q1+Q2.

Observe that in this case xRk, yBi and ik − 3. So we know that |xy| ≈ |x| ≈ 2k. By (3.2), we have

Q1BiΩ(x-y)x-yn-1y12x-y32ai(y)dy,

by the m-order vanishing moments of ai with m[α+-n(1-1p1-)], we can subtract the Taylor expansion of x-y12-n at x, we obtain

Q1Biym+1xn+m+12ai(y)Ω(x-y)dyc2-k(n+m+12)+i(m+1)Biai(y)Ω(x-y)dy,

The estimation of Q2 is the same as before since we never use |xy| ≈ |x|, we have

μ(ai)(x)c2-k(n+m+12)+i(m+1)Biai(y)Ω(x-y)dy,

as the same reason in the proof of (3.3), we get

μ(ai)χkp2(·)2-(n-12)k2β(i-k)aip1(·)χip1(·)χkp2(·)ΩLs(Sn-1)2-(n-12)k2β(i-k)aip1(·)χip1(·)χkp2(·),

where β=(1+m-ns-τ).

On the other hand (see [7, p.350] for σ=12), we have

BkχBk(y)x-yn-12dyBkdyx-yn-12χBk(x)c2k2χBk(x).

(3.5), (3.6) and Lemma 2.5, gives

μ(ai)χkp2(·)2-(n-12)k2β(i-k)aip1(·)χBip1(·)χkp2(·)2-nk2β(i-k)aip1(·)χBip1(·)BkχBk(y)dy·-yn-12p2(·)2-nk2β(i-k)aip1(·)χBip1(·)χBkp1(·),

by Lemma 2.1, we have

F1={k=--12kα(0)q2(0)(i=-k-3λiμ(ai)χkp2(·))q2(0)}1/q2(0)c{k=--1(i=-k-3λi2(i-k)(β-(α+n/p1)(0)))q2(0)}1/q2(0),

since we can choose m large enough such that β-α++n(1-1p1-)>0, by Lemma 2.2, we obtain

F1c(k=--1λkq2(0))1/q2(0)c(k=--1λkq1(0))1/q1(0)cfHK˙p1(·)α(·),q1(·).

Let us estimate F2. By Lemma 2.4 and applying the size condition of ai (conditions (ii) and (iii) in Definition 3.4), we have

F2={k=--12kα(0)q2(0)(i=k-2+λiμ(ai)χkp2(·))q2(0)}1/q2(0)c{k=--12kα(0)q2(0)(i=k-2+λiaip2(·))q2(0)}1/q2(0)c{k=--12kα(0)q2(0)(i=k-2-1λiaip2(·))q2(0)}1/q2(0)+c{k=--12kα(0)q2(0)(i=0+λiaip2(·))q2(0)}1/q2(0)c{k=--1(i=k-2-1λi2(k-i)α(0))q2(0)}1/q2(0)+c{k=--1(i=0+λi2(k-i)α-+k(α(0)-α-)+i(α--α))q2(0)}1/q2(0)

for k < 0 ≤ i and since α ≤ min(α (0), α), we have

k(α(0)-α-)+i(α--α)0.

By Lemma 2.2, we obtain

F2c(k=--1λkq2(0))1/q2(0)c(k=--1λkq1(0))1/q1(0)cfHK˙p1(·)α(·),q1(·).

We can estimate F3 and F4 by the same arguments used in the estimation of F1 and F2 if we replace α(0), p2(0) and q2(0) by α, (p2) and (q2) respectively.

A combination of estimations of F1, F2, F3 and F4 completes the proof of Theorem 3.6.

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