# Strongly Singular Convolution Operators on Herz-Type Hardy Spaces with Variable Exponent

Authors: Hongbin Wang; Dunyan Yan
Journal: Journal of Function Spaces
 (2019)

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## Abstract

We investigate the boundedness of the strongly singular convolution operators on Herz-type Hardy spaces with variable exponent.


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## Body

## 1. Introduction

The theory of function spaces with variable exponents has been extensively studied by researchers since the work of Kovácˇik and Rákosník [1] appeared in 1991. In [2, 3] the authors defined the Herz-type Hardy spaces with variable exponent and gave some characterizations for them. In [4–7], the authors proved the boundedness of some integral operators on variable function spaces.Given an open setE⊂Rn and a measurable function p(·):E→[1,∞), Lp(·)(E) denotes the set of measurable functions f defined on E such that(1)∫Efxλpxdx<∞holds for some λ>0.The setLp(·)(E) is a Banach function space when it is equipped with the Luxemburg-Nakano norm as follows:(2)fLp·E=infλ>0:∫Efxλpxdx≤1. The space is regarded as the variable Lp space, since it generalized the standard Lp space: if p(x)=p is constant, then Lp(·)(E) is isometrically isomorphic to Lp(E).The spaceLlocp(·)(E) is defined by (3)Llocp·E≔f:f∈Lp·FforallcompactsubsetsF⊂E. Define P0(E) to be the set of p(·):E→(0,∞) such that(4)p-=essinfpx:x∈E>0,p+=esssuppx:x∈E<∞.Define P(E) to be the set of p(·):E→[1,∞) such that(5)p-=essinfpx:x∈E>1and(6)p+=esssuppx:x∈E<∞.Denote p′(x)=p(x)/(p(x)-1).Letf∈Lloc1(Rn). The Hardy-Littlewood maximal operator is defined by(7)Mfx=supr>01Brx∫Brxfydy,where Br(x)=y∈Rn:x-y<r. Let B(Rn) be the set of p(·)∈P(Rn) such that the Hardy-Littlewood maximal operator M is bounded on Lp(·)(Rn).Lemma 1 (see [8]).
Ifp(·)∈P(Rn) and satisfies(8)px-py≤C-log⁡x-y,x-y≤12and (9)px-py≤Clog⁡x+e,y≥x,then (·)∈B(Rn), that is, the Hardy-Littlewood maximal operator M is bounded on Lp(·)(Rn).In addition, we denote the Lebesgue measure and the characteristic function of a measurable setA⊂Rn by A and χA, respectively. The notation f≈g means that there exist two constants C1,C2>0 such that C1g≤f≤C2g.Next we recall the definition of the Herz spaces with variable exponent. LetBk=x∈Rn:x≤2k and Ak=Bk∖Bk-1 for k∈Z. Denote Z+ and N as the sets of all positive and nonnegative integers, respectively, χk=χAk for k∈Z, χ~k=χk if k∈Z+, and χ~0=χB0.Definition 2 (see [9]).
Letα∈R,0<p≤∞, and q(·)∈P(Rn). The homogeneous Herz space with variable exponent K˙q(·)α,p(Rn) is defined by(10)K˙q·α,pRn=f∈Llocq·Rn∖0:fK˙q·α,pRn<∞,where(11)fK˙q·α,pRn=∑k=-∞∞2kαpfχkLq·Rnp1/p.The nonhomogeneous Herz space with variable exponent Kq(·)α,p(Rn) is defined by(12)Kq·α,pRn=f∈Llocq·Rn:fKq·α,pRn<∞, where(13)fKq·α,pRn=∑k=0∞2kαpfχ~kLq·Rnp1/p.
In [2], the authors gave the definition of the Herz-type Hardy space with variable exponent HK˙q(·)α,p(Rn) and the atomic decomposition characterizations. S(Rn) denotes the space of Schwartz functions, and S′(Rn) denotes the dual space of S(Rn). Let GN(f) be the grand maximal function of f defined by(14)GNfx=supϕ∈ANϕ∇∗fx,where(15)AN=ϕ∈SRn:supα,β≤NxαDβϕx≤1and N>n+1; ϕ∇∗ is the nontangential maximal operator defined by(16)ϕ∇∗fx=supy-x<tϕt∗fywith ϕt(x)=t-nϕ(x/t).Definition 3 (see [2]).
Letα∈R,0<p<∞,q(·)∈P(Rn), and N>n+1.
(i) The homogeneous Herz-type Hardy space with variable exponentHK˙q(·)α,p(Rn) is defined by(17)HK˙q·α,pRn=f∈S′Rn:GNfx∈K˙q·α,pRnand(18)fHK˙q·α,pRn=GNfK˙q·α,pRn.
(ii) The nonhomogeneous Herz-type Hardy space with variable exponentHKq(·)α,p(Rn) is defined by(19)HKq·α,pRn=f∈S′Rn:GNfx∈Kq·α,pRnand(20)fHKq·α,pRn=GNfKq·α,pRn.Forx∈R, we denote by x the largest integer less than or equal to x. δ2 is the same as in Lemma 9.Definition 4 (see [2]).
Letnδ2≤α<∞,q(·)∈P(Rn), and nonnegative integer s≥[α-nδ2].(i)
A functiona on Rn is said to be a central (α,q(·))-atom, if it satisfies
(1)
suppa⊂B(0,r)={x∈Rn:x<r}(2)
aLq(·)(Rn)≤B0,r-α/n(3)
∫Rna(x)xβdx=0,β≤s(ii)
A functiona on Rn is said to be a central (α,q(·))-atom of restricted type, if it satisfies conditions (2), (3) and
(1′)
suppa⊂B(0,r),r≥1Ifr=2k for some k∈Z in Definition 4, then the corresponding central (α,q(·))-atom is called a dyadic central (α,q(·))-atom.Lemma 5 (see [2]).
Letnδ2≤α<∞,0<p<∞ and q(·)∈B(Rn). Then f∈HK˙q(·)α,p(Rn) (or HKq(·)α,p(Rn)) if and only if(21)f=∑k=-∞∞λkakor∑k=0∞λkak,inthesenseofS′Rn,where each ak is a central (α,q(·))-atom (or central (α,q(·))-atom of restricted type) with support contained in Bk and ∑k=-∞∞λkp<∞ (or ∑k=0∞λkp<∞). Moreover,(22)fHK˙q·α,pRn≈inf∑k=-∞∞λkp1/porfHKq·α,pRn≈inf∑k=0∞λkp1/p,where the infimum is taken over all above decomposition of f.
Letθ be a smooth radial cut-off function such that θ(ξ)=1 if ξ≥1 and θ(ξ)=0 if ξ≤1/2. Define the multipliers(23)Tb:Tbf^ξ=θξeiξbξnb/2f^ξ,where 0<b<1. The kernel for Tb is very singular. Roughly speaking, it looks like(24)Kb′x=eix-b′xn,where b′=b/(1-b). Indeed the cancellation is minimal and if one makes a quick computation for x≥2y, we have(25)Kb′x-y-Kb′x≤Cyxn+b′+1.
The study of these operators in the context ofLq spaces was carried out by Hirschman [10] and Wainger [11]. Sharp endpoint estimates were obtained by Fefferman and Stein in [12] via the duality of H1 and BMO. Weighted Lq norm and weak(1,1) estimates were established by Chanillo in [13]. The boundedness of these operators on the weighted Herz-type Hardy spaces was proved by Xiaochun Li and Shanzhen Lu in [14].
Motivated by [2, 14], we will study the boundedness of the strongly singular convolution operators Tb on Herz-type Hardy spaces with variable exponent. The main results are as follows.Theorem 6.
Suppose that0<p≤1,q(·)∈P(Rn) satisfies conditions (8) and (9) in Lemma 1 and α=nδ2. Then we have (26)TbfK˙q·α,pRn≤CfHK˙q·α,pRn, where C is independent of f.Theorem 7.
Suppose that0<p≤1,q(·)∈P(Rn) satisfies conditions (8) and (9) in Lemma 1 and nδ2≤α≤nδ2+1. Then we have(27)TbfHKq·α,pRn≤CfHKq·α,pRn,where C is independent of f.


## 2. Preliminary Lemmas

Referring to the variableLp(·) space, there are some important lemmas as follows.Lemma 8 (see [1]).
Letp(·)∈P(Rn). If f∈Lp(·)(Rn) and g∈Lp′(·)(Rn), then fg is integrable on Rn and(28)∫Rnfxgxdx≤rpfLp·RngLp′·Rn,where (29)rp=1+1p--1p+.
The above inequality is named generalized Hölder’s inequality with respect to the variableLp space.Lemma 9 (see [9]).
Letq(·)∈B(Rn). Then there exists a positive constant C such that, for all balls B in Rn and all measurable subsets S⊂B,(30)χBLq·RnχSLq·Rn≤CBS,χSLq·RnχBLq·Rn≤CSBδ1,and(31)χSLq′·RnχBLq′·Rn≤CSBδ2hold, where δ1 and δ2 are constants with 0<δ1,δ2<1.Throughout this paperδ2 is the same as in Lemma 9.Lemma 10 (see [9]).
Supposeq(·)∈B(Rn). Then there exists a constant C>0 such that, for all balls B in Rn,(32)1BχBLq·RnχBLq′·Rn≤C.Lemma 11 (see [15]).
Define a variable exponentq~(·) by 1/p(x)=1/q~(x)+1/q for x∈Rn. Then we have(33)fgLp·Rn≤CfLq~·RngLqRn,for all measurable functions f and g.Lemma 12 (see [16]).
Letp(·)∈P(Rn) satisfy conditions (8) and (9) in Lemma 1. Then(34)χQLp·Rn≈Q1/pxifQ≤2nandx∈Q,Q1/p∞ifQ≥1for every cube (or ball) Q⊂Rn, where p(∞)=limx→∞⁡p(x).
A nonnegative locally integrable functionω(x) on Rn is said to belong to Ap(1<p<∞), if there is a constant C>0 such that (35)supQ1Q∫Qωxdx1Q∫Qωx1-p′dxp-1≤C<∞,where p′=p/(p-1); Q denotes a cube in Rn with its sides parallel to the coordinate axes.
The weighted(Lp,Lp) boundedness of Tb has been proved by Chanillo [13].Lemma 13 (see [13]).
Letω∈Ap,1<p<∞. Then(36)∫RnTbfxpωxdx≤C∫Rnfxpωxdx.Lemma 14 (see [5]).
Given a familyF and an open set E⊂Rn, assume that for some p0,0<p0<∞ and for every ω∈A∞,(37)∫Efxp0ωxdx≤C0∫Egxp0ωxdx,f,g∈F.Given p(·)∈P0(E) such that p(·) satisfies (8) and (9) in Lemma 1, then for all (f,g)∈F such that f∈Lp(·)(E)(38)fLp·E≤CgLp·E.Since Ap⊂A∞, by Lemmas 13 and 14 it is easy to get the (Lp(·)(Rn),Lp(·)(Rn))-boundedness of the strongly singular convolution operators Tb.To prove our main results, we also need the following lemmas.Lemma 15 (see [11]).
The kernel for the multiplier operatorTb(f)(x) is given by(39)Ceiαbx-b′xnχx≤1+hx,b′=b1-b,with hx≤C(1+x)-n+1+Cx-n+εχ(x≤1),ε>0. Here αb=bb/(1-b)-b1/(1-b) and ε depend only on b.Lemma 16 (see [13]).
LetK~b′,s(x)=eiαbx-b′/xn(b′+2)/s and (b′+2)/s<1. Then(40)K~b′,s∗fs≤Cfs′,1s+1s′=1.


## 3. The Proof of Main Results

Firstly we give the proof of Theorem6.Proof of Theorem6.
Letf∈HK˙q(·)α,p(Rn). By Lemma 5, we have(41)fx=∑j=-∞∞λjaj,where(42)fHK˙q·α,pRn≈inf∑j=-∞∞λjp1/p,the infimum is taken over the above decomposition of f, and aj is a dyadic central α,q·-atom with the support Bj. Then we have(43)TbfK˙q·α,pRnp=∑k=-∞∞2kαpTbfχkLq·Rnp≤C∑k=-∞∞2kαp∑j=-∞k-2λjTbajχkLq·Rnp+C∑k=-∞∞2kαp∑j=k-1∞λjTbajχkLq·Rnp≕CI1+CI2.
We first estimateI2; by 0<p≤1 and the (Lq(·)(Rn),Lq(·)(Rn))-boundedness of Tb we have(44)I2≤C∑k=-∞∞2kαp∑j=k-1∞λjajLq·Rnp≤C∑k=-∞∞∑j=k-1∞λj2k-jαp≤C∑j=-∞∞λjp∑k=-∞j+12k-jαp≤C∑j=-∞∞λjp≤CfHK˙q·α,pRn.
Now we estimateI1. Let(45)Kb′x=Ciαbx-b′xnχx≤1.By Lemma 15 and the Minkowski inequality, we have (46)I1≤∑k=-∞∞2kαp∑j=-∞k-2λjKb′∗ajχkLq·Rnp+∑k=-∞∞2kαp∑j=-∞k-2λjh∗ajχkLq·Rnp≕I11+I12.
To estimate the termI12, we need the pointwise estimate for h∗aj(x).
Letx≥2j. By generalized Hölder’s inequality we have (47)h∗ajx≤∫t≤rhx-tajtdt≤C∫t≤rajt11+x-tn+1+χx-t≤1x-tn-εdt≤C∫t≤rajtdt11+xn+1+χx≤2xn-ε≤CajLq·RnχjLq′·Rn11+xn+1+χx≤2xn-ε.Therefore, by nδ2=α, 0<p≤1, Lemmas 9 and 10, the Minkowski inequality, and generalized Hölder’s inequality we have(48)
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What remains is estimatingI11. Let 2j0-1<2j(1-b)≤2j0 for some j0∈Z, where b is the same as the above. Then it follows that(49)I11=∑k=-∞j02kαp∑j=-∞k-2λjKb′∗ajχkLq·Rnp+∑k=j0+1∞2kαp∑j=-∞k-2λjKb′∗ajχkLq·Rnp≕I13+I14.
To estimate the termI14, we need the pointwise estimate for Kb′∗aj(x). Let x≥2j. Then, by the vanishing moment condition on aj(x), we have(50)Kb′∗ajx≤∫BjKb′x-y-Kb′xajydy.From the condition of Kb′x, Kb′x-y-Kb′x≤Cy/xn+b′+1, if x≥2y, it follows that(51)Kb′∗ajx≤C2jxn+b′+1∫Bjajydy≤C2jxn+b′+1ajLq·RnχBjLq′·Rn.Note that b′=b/(1-b); that is, (1-b)(b′+1)=1. Since nδ2=α, 0<p≤1, and 2j0-1<2j(1-b)≤2j0, then by Lemmas 9 and 10 we have(52)I14≤C∑k=j0+1∞2kαp∑j=-∞k-2λj2j2kn+b′+1ajLq·RnχBjLq′·RnχkLq·Rnp≤C∑k=j0+1∞∑j=-∞k-2λj2kα-jα2j2kn2kn+b′+1χBjLq′·RnχBkLq′·Rnp≤C∑j=-∞∞λjp∑k=j0+1∞2jp2kb′+1p≤C∑j=-∞∞λjp2jp2j0b′+1p≤C∑j=-∞∞λjp2jp2j1-bb′+1p≤C∑j=-∞∞λjp.Now to estimate I13, we split Kb′∗aj(x) as follows: (53)Kb′∗ajx=C∫Bjeiαbx-y-b′x-ynb′+2/s1x-yn1-b′+2/s-1xn1-b′+2/sajydy+CK~b′,s∗ajx1xn1-b′+2/s≕Ex+Fx,where K~b′,s is the same as in Lemma 16 and let s>max{q+,2} satisfy (b′+2)/s<1.
Applying the mean value theorem to the term brackets in the integrand ofE(x), then for x≥2j we have the pointwise estimate for E(x) as follows:(54)Ex≤C∫Bjyxn+1ajydy≤C2jxn+1ajLq·RnχBjLq′·Rn.On the other hand, since 0<p≤1, by the Minkowski inequality we get(55)I13≤∑k=-∞j02kαp∑j=-∞k-2λjpEχkLq·Rnp+∑k=-∞j02kαp∑j=-∞k-2λjpFχkLq·Rnp≕I15+I16.For I15, using nδ2=α, the pointwise estimate for E(x), and Lemmas 9 and 10 we have(56)I15≤C∑k=-∞j02kαp∑j=-∞k-2λjp2jp2kn+1pajLq·RnpχBjLq′·RnpχkLq·Rnp≤C∑j=-∞∞λjp∑k=j+2j02jp2kn+1p2k-jαp2knχBjLq′·RnχBkLq′·Rnp≤C∑j=-∞∞λjp.Finally, we estimate I16. Noting that x∈Ak, we get (57)I16=∑k=-∞j02kαp∑j=-∞k-2λjpFχkLq·Rnp≤C∑k=-∞j02kαp∑j=-∞k-2λjpK~b′,s∗aj·1·n1-b′+2/sχk·Lq·Rnp≤C∑k=-∞j02kαp∑j=-∞k-2λjp12kn1-b′+2/spK~b′,s∗aj·χk·Lq·Rnp.Noting s>max{q+,2},s′<min{q-,2},1/s+1/s′=1, we denote q~(·)>s/s-2 and 1/q(x)=1/q~(x)+1/s.
WhenBk≤2n and xk∈Bk, by Lemma 12 we have(58)χBkLq~·Rn≈Bk1/q~xk≈χBkLq·RnBk-1/s.When Bk≥1 we have(59)χBkLq~·Rn≈Bk1/q~∞≈χBkLq·RnBk-1/s.So we obtain(60)χBkLq~·Rn≈χBkLq·RnBk-1/s.
In similar method we can obtain(61)χBjLq·/q·-s′Rn≈Bjq·-s′/q·and(62)χBjLq′·Rn≈Bj1/q′·.
Thus by Lemmas11, 12, and 16 we have (63)K~b′,s∗aj·χk·Lq·Rn≤CK~b′,s∗aj·χk·LsRnχkLq~·Rn≤CajLs′RnχBkLq~·Rn≤CajLs′RnχBkLq·RnBk-1/s≤Cajs′Lq·/s′Rn1/s′χBjLq·/q·-s′Rn1/s′χBkLq·RnBk-1/s≤CajLq·RnBjq·-s′/q·1/s′χBkLq·RnBk-1/s=CajLq·RnBj1-1/q·Bj-1/sχBkLq·RnBk-1/s≤CajLq·RnχBjLq′·RnBj-1/sχBkLq·RnBk-1/s.So by using 0<p≤1, nδ2=α, 2j0-1<2j(1-b)≤2j0, (1-b)(b′+1)=1, and Lemmas 9 and 10 we have (64)I16≤C∑k=-∞j02kαp∑j=-∞k-2λjp12kn1-b′+2/sp×ajLq·RnχBjLq′·RnBj-1/sχBkLq·RnBk-1/sp≤C∑k=-∞j0∑j=-∞k-2λjp2k-jαp2knpb′+1/sχBjLq′·RnχBkLq′·RnpBj-p/s≤C∑j=-∞∞λjp∑k=j+2j02knpb′+1/sBj-p/s≤C∑j=-∞∞λjp2j0npb′+1/s2jnp/s≤C∑j=-∞∞λjp2jnp1-bb′+1/s2jnp/s≤C∑j=-∞∞λjp.
Therefore, by (43), (44), (46), (48), (49), (52), (55), (56), and (64) we complete the proof of Theorem 6.Similar to the method of Theorem6, next we give the proof of Theorem 7.Proof of Theorem7.
Letf∈HKq(·)α,p(Rn). By Lemma 5, we have(65)fx=∑j=0∞λjaj,where(66)fHKq·α,pRn≈inf∑j=0∞λjp1/p,the infimum is taken over the above decomposition of f, and aj is a dyadic central α,q·-atom of restricted type with the support Bj. Then we have(67)TbfHKq·α,pRnp=GNTbfKq·α,pRnp=∑k=0∞2kαpGNTbfχkLq·Rnp≤C∑k=0∞2kαp∑j=0k-1λjGNTbajχkLq·Rnp+C∑k=0∞2kαp∑j=k∞λjGNTbajχkLq·Rnp≕CJ1+CJ2.
We first estimateJ2; by 0<p≤1 and the (Lq(·)(Rn),Lq(·)(Rn))-boundedness of M and Tb we have(68)J2=∑k=0∞2kαp∑j=k∞λjGNTbajχkLq·Rnp≤C∑k=0∞2kαp∑j=k∞λjMTbajLq·Rnp≤C∑k=0∞2kαp∑j=k∞λjTbajLq·Rnp≤C∑k=0∞2kαp∑j=k∞λjajLq·Rnp≤C∑j=0∞λjp∑k=0j2k-jαp≤CfHKq·α,pRn.
To estimateJ1, we need the pointwise estimate for GN(Tbaj)(x).
Suppose thaty,t satisfy x-y<t. Let x>2j+2 and ϕ∈Km, where m∈N and Km={ϕ∈S(Rn):supu∈Rn,α≤m(1+u)m+nDαϕu≤1}. By the vanishing moment condition on aj(x), it is easy to prove that ∫RnTbaj(x)dx=0. So we have(69)Tbaj∗ϕty=∫Rnt-nTbajzϕy-zt-ϕytdz≤∫z≤2j+1t-nTbajzϕy-zt-ϕytdz+∫z>2j+1t-nTbajzϕy-zt-ϕytdz≕J11+J12.
ForJ11, by Lemma 9, the generalized Hölder inequality, and the mean value theorem, we obtain(70)J11≤CTbajLq·Rnt-nϕy-·t-ϕytχBj+1Lq′·Rn≤CajLq·Rnt-n∑β=1Dβϕy-θ·t·tχBjLq′·Rn≤CajLq·Rn·χBjx-y+y-θ·n+1Lq′·Rn≤C2-jα1xn+1·χBjLq′·Rn≤C2-jα+j1xn+1χBjLq′·Rn,where 0≤θ≤1.
ForJ12, by Lemma 15 we have(71)J12≤t-n∫z>2j+1Kb′∗ajzϕy-zt-ϕytdz+t-n∫z>2j+1h∗ajzϕy-zt-ϕytdz≕J13+J14.Noting that 2j≥1, then z>2j+1≥2. Since z-w≥z-w>2j≥1 for w≤2j, we obtain(72)Kb′∗ajz=∫Bjeiαbz-w-b′z-wnχz-w≤1ajwdw=0.So we have J13=0.
ForJ14, by the pointwise estimate for h∗aj(z) in the proof of Theorem 6, we obtain (73)J14≤CajLq·RnχjLq′·Rnt-n×∫z>2j+1ϕy-zt-ϕyt11+zn+1+χz≤2zn-εdz≤CajLq·RnχjLq′·Rnt-n∫x/2>z>2j+1ϕy-zt-ϕyt11+zn+1dz+CajLq·RnχjLq′·Rnt-n∫z≥x/2ϕy-zt-ϕyt11+zn+1dz≕J15+J16.Using the mean value theorem, we get (74)J15≤CajLq·RnχjLq′·Rn∫x/2>z>2j+1∑β=1Dβϕy-θztztt-n1+zn+1dz≤CajLq·RnχjLq′·Rn∫x/2>z>2j+1zx-y+y-θzn+111+zn+1dz≤CajLq·RnχjLq′·Rn1xn+1∫x/2>z>2j+1z1+zn+1dz≤CajLq·RnχjLq′·Rnln⁡xxn+1,where 0≤θ≤1.
ForJ16, noting that ϕ∈Km, we get (75)J16≤CajLq·RnχjLq′·Rnt-n∫z≥x/2ϕy-zt+ϕyt11+zn+1dz≤CajLq·RnχjLq′·Rn1xn+1+t-n1+y/tn∫z≥x/211+zn+1dz≤CajLq·RnχjLq′·Rn1xn+1+1xx-y+yn≤CajLq·RnχjLq′·Rn1xn+1.
Thus, forx>2j+2, we get(76)GNTbajx≤CajLq·RnχBjLq′·Rn2j+lnx1xn+1.So by using 0<p≤1, α≤1+nδ2, and Lemmas 9 and 10 we have (77)J1=∑k=0∞2kαp∑j=0k-1λjGNTbajχkLq·Rnp≤C∑k=0∞∑j=0k-1λj2kαajLq·RnχBjLq′·Rn2j12kn+1χBkLq·Rnp+C∑k=0∞∑j=0k-1λj2kαajLq·RnχBjLq′·Rnln⁡2k12kn+1χBkLq·Rnp≤C∑k=0∞∑j=0k-1λj2j-k1+nδ2-αp+C∑k=0∞∑j=0k-1λj2jnδ2-αk2k1+nδ2-αp≤C∑j=0∞λjp∑k=j+1∞2j-k1+nδ2-αp+C∑j=0∞λjp2jnδ2-αp∑k=j+1∞kp2k1+nδ2-αp≤C∑j=0∞λjp+C∑j=0∞λjp2jnδ2-αpjp2j1+nδ2-αp≤C∑j=0∞λjp≤CfHKq·α,pRn.
Therefore, by (67), (68), and (77) we complete the proof of Theorem 7.



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*Source: 6821464-2019-08-08.xml*