# Weighted Estimates for Rough Maximal Functions with Applications to Angular Integrability

Authors: Feng Liu; Fangfang Xu
Journal: Journal of Function Spaces
 (2019)

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## Abstract

In this note we establish certain weighted estimates for a class of maximal functions with rough kernels along “polynomial curves” onRn. As applications, we obtain the bounds of the above operators on the mixed radial-angular spaces, on the vector-valued mixed radial-angular spaces, and on the vector-valued function spaces. Particularly, the above bounds are independent of the coefficients of the polynomials in the definition of the operators.


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## Body

## 1. Introduction

During the last several years, a considerable amount of attention has been given to the investigation of the boundedness for various kinds of integral operators on the Lebesgue spaces and other more general function spaces (see [1–8], for examples). The primary aim of this article is to establish the boundedness for maximal operators related to singular integrals on the mixed radial-angular spaces.LetRn(n≥2) be the n-dimensional Euclidean space and Sn-1 denote the unit sphere in Rn equipped with the induced Lebesgue measure dσ. Assume that Ω∈L1(Sn-1) is a homogeneous function of degree zero and satisfies(1)∫Sn-1Ωudσu=0.Let PN(t) be a real polynomial on R of degree N satisfying PN(0)=0 and h be a suitable function defined on R; we define the singular integral operator Th,Ω,PN along the “polynomial curve” PN on Rn by(2)Th,Ω,PNfx=p.v.∫Rnfx-PNyy′hyΩy′yndy,where y′=y/y for y≠0. Let R+≔(0,∞) and Hγ(R+)(γ>0) denote the set of all measurable functions h:R+→R satisfying(3)hHγR+≔∫0∞htγdtt1/γ<∞.Let Th,Ω,PN be defined as in (2) and let U¯(R+) denote the closed unit ball in H2(R+). Define the maximal function SΩ,PN by(4)SΩ,PNfx=suph∈U¯R+⁡Th,Ω,PNfx.IfPN(t)=t and h(t)≡1, we denote Th,Ω,PN=TΩ and SΩ,PN=SΩ.As a formal extension of the Lebesgue spaceLp, the mixed radial-angular space L|x|pLθp~(Rn) has already been successfully used in studying Strichartz estimates and dispersive equations (see [9–16]) over the last several years. Recall that the mixed radial-angular spaces L|x|pLθp~(Rn), 1≤p,p~≤∞, consist of all functions u satisfying uLxpLθp~(Rn)<∞, where (5)uLxpLθp~Rn≔∫0∞uρ·Lp~Sn-1pρn-1dρ1/pand (6)uLx∞Lθp~Rn≔supρ>0⁡uρ·Lp~Sn-1.It is clear that the spaces L|x|pLθp~(Rn) have the following easy properties.(i) If1≤p≤∞ and p~=p, then(7)uLxpLθp~Rn=uLpRn.(ii) Ifu is a radial function on Rn and 1≤p≤∞, 1≤p~≤∞, then (8)uLxpLθp~Rn≃uLpRn.(iii) If1≤p≤∞ and 1≤p~1≤p~2≤∞, then (9)uLxpLθp~1Rn≤Cn,p,p~1,p~2uLxpLθp~2Rn.Here the notation A≃B means that there are two positive constants C,C′ such that A≤CB and B≤C′A. Throughout this paper, we use Cα,β,… to denote positive constants that depend on parameters α,β,….Recently, the mixed radial-angular spaceL|x|pLθp~(Rn) is also playing active roles in the theory of singular integral operator (see [17–19]). In [18], among other things, Córdoba proved that TΩ is bounded on L|x|pLθp~(Rn) for all 1<p<∞ and p~=2, provided that Ω∈C1(Sn-1). Later on, D’Ancona and Lucà [19] extended the above index p~=2 to the range 1<p~<∞ by applying the same argument in [18, Theorem 2.1] (see also [17, Theorem 1.1] for the weighted case). Recently, Liu and Fan [20] improved the above result to the case Ω∈Lq(Sn-1) and extended the above results to the singular integral operators along polynomial curves. Precisely, let Δγ(R+)(γ>0) be the set of all measurable functions h defined on R+ satisfying (10)hΔγR+≔supR>0⁡R-1∫0Rhtγdt1/γ<∞.Liu and Fan [20] proved the following result. Our main results can be formulated as follows.Theorem A (see [20]).
LetPN(t) be a real polynomial on R of degree N and satisfy PN(0)=0. Suppose that Ω∈Lq(Sn-1) satisfies (1) and h∈Δγ(R+) for some 1<q,γ≤∞. Then for 1<p,p~<∞, the following inequalities hold: (11)Th,Ω,PNfLxpLθp~Rn≤Ch,Ω,q,γ,p,p~,NfLxpLθp~Rn;∑j∈ZTh,Ω,PNfjp~1/p~LxpLθp~Rn≤Ch,Ω,q,γ,p,p~,N∑j∈Zfjp~1/p~LxpLθp~Rn;∑j∈ZTh,Ω,PNfjp~1/p~LpRn≤Ch,Ω,q,γ,p,p~,N∑j∈Zfjp~1/p~LpRn.Here the above constants Ch,Ω,q,γ,p,p~,N>0 are independent of the coefficients of PN.On the other hand, the classical maximal operatorSΩ was originally introduced by Chen and Lin [21] who proved that if Ω∈C1(Sn-1), then SΩ is of type (p,p) for any p>2n/2n-1 and the range of p is best possible. Subsequently, the Lp mapping properties of SΩ have been discussed extensively by many authors (see [22–26], for example). Particularly, Al-Salman [23] proved the following result.Theorem B (see [23]).
LetΩ satisfy (1) and Ω∈L(log⁡L)1/2(Sn-1). Then(12)SΩ,PNfLpRn≤CpfLpRnfor all 2≤p<∞. Here Cp>0 is independent of the coefficients of PN.It is well known that the following are valid.(13)H∞R+=Δ∞R+=L∞R+;Δγ1R+⊊Δγ2R+,∀1<γ2<γ1<∞;HγR+⊊ΔγR+,∀1<γ<∞;(14)C1Sn-1⊊⋃1<q≤∞LqSn-1⊊Llog⁡L1/2Sn-1.It follows from (14) and Theorem B that the condition Ω∈Lq(Sn-1) implies the Lp-boundedness of SΩ,PN for 2≤p<∞. A question that arises naturally is the following.Question A.Is the operator SΩ,PN bounded on L|x|pLθp~(Rn) for 1<p,p~<∞(p≠p~) if Ω∈Lq(Sn-1) for some q>1?In this paper we will give an affirmative answer to Question A. In order to obtain theL|x|pLθp~(Rn) boundedness for SΩ,PN, we shall establish the following weighted estimates for SΩ,PN.Theorem 1.
LetPN(t) be a real polynomial on R of degree N and satisfy PN(0)=0. Suppose that Ω satisfies (1) and Ω∈Lq(Sn-1) for some 1<q≤∞. Then, for any nonnegative measurable function u on Rn, it holds that (15)SΩ,PNfLpu≤CΩ,q,p,s,NfLpΛN,su,provided that one of the following conditions holds:(i)
1<p<2 and s>2/p;(ii)
2≤p<∞ and s>1.
HereΛN,su=∑i=0NMsN+1-iMi,sσ~Msu and Mλ,sσ~u=(Mλσ~(us))1/s, Msku=(Mkus)1/s for any k∈N. Mk is the Hardy-Littlewood maximal operator M iterated k times for all k=1,2,⋯. Specially, Mk=M when k=1. Mλσ~ is a maximal operator defined as in Section 2.As applications of Theorem1, we can obtain the following result.Theorem 2.
LetSΩ,PN be the maximal operator defined by (4). Let PN(t) be a real polynomial on R of degree N and satisfy PN(0)=0. Suppose that Ω satisfies (1) and Ω∈Lq(Sn-1) for some 1<q≤∞. Then, for 1<p~<2 and p~<p<2p~/2-p~ or 2≤p~≤p<∞, the following inequalities hold:(16)SΩ,PNfLxpLθp~Rn≤CΩ,q,p,p~,NfLxpLθp~Rn;(17)∑j∈ZSΩ,PNfjp~1/p~LxpLθp~Rn≤CΩ,q,p,p~,N∑j∈Zfjp~1/p~LxpLθp~Rn;(18)∑j∈ZSΩ,PNfjp~1/p~LpRn≤CΩ,q,p,p~,N∑j∈Zfjp~1/p~LpRn.Here the constants CΩ,q,p,p~,N>0 are independent of the coefficients of PN.Remark 3.
We remark that our main results are new even in the special casePN(t)=t.The rest of this paper is organized as follows. In Section2 we shall present some notations and auxiliary lemmas. The proofs of Theorems 1 and 2 will be given in Section 3. It should be pointed out that the main idea in the proof of Theorem 1 is a combination of ideas and arguments from [18–20, 27–29]. The proof of Theorem 2 is based on Theorem 1 and a criterion established in Section 2 (see Proposition 7). Throughout this note, for any p∈(1,∞), we let p′ denote the dual exponent to p defined as 1/p+1/p′=1. We also use the convention ∏j∈∅aj=1. In what follows, for any function f, we define f~ by f~(x)=f(-x). For f∈Lp(w), we set(19)fLpw=∫Rnfxpwxdx1/p.


## 2. Preliminary Lemmas

In this section we shall give some notations and necessary lemmas, which will play key roles in the proof of our main results. In what follows, we assume thatPN(t)=∑i=1Naiti with ai≠0. Let Pλ(t)=∑i=1λaiti for λ=1,2,…,N and P0(t)=0. For λ∈{1,2,…,N}, we define a family of measures {σλ,r}r>0 and the related maximal operator Mλσ by (20)σλ,r^x=∫Sn-1e-2πiPλrx·y′Ωy′dσy′,Mλσfx=supk∈Z⁡∫2k2k+1σλ,r∗fxdrr,where |σλ,r| is defined in the same way as σλ,r, but with Ω replaced by Ω. We also define the maximal operator Mλσ~ by (21)Mλσ~fx=supk∈Z⁡∫2k2k+1σ~λ,r∗fxdrr,where σ~λ,r is defined in the following way (22)∫Rnfxdσ~λ,rx=∫Rnf-xdσλ,rx.Lemma 4 ([30, p.186, corollary]).
LetΦ(t)=tα1+μ2tα2+⋯+μntαn and Ψ∈C1[a,b], where μ2,…,μn are real parameters and α1,…,αn are distinct positive (not necessarily integer) exponents. Then(23)∫abe-iλΦtΨtdt≤Cλ-ϵsupt∈a,b⁡Ψt+∫abΨ′tdt,with ϵ=min{1/α1,1/n} and C does not depend on μ2,…,μn as long as 0≤a<b≤1.Lemma 5.
LetΩ satisfy (1) and Ω∈Lq(Sn-1) for some q>1. Then for λ∈1,2,…,N and k∈Z, the following estimates hold:(24)supr>0⁡max⁡σλ,r,σλ,r^ξ,σλ,r^ξ≤ΩL1Sn-1;(25)∫2k2k+1σλ,r^ξ2drr≤CΩ,qmin⁡1,2kλaλξ-1/2q′λ;(26)∫2k2k+1σλ,r^ξ2drr≤CΩ,qmin⁡1,2kλaλξ-1/2q′λ;(27)∫2k2k+1σλ,r^ξ-σλ-1,r^ξ2drr≤CΩ,qmin⁡1,2k+1λaλξ2;(28)∫2k2k+1σλ,r^ξ-σλ-1,r^ξ2drr≤CΩ,qmin⁡1,2k+1λaλξ2.Proof.
Estimate (24) is trivial. By Lemma 4, Hölder’s inequality, and the changes of variable, we have (29)∫2k2k+1σλ,r^ξ2drr=∫2k2k+1∬Sn-12ΩuΩy′¯e-2πiPλrξ·u-y′dσudσy′dtt≤∬Sn-12ΩuΩy′¯∫2k2k+1e-2πiPλrξ·u-y′dttdσudσy′≤∬Sn-12ΩuΩy′¯∫12e-2πiPλ2krξ·u-y′dttdσudσy′≤∬Sn-12ΩuΩy′¯min⁡1,2kλaλξ·u-y′-1/λdσudσy′≤CΩLqSn-122kλaλξ-1/ϵ,for any ϵ∈(0,min{1/q′,1/λ}). This together with (24) yields (25). Similarly, we can prove (26). It is clear that (30)max⁡σλ,r^ξ-σλ-1,r^ξ,σλ,r^ξ-σλ-1,r^ξ≤∫Sn-1Ωy′e-2πiPλrξ·y′-e-2πiPλ-1rξ·y′dσy′≤2πΩL1Sn-1rλaλξ≤CΩ,qrλaλξ,which together with (24) and the fact that σ0,r(ξ)=0 yields (27)-(28) and completes the proof.Lemma 6.
LetΩ satisfy (1) and Ω∈Lq(Sn-1) for some q>1. Then for λ∈{0,1,…,N}, we have(31)MλσLpRn≤CΩ,q,pfLpRnfor all 1<p<∞.Proof.
We shall prove (31) by induction on λ. It is easy to see that(32)M0σfx≤ΩL1Sn-1fx.This yields (31) for λ=0. Suppose that (31) holds for λ=l-1 with l∈{1,2,…,N}. We shall prove (31) for λ=l. Let ϕ be a nonnegative Schwartz function supported in {|t|≤1} satisfying ϕ(t)=1 when |t|<1/2. Define the measures {τk}k∈Z by (33)τk^ξ=∫2k2k+1σl,r^ξdrr-∫2k2k+1σl-1,r^ξdrrϕ2klalξ.By Lemma 5 and the definition of τk, we have(34)τk^ξ≤CΩ,qmin⁡1,2kλaλξ,2kλaλξ-1/2q′λ;(35)MlσfLpRn≤GfLpRn+CMl-1σfLpRn,1<p<∞;(36)τ∗fLpRn≤GfLpRn+CMl-1σfLpRn,1<p<∞,where(37)Gfx=∑k∈Zτk∗fx21/2andτ∗fx=supk∈Z⁡τk∗fx.By (34) and Plancherel’s theorem, we get(38)GfL2Rn≤CΩ,qfL2Rn.From (36)-(38) and our assumption, we have(39)τ∗fL2Rn≤CΩ,qfL2Rn.By the lemma on page 544 of [27] (q=2, p0=4), we have(40)GfLpRn≤CΩ,q,pfLpRn,43<p<4.From our assumption, (35) and (40), we get(41)τ∗fLpRn≤CΩ,q,pfLpRn,43<p<4.By the lemma on page 544 of [27] (q=4/3, p0=8), we have(42)GfLpRn≤CΩ,q,pfLpRn,87<p<8.By using this argument repeatedly, one can obtain ultimately(43)GfLpRn≤CΩ,q,pfLpRn,1<p<∞.Combining (43) with (35) and assumption yields that(44)MlσfLpRn≤CΩ,q,pfLpRn,1<p<∞.This completes the proof of Lemma 6.To prove Theorem1, we need the following proposition, which is of interest in its own right.Proposition 7.
Let1<p<∞ and s0≥1. Suppose that T is a linear or sublinear operator such that(45)TfLpu≤Cp,s,s0fLpGsu,∀s>s0for any nonnegative measurable function u on Rn, where Gs is a bounded operator from Lr(Rn) to itself for all r∈(s,∞). Then for any q∈(p,ps0/s0-1), the following inequalities hold:(46)TfLxqLθpRn≤Cp,qfLxqLθpRn;(47)∑j∈ZTfjp1/pLxqLθpRn≤Cp,q∑j∈Zfjp1/pLxqLθpRn;(48)∑j∈ZTfjp1/pLqRn≤Cp,q∑j∈Zfjp1/pLqRn.Proof.
We only prove (46) since (47) and (48) can be proved similarly. The argument of the proof for (46) is similar to those of the proof in [19, Theorem 2.6] essentially. Fix q∈(p,ps0/s0-1) and write r=q/q-p. We can choose a number s such that s0<s<r. Let X denote the set of all g∈S(R) with ∫0∞gr(ρ)ρn-1dρ≤1. By changes of variables, one has(49)TfLxqLθpRnp=∫0∞∫Sn-1Tfρθpdσθq/pρn-1dρp/q=supg∈X⁡∫0∞∫Sn-1Tfρθpgρρn-1dσθdρ=supg∈X⁡∫RnTfxpgxdx.Fix g∈X. Let I(g)≔∫Rn|Tf(x)|pg(|x|)dx and h(x)=g(|x|). By (45), Hölder’s inequality, and changes of variables, (50)Ig≤Cp,s,s0∫RnfxpGshxdx≤Cp,s,s0∫0∞∫Sn-1fρθpdσθGsgρρn-1dρ≤Cp,s,s0∫0∞∫Sn-1fρθpdσθq/pρn-1dρp/q×∫0∞Gsgρrρn-1dρ1/r≤Cp,qfLxqLθpRnpGshLrRn≤Cp,qfLxqLθpRnp,which together with (49) leads to (46).


## 3. Proofs of Theorems1 and 2

We begin with the proof of Theorem1.Proof of Theorem1.
By the duality we can write(51)SΩ,PNfx=∫0∞σN,r∗fx2drr1/2.
Letϕ be a nonnegative Schwartz function supported in {|t|≤1} satisfying ϕ(t)=1 when |t|<1/2. For 1≤λ≤N, we define the Borel measures {μλ,r}r>0 on Rn by (52)μλ,r^ξ=σλ,r^ξ∏j=λ+1Nϕrjajξ-σλ-1,r^ξ∏j=λNϕrjajξ,∀ξ∈Rn.One can verify that(53)σN,r=∑λ=1Nμλ,r;(54)∫2k2k+1μλ,r^ξ2drr≤CΩ,qmin⁡1,2kλaλξ,2kλaλξ-11/2q′λ;(55)Mλμfx≤MN-λMλσfx+MN-λ+1Mλ-1σfx.Equation (51) together with (53) and Minkowski’s inequality yields that(56)SΩ,PNfx≤∑λ=1N∫0∞μλ,r∗fx2drr1/2≕∑λ=1NTλfx.It follows from [31] that u≤Msu and Msu∈A1 for s>1. From (55) we have(57)∑λ=1NMsMλ,sμ~Msu≤∑λ=1NMsN+1-λMλ,sσ~Msu+MsN+2-λMλ-1,sσ~Msu≤ΛN,su.By (56) and (57), to prove Theorem 1, it suffices to show that(58)TλfLpu≤CΩ,q,p,λ,sfLpMsMλ,sμ~uholds for all 1≤λ≤N and any nonnegative measurable function u on Rn, provided that one of the following conditions holds:(i)
1<p<2 and s>2/p;(ii)
2≤p<∞ and s>1.
We now prove (58) for the case 1<p<2 and s>2/p. For 1≤λ≤N, let Ψλ(t)∈Cc∞((1/4,1)) such that 0≤Ψλ≤1 and ∑k∈Z(Ψλ(2kλ|aλξ|))3=1. Define the Fourier multiplier operators {Γk,λ}k∈Z by Γk,λf(x)=Θk,λ∗f(x), where Θk,λ^(ξ)=Ψλ(2kλ|aλξ|). It was shown in [28] that(59)∑k∈ZΓk,λf21/2Lpw≤Cp,w,λfLpwand(60)∑k∈ZΓk,λfkLpw≤Cp,w,λ∑k∈Zfk21/2Lpwfor all 1<p<∞ and w∈Ap.
By the changes of variables and Minkowski’s inequality, we can write(61)Tλfx=∑k∈Z∫2k2k+1μλ,r∗fx2drr1/2=∑k∈Z∫12μλ,2kr∗fx2drr1/2=∑k∈Z∫12∑j∈ZΓj+k,λ3μλ,2kr∗fx2drr1/2≤∑j∈Z∑k∈Z∫12Γj+k,λ3μλ,2kr∗fx2drr1/2≕∑j∈ZTλ,jfx.Hence, by (61), to prove (58) for the case 1<p<2 and s>2/p, it suffices to show that for any 1<p<2 and s>2/p, there exists α(p,q,γ)>0 independent of j such that(62)Tλ,jfLpu≤CΩ,q,p,λ2-αp,q,γjfLpMsMλ,sμ~u.
We now prove (62). Fix a nonnegative measurable function u on Rn. By (54) and Plancherel’s theorem, we have(63)∫Rn∫12μλ,2kr∗Γj+k,λwx2drrdx≤CΩ,q2-j/2q′γ′∫Rnwx2dxfor arbitrary function w on Rn. One can easily check that(64)∫Rn∫12μλ,2kr∗Γj+k,λwx2drrusxdx≤supr>0⁡μλ,rΘj+k,λL1Rn∫12∫Rnμλ,2kr∗Θj+k,λ∗w2xusxdxdrr≤CΩ,q∫Rnwx2MMλμ~usxdxfor any s>1. By (63)-(64) and the interpolation of L2-spaces with change of measure ([32, Theorem 5.4.1]), we obtain(65)∫Rn∫12μλ,2kr∗Γj+k,λwx2drruxdx≤CΩ,q,s2-1-1/s/2q′γ′j∫Rnwx2MsMλ,sμ~uxdxfor any s>1. By (65) with w=Γj+k,λf, (60), and the well-known property of the Rademacher’s function, we obtain (66)Tλ,jfL2u2=∫Rn∑k∈Z∫12Γj+k,λ3μλ,2kr∗fx2drruxdx=∑k∈Z∫12∫RnΓj+k,λ3μλ,2kr∗fx2uxdxdrr≤Cλ∑k∈Z∫12∫Rnμλ,2kr∗Γj+k,λ2fx2uxdxdrr≤CΩ,q,λ,s2-1-1/s/2q′γ′j∫Rn∑k∈ZΓj+k,λfx2MsMλ,sμ~uxdx≤CΩ,q,λ,s2-1-1/s/2q′γ′jfL2MsMλ,sμ~u2.It follows that(67)Tλ,jfL2u≤CΩ,q,λ,s2-1-1/s/4q′γ′jfL2MsMλ,sμ~ufor any s>1.
On the other hand, fixk∈Z, and it is easy to see that(68)supr>0⁡μλ,2kr∗fL∞Rn≤CΩ,qfL∞Rn;(69)∫Rn∫12μλ,2kr∗fxdrruxdx≤fL1Mλμ~u.The interpolation between (68) and (69) implies that(70)∫Rn∫12μλ,2kr∗fkxpdrruxdx≤CΩ,q,p∫RnfkxpMλμ~uxdxfor all 1<p<2. It follows from (70) that(71)∫Rn∑k∈Z∫12μλ,2kr∗fkxpdrruxdx≤CΩ,q,p∫Rn∑k∈ZfkxpMλμ~uxdxfor all 1<p<2. On the other hand, we have(72)∫Rnsupk∈Zsupr∈1,2μλ,2kr∗fkxpdx≤CΩ,q,p∫Rnsupk∈Zfkxpdxfor all 1<p<2. By interpolating between (71) and (72),(73)∫Rn∑k∈Z∫12μλ,2kr∗fkx2drrp/2u1/t1xdx≤CΩ,q,p,t1∫Rn∑k∈Zfkx2p/2Mλμ~u1/t1xdxfor all 1<p<2, where t1=2/p. Substituting ut1 for u in (73), we obtain(74)∫Rn∑k∈Z∫12μλ,2kr∗fkx2drrp/2uxdx≤CΩ,q,p,t1∫Rn∑k∈Zfkx2p/2Mλ,t1μ~uxdx.Since Mλ,t1μ~u∈A1, by the weighted Littlewood-Paley theory and (74),(75)Tλ,jfLpu=∑k∈Z∫12μλ,2kr∗Γj+k,λ3f2drr1/2Lpu≤CΩ,q,p,t1∑k∈ZΓj+k,λ3f21/2LpMλ,t1μ~u≤CΩ,q,p,λ,t1fLpMλ,t1μ~ufor all 1<p<2. Letting s=t1, (67) gives us that(76)Tλ,jfL2u≤CΩ,q,λ,t12-1-1/t1/4q′γ′jfL2Mt1Mλ,t1μ~u.By an interpolation between (75) and (76), one has(77)Tλ,jfLpu≤CΩ,q,p,λ2-αp,q,γjfLpMt1Mλ,t1μ~ufor all 1<p<2, where α(p,q,γ)>0. Note that Mt1Mλ,t1μ~u≤CMsMλ,sμ~u for s>t1 by Hölder’s inequality. This together with (61) yields (58).
It remains to show (58) for the case 2≤p<∞ and s>1. We want to show that(78)Tλ,jfLpu≤CΩ,q,p,λ,sfLpMsMλ,sμ~ufor all 2<p<∞. Actually, by (67), (78), and an interpolation (see [32, Corollary 5.5.4]), one has(79)Tλ,jfLpu≤CΩ,q,p,λ,s2-βp,q,γ,sjfLpMsMλ,sμ~ufor 2≤p<∞ and s>1. Here β(p,q,γ,s)>0 depends only on p,q,γ, and s. Combining (79) with (61) yields (58) for the case 2≤p<∞ and s>1.
Next we shall prove (78). Fix 2<p<∞. By duality we can choose a function v∈L(p/2)′(u) with unit norm such that (80)∑k∈Z∫12μλ,2kr∗gk2drr1/2Lpu2=∫Rn∑k∈Z∫12μλ,2kr∗gkx2drr·vxuxdx.This together with the fact that μk,tλ≤CΩ,q shows that(81)∑k∈Z∫12μλ,2kr∗gk2drr1/2Lpu2≤CΩ,q∫Rn∑k∈Zgkx2∫12μ~λ,2kr∗vuxdrrdx.Fix s>1 and let t=ps/2. By Hölder’s inequality,(82)μ~λ,2kr∗vu≤μ~λ,2kr∗us1/tμ~λ,2kr∗ut′/p/2′vt′1/t′.By Hölder’s inequality with exponents p/2 and (p/2)′ again and (81)-(82),(83)∑k∈Z∫12μλ,2kr∗gk2dtt1/2Lpu2≤Ch,Ω,q,γ∫Rn∑k∈Zgkx2Mλμ~us1/tMλμ~ut′/p/2′vt′1/t′xdx≤Ch,Ω,q,γ∑k∈Zgk21/2LpMλ,sμ~u2Mλμ~ut′/p/2′vt′Lp/2′/t′Rn1/t′.On the other hand, by Lemma 6 and (55), we have(84)Mλμ~fLrRn≤Ch,Ω,q,γ,rfLrRn,∀1<r<∞.Since p/2=t/s<t, then (p/2)′>t′. (84) leads to (85)Mλμ~ut′/p/2′vt′Lp/2′/t′Rn1/t′≤Ch,Ω,q,γ,p,sut′/p/2′vt′Lp/2′/t′Rn1/t′≤CΩ,q,p,s.This together with (84) yields that(86)∑k∈Z∫12μλ,2kr∗gk2drr1/2Lpu≤CΩ,q,p,s∑k∈Zgk21/2LpMλ,sμ~ufor all 2<p<∞ and any s>1.
It follows from (59), (60), (86), and the fact that Mλ,sμ~u≤MsMλ,sμ~u that (87)Tλ,jfLpu=∑k∈Z∫12μλ,2kr∗Γj+k,λ3f2drr1/2Lpu≤CΩ,q,p,λ,s∑k∈ZΓj+k,λ3f21/2LpMλ,sμ~u≤CΩ,q,p,λ,sfLpMsMλ,sμ~ufor all 2<p<∞ and any s>1. This gives (78) and completes the proof of Theorem 1.Proof of Theorem2.
By Lemma6, one has(88)ΛN,suLrRn≤Ch,Ω,q,γuLrRnfor any r>s. By (88), Theorem 1, and Proposition 7, we obtain (16)-(18) for the case 1<p~<2 and p~<p<2p~/2-p~ or 2≤p~<p<∞. It was known that MΩ,PN is bounded on Lp(Rn) for 2≤p<∞. This together with (7) yields (16)-(18) for the case of 2≤p=p~<∞.



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*Source: 4976351-2019-07-16.xml*