# A

Authors: Ruimin Wu; Songbai Wang
Journal: Journal of Function Spaces
 (2018)

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## Abstract

We introduce new classes of weights andBMO functions associated with a nondecreasing function ϕ of upper type β with β>0 and obtain the weighted norm inequalities for Calderón-Zygmund operators of ϕ-type and their commutators.


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## Body

## 1. Introduction

For a nonnegative and nondecreasing functionϕ mapping from [0,∞) to [1,∞), we shall mean that it is of upper type β with β>0, if there exists a constant c0 such that(1)ϕθt≤c0θβϕt, for all θ≥1 and t≥0. In this paper, we always assume that ϕ(1)>1.LetT be a continuous associated with the kernel function K(x,y) from S(Rn) to S(Rn) that satisfies (2)Tfx=∫RnKx,yfydy for all f∈C0∞ and x not in the support of f is said to be associated with K. Here, the kernel K(x,y) is defined on Rn×Rn\{x,x:x∈Rn} and satisfies that for any N≥0 and some A>0 the size condition(3)Kx,y≤Ax-yn1ϕx-ynN,and for some δ>0 the regularity conditions(4)Kx,y-Kx′,y≤Ax-x′δx-y+x′-yn+δ1ϕx-y+x′-ynN,whenever x-x′≤1/2max⁡x-y,x′-y and(5)Kx,y-Kx,y′≤Ax-x′δx-y+x-y′n+δ1ϕx-y+x-y′nN,whenever y-y′≤(1/2)max⁡(x-y,x-y′). We also suppose that T is associated with K and it can be extended to be a bounded extension on L2(Rn),(6)TfL2≤BfL2.In this paper, we will consider weighted estimates for these operators and their commutators.Remark 1.
It is clear that ifT satisfies (3), (4), (5), and (6) then T falls within the scope of the Calderón-Zygmund theory. Then T has an extension that maps L1(Rn) into L1(Rn), and by interpolation, T also maps Lp(Rn) into itself for 1<p<∞.Remark 2.
Whenϕ(t)=(1+t), from [1–3], we know that the pseudodifferential operators Tσ with the symbol σ∈S1,00 defined by (7)Tσfx=∫Rnσx,ξe2πix·ξf^ξdξ satisfy (3), (4), (5), and (6), where S1,δm denotes the set of all smooth functions σ(x,ξ) on Rn×Rn satisfying (8)DxαDξβσx,ξ≤Cα,β1+ξm-β+δα for all multi-indices α and β.Definition 3.
A weight will always mean a positive function which is locally integrable. We say that a weightω belongs to the class Apθ(ϕ) for θ≥0 and 1<p<∞, if there is a positive constant C such that, for all cubes Q.(9)1Q∫Qωxdx1/p1Q∫Qωx1-p′dx1/p′≤CϕQθ. We also say that a nonnegative function satisfies the A1θ(ϕ) condition if there exists a constant C>0 for all cubes Q,(10)1Q∫Qωxdx≤CϕQθinfx∈Q⁡ωx. We also write A∞θ(ϕ)=∪p≥1Apθ(ϕ), Ap∞(ϕ)=∪θ≥0Apθ(ϕ), and A∞∞(ϕ)=∪θ≥0A∞θ(ϕ)=∪p≥1Ap∞(ϕ).Remark 4.
We remark thatAp0(ϕ) coincides with Muckenhoupt’s class of weights Ap for all 1≤p<∞. However, in general, the class Ap∞(ϕ) is strictly larger than the class Ap for all 1≤p<∞. On the other hand, when ϕ is a constant function, Apθ(ϕ) also coincides with Ap for any θ∈(0,∞). In particular, if ϕ≡1, A∞∞(ϕ)=A∞.Remark 5.
It should be noted thatω(x)dx may not be a doubling measure; that is, for each positive constant C, there exists at least a cube Q such that (11)ω2Q≥CωQ. For example, let ϕ=1+|Q|. If 0≤γ≤nθ, then ω=(1+|x|)-(n+γ)∉A∞ and ω(x)dx is a nondoubling measure, but ω=(1+x)-n+γ∈A1θϕ; see [1, 4]. We also need to point out that these weights have many differences from Orobitg-Pérez’s undoubling weights [5] because they are against the reverse Hölder inequality by Proposition 15 (v).Next we state our main theorem for these operators satisfying (3)–(6) as follows.Theorem 6.
Assume thatT satisfies (3), (4), (5), and (6). Let ω∈Ap∞(ϕ) with 1≤p<∞. Then T is bounded from Lp(ω) to Lp(ω) for 1<p<∞ and L1(ω) to L1,∞(ω).We also consider the commutator of Coifman-Rochberg-Weiss[b,T] defined by (12)b,Tfx=bxTfx-Tbfx. Since our operator T is stronger than the standard Carlderón-Zygmund operator, we will generalize the symbol b from usual BMO to BMOθ(ϕ).Definition 7.
A locally integrable functionf is said to be in BMOpθ(ϕ) with p≥1 and θ≥0, if there exists a positive constant C such that for any cube Q(13)1Q∫Qfx-fQpdx1/p≤CϕQθ,where fQ is the average of f on Q. A norm for b∈BMOθ(ϕ), denoted by fBMOθ(ϕ), is given by the infimum of the constants satisfying (13).Remark 8.
ClearlyBMOθ1(ϕ)⊂BMOθ2(ϕ) for θ1≤θ2 and BMOθ(ϕ)=BMO if θ=0 or ϕ≡constant. We define BMO∞(ϕ)=∪θ≥0BMOθ(ϕ). In [6], Morvidone proved that these spaces are independent of the scale p; that is, BMO1θ(ϕ)=NMOpθ(ϕ) in the sense of norm. So we denote BMOθ(ϕ) simply.Remark 9.
Recently Harboure et al. [7] established an intimate relationship between these spaces and ϕ-Carleson measures similar to well-known Fefferman-Stein’s theorem. Precisely, let Ψ be a function in S with a null integral. f∈BMO1(ϕ) if and only if dμ=Ψt∗f2(x)dx(dt/t) is a ϕ-Carleson measure; that is, dμ(Q^)≤|Q|ϕ2(|Q|), where Q^ is a Carleson box.Theorem 10.
Assume thatT satisfies (3), (4), (5), and (6). Let b∈BMOθ(ϕ)  for θ>0 and ω∈Ap∞(ϕ) with 1≤p<∞. Then there exists a positive constant C such that for any 1<p<∞(14)b,TfLpω≤CfLpω.To consider the endpoint case, we introduce the Orlicz norm. ForΦ(t)=t(1+log+t) and a cube Q on Rn, by the Luxemburg norm of f it means that (15)fΦ,Q=inf⁡λ>0:1Q∫QΦfxλdx≤1. We recall that fΦ,Q>1 if and only if 1/Q∫QΦfxdx>1.Theorem 11.
Assume thatT satisfies (3), (4), (5), and (6). Let b∈BMOθ(ϕ)  for θ≥0 and ω∈A1∞(ϕ) with 1≤p<∞. Then there exists a positive constant C such that for any 1<p<∞(16)ωx∈Rn:b,Tf>λ≤C∫RnΦfxλωxdx.Remark 12.
One of our original motivations is to procure some detailed operators that satisfy our conditions like pseudodifferential operators. Restricted with our knowledge, we have found nothing to date.


## 2. Some Preliminaries and Notations

In this section, we begin with defining the weighted maximal function by(17)Mωfx=supQ∋x⁡1ω5Q∫Qfyωydy.Proposition 13.
Letω∈A∞∞(ϕ). Then(18)ωx∈Rn:Mωfx>λ≤CλfL1ω,∀λ>0,f∈L1ω,and for 1<p<∞,(19)MωfLpω≤CpfLpω.Proof.
By the Marcinkiewicz interpolation theorem and the fact thatMϕ,ω is bounded on L∞(ω), we only need to prove (18). For any λ>0, let Eλ=x∈Rn:Mωf(x)>λ. Fixing x∈Eλ, then there exists a cube Qx∋x such that (20)1ω5Qx∫Qxfyωydy>λ. Thus, {Qx}x∈Eλ covers Eλ. By Vitali’s lemma, there exists a class of disjoint cubes {Qxj} such that Qxj⊂Eλ⊂5Qxj and (21)ωEλ≤∑jω5Qxj≤Cλ∑j∫Qxjfyωydy≤CλfL1ω.By the boundedness ofMω and the following covering lemma, we obtain some properties for Ap(ϕ).Lemma 14 (see [8]).
There exists a sequence of pointsxj,j≥1 in Rn so that the family of cubes Qjj where Qj≔B(xj,1), j≥1, satisfies the following: (a)
∪jQj=Rn.(b)
There exists a constantC such that for any σ>1,∑jχσQj≤Cσn.Proposition 15.
The following statements hold:(i)
If1≤p1<p2<∞, then Ap1θ(ϕ)⊂Ap2θ(ϕ).(ii)
ω∈Apθ(ϕ) if and only if ω1-p′∈Ap′θ(ϕ), where (1/p)+(1/p′)=1.(iii)
Ifω1,ω2∈Apθ(ϕ), p≥1, then ω1αω21-α∈Apθ(ϕ) for any 0<α<1.(iv)
Ifω∈Apθ(ϕ) for 1≤p<∞, then (22)1ϕQθQ∫Qfydy≤Cθ1ω5Q∫5Qfypωydy1/p. Particularly, let f=χE  for any measurable set E⊂Q,(23)EϕQθQ≤CθωEω2Q1/p.(v)
Ifω∈Apθ(ϕ) with p≥1, then there exist positive numbers δ, η, and C so that for all cubes Q(24)1Q∫Qωx1+δdx1/1+δ≤C1Q∫QωxdxϕQη.(vi)
Ifω∈Ap∞(ϕ) with p>1 then there exists ϵ>0 such that ω∈Ap-ϵ∞(ϕ).Proof.
(i), (ii), and (iii) are easy from the definition. We only prove (iv). In fact, note thatϕ5Q≤5nβϕQ; by Hölder’s inequality and Apθ(ϕ) condition we then have (25)1ϕQθQ∫Qfydy≤1ϕQθQ∫Qfypωydy1/p1ϕQθQ∫Qωy1-p′dy1/p′≤C1ϕQθQ∫Qfypωydy1/p1ϕ5Qθ5Q∫5Qωydy-1/p≤C1ω5Q∫5Qfypωydy1/p, for all p>1. For p=1, we note that (26)supy∈2Q⁡ωy-1≤1ϕ5Qθ5Q∫5Qωydy-1; thus it concludes (iv).
For (v), by Lemma  2.4 [1], there exist positive constants C>0 and δ>0, such that(27)1Q∫Qωx1+δdx1/1+δ≤C1Q∫Qωxdxfor any Q=Q(x0,r)⊂Rn with r<1. On the other hand, let Q=Q(x,r), with r>1 and F=j:Qj∩B≠∅, where Qj=Q(xj,1) and xjj is the sequence of Lemma 14. It is easy to see that ∪jQj⊂5Q. Now let δ be the constant in (27). By using Lemma 14, Apθ(ϕ) condition, and Hölder’s inequality, (28)∫Qωx1+δdx1/1+δ≤∑j∈F∫Qjωx1+δdx1/1+δ≤C∑j∈FωQjQj-δ/1+δ≤Cω5Q≤CQpϕ5Qp∫Qωx1-p′dx-1/p′≤CωQϕ5Qp≤CωQϕQp-δ/1+δ1/θβQδ/1+δϕ1, we obtain the desired inequality with η=p-δ/1+δθβ; here β is sufficiently big. And (vi) is a consequence of (v).Then we also define maximal functions associated withϕ and η>0 by (29)Mϕ,ηfx=supQ∋x⁡1ϕQηQ∫Qfydy. From (iv) in Proposition 15, we have that (30)Mϕ,θfx≤Mωfpx1/p,a.e.x∈Rn, for ω∈Apθ(ϕ). From this and using (18) and (19), we can get the following result.Proposition 16.
Let1≤p1<∞ and suppose that ω∈Ap1θ(ϕ). If p1<p<∞, then the equality (31)∫RnMϕ,θfxpωxdx≤Cp∫Rnfxpωxdx. Furthermore, let 1≤p<∞ and ω∈Apθ(ϕ) if and only if (32)ωx∈Rn:Mϕ,θfx>λ≤Cpλp∫Rnfxpωxdx.Proposition 17.
Let1<p<∞ and p′=p/(p-1) and suppose that ω∈Apθ(ϕ). There exists a constant C>0 such that (33)Mϕ,p′θfLpω≤CfLpω.As a consequence of Proposition17, we have the following.Corollary 18.
Let1≤p<∞ and ω∈Apθ(ϕ). Let ψ be a radial, positive function with compact support and total integral 1. Set ψt(x)=t-nψ(x/t). Then (i)
sup0<t<1⁡f∗ψtx≤CηMϕ,ηfx for f∈Lpω and 0<η<∞;(ii)
f∗ψtx→fx, as t→0, almost every for f∈Lp(ω);(iii)
f∗ψt-fLp(ω)→0, as t→0, almost every for f∈Lp(ω).Next we define the dyadic maximal function and sharp maximal functionMϕ,η∆ and Mϕ,η♯,∆ by for η>0(34)Mϕ,η∆fx=supQ∋xQdyadic⁡1ϕQηQ∫Qfydy;Mϕ,η♯,∆fx=supQ∋x,lQ<1Qdyadic⁡1Q∫Qfy-fQdy+supQ∋x,lQ≥1Qdyadic⁡1ϕQηQ∫Qfydy=supQ∋x,lQ<1Qdyadic⁡infc⁡1Q∫Qfy-cdy+supQ∋x,lQ≥1Qdyadic⁡1ϕQηQ∫Qfydy Then we have the following.Lemma 19.
Letf be a locally integrable function on Rn, η, λ>0, and Ωλ={x∈Rn:Mϕ,ηf(x)>λ}. Then Ωλ may be written as a disjoint union of maximal dyadic cubes {Qj} with (35)λ<1ϕQjηQj∫Qjfxdx<c02n1+βηλ. Moreover, we have that |f(x)|≤λ for almost all x∈Rn\∪jQj and |Ωλ|≤λ-1fL1.Proof.
The proof is standard; see Lemma  1 on P. 150 of [9].Lemma 20.
Letω∈Apθ(ϕ) and 0<η<∞. For a locally integrable function f, and for b and γ positive γ<b<b0=1/ϕ(2n), then there exist positive constants C and δ1 such that we have the following inequality:(36)ωx∈Rn:Mϕ,η∆fx>λ,Mϕ,η♯,∆fx<γλ≤Caδ1ωx∈Rn:Mϕ,η∆fx>bλ,for all λ>0, where a=2nγ/(1-b/b0).Proof.
The proof is very similar to that of Lemma2.6 [1].Corollary 21.
Let1<p<∞, ω∈A∞θ(ϕ), and 0<θ<∞. Then there exists a constant C>0 such that (37)Mϕ,θ∆fLpω≤CMϕ,θ♯,∆fLpω.Note that|f(x)|≤Mϕ,θf(x) a.e. x∈Rn for any θ>0. By Corollary 21, we have the following.Corollary 22.
Let1<p<∞, ω∈Apθ(ϕ), 0<θ<∞, and f∈Lp(ω), and then (38)fLpω≤Mϕ,θ∆fLpω≤CMϕ,θ♯,∆fLpω.Forη>0 and the Young function Φ(t)=t(1+log+t), we define the maximal function MLlog⁡L,ϕ,η by (39)MLlog⁡L,ϕ,ηfx=supQ∋x⁡ϕQ-ηfΦ,Q.Lemma 23.
Let0<η<∞ and Mϕ,η/2f be locally integrable. Then there exist positive constants C1 and C2 independent of f and x such that (40)C2Mϕ,ηMϕ,ηfx≤MLlog⁡L,ϕ,ηfx≤C1Mϕ,η/2Mϕ,η/2fx.Proof.
On the one hand, we takef with fLlog⁡L,Q=1 which implies (41)fLlog⁡L,Q=1≤CQ∫Qfy1+log+⁡fydy≤CQ∫Qfy∫0fy+edttdy≤CQ∫Qfy∫0fy+edttdy≤CQ∫1e∫Qfydydtt+CQ∫e∞∫y∈Q:fy>t-efydydtt≤CQ∫QMfχQxdx≤CϕQη/2Q∫QMϕ,η/2fxdx≤CϕQηMϕ,η/2Mϕ,η/2fx.On the other hand, for any x∈Rn and any fixed cube Q∋x, write f=f1+f2, with f1=fχ3Q. Thus (42)1ϕQθQ∫QMϕ,ηfydy≤1ϕQθQ∫QMϕ,ηf1ydy+1ϕQθQ∫QMϕ,ηf2ydy≔I+II. Since M(g)χQL1≤CQgLlog⁡L,Q  for any g with suppg⊂Q, we have (43)I≤1ϕQθ3Q∫3QMϕ,ηf1ydy≤CMLlog⁡L,ϕ,ηfx. Noting that Mϕ,ηf2y≤CMfz for any two y,z∈Q, then (44)II≤CϕQθQ∫Qinfz∈Q⁡Mfzdy≤CMϕ,ηfx≤CMLlog⁡L,ϕ,ηfx.The following proposition is a technical result. Its proof copies almost verbatim the proof of Proposition  3 [10] and is omitted.Proposition 24.
Letq≥1. If b∈BMOθ(ϕ) then for all cube Q(45)1Q∫Qfx-fQqdx1/q≤CϕQθ,12kQ∫2kQfx-fQqdx1/q≤Ckϕ2kQθ for all k∈N.Proposition 25.
Supposing thatf is in BMOθ(ϕ), there exist positive constants c1 and c2 such that(46)supQ1Q∫Qexpc1fx-fQfBMOθϕϕQθdx≤c2.Proof.
The proof is classical; we refer to Proposition4.3 [11].As we know thatΨ(t)=et-1 is also a Young function, the corresponding average is denoted by (47)fΨ,Q=fexp⁡L,Q. Then there is a generalized Hölder inequality(48)1Q∫Qfxgxdx≤2fexp⁡L,QgLlog⁡L,Q.By Proposition 25 and (48), we can get(49)1Q∫Qfx-fQgxdx≤2fBMOθϕgLlog⁡L,QϕQθ.


## 3. Proofs of Theorems

The proof of Theorem6 needs the following technical lemma.Lemma 26.
Assume thatT satisfies (3), (4), (5), and (6). Let 0<ϵ<1 and η>0. Then there exists a constant C>0 such that for all x∈Rn and all f∈S(50)Mϕ,ϵ,η♯,∆Tfx≤CMϕ,ηfx, where Mϕ,ϵ,η♯,∆ is a variant of dyadic sharp maximal operator (51)Mϕ,ϵ,η♯,∆f=Mϵ,η♯,∆fϵ1/ϵ.Proof.
Fixx∈Rn and let a dyadic cube Q=Q(x0,r)∋x, and we write f(x)=fχ4Q(x)+fχRn\4Q(x)=f1(x)+f2(x). When 0<r≤1, since 0<ϵ<1, we have the inequality (52)1Q∫QTfxϵ-cϵdx1/ϵ≤1Q∫QTf1xϵdx1/ϵ+1Q∫QTf2x-cϵdx1/ϵ≔I+II. Applying Kolmogorov’s inequality to the term I and the weak (1,1) boundedness of T, we have (53)I≤CTf1L1,∞Q,dx/Q≤C14Q∫4Qfxdx≤C1ϕ4Qη4Q∫4Qfxdx≤CMϕ,ηfx. Set c=T(f)(x0), for N=η in the regularity in (4) and (5); we have (54)Tf2x-Tf2x0≤C∫Rn∖4Qx-x0δx-y+x0-yn+δfyϕx-y+x0-ynηdy≤C∑k=1∞4-kδ/n4k+1Q1ϕ4k+1Qη∫4k+1Qfydy≤CMϕ,ηfx.Thus II≤CMϕ,η(f)(x).
Whenr>1, by Hölder’s inequality with 0<ϵ<1, we have (55)1ϕQηQ∫QTfxϵdx1/ϵ≤1ϕQηQ∫QTfxdx≤1ϕQηQ∫QTf1xdx+1Q∫QTf2xdx≔III+IV. Similar to I, it follows that III≤CMϕ,η(f)(x) with Kolmogorov’s inequality to the term I and the weak (1,1) boundedness of T. For IV, taking N=η+1 in the size condition (3), we have (56)IV≤C∫Rn∖4Q1x-y+x0-ynfyϕx-y+x0-ynη+1dy≤C∑k=1∞14k+1Q4-nkϕ4k+1Qη∫4k+1Q∖4kQfydy≤CMϕ,ηfx.
The proof is complete.Proof of Theorem6.
From Proposition17 and Lemmas 20 and 26, the standard statement gets the proof of Theorem 6.To prove Theorem10, we establish the following pointwise estimate firstly.Lemma 27.
Assume thatT satisfies (3), (4), (5), and (6). Let b∈BMOθ(ϕ) for θ>0, 0<ϵ<γ<1, and η>θϵ/1-ϵ. Then for almost all x∈Rn(57)Mϕ,ϵ,η♯,∆b,Tfx≤CbBMOθϕMϕ,γ,η∆Tfx+MLlog⁡L,ϕ,ηfx, where Mϕ,γ,η∆ is a variant of dyadic maximal operator (58)Mϕ,γ,η∆f=Mϕ,η∆fγ1/γ.Proof.
Fixx∈Rn and a dyadic cube Q=Q(x0,r)∋x, we decompose the function f into f=f1+f2 with f1=fχQ~, where Q~=4Q. For b∈BMOθ(ϕ), we can write (59)b,Tfx=b-bQ~Tfx-Tb-bQ~fx. When 0<r<1, Since 0<ϵ<1, we have (60)1Q∫Qb,Tfxϵ-cϵdx1/ϵ≤1Q∫Qb,Tfx-cϵdx1/ϵ≤1Q∫Qbx-bQ~Tfxϵdx1/ϵ+1Q∫QTb-bQ~f1xϵdx1/ϵ+1Q∫QTb-bQ~f2x-cϵdx1/ϵ≔I1+I2+I3. Choosing any q∈(1,γ/ϵ), by the Hölder inequality and Proposition 24, it estimates that (61)I1≤1Q∫Qbx-bQ~q′ϵdx1/q′ϵ1Q∫QTfxqϵdx1/qϵ≤1Q∫Qbx-bQ~q′ϵdx1/q′ϵ1ϕQηQ∫QTfxγdx1/γ≤CbBMOθϕMγ,ϕ,ηTfx. For I2, we recall that T is a Calderón-Zygmund operator and is of weak type (1,1). By Kolmogorov’s inequality, (49), and Proposition 24, we then have(62)I2≤CTb-bQ~f1L1,∞Q,dx/Q≤C1Q~∫Q~bx-bQ~fxdx≤CbBMOθϕfLlog⁡L,QϕQθ≤CbBMOθϕMLlog⁡L,ϕ,ηfx, and the last inequality holds by 1≤ϕ(Q)≤ϕ(1). It turns to the term I3. Choosing c=T((b-bQ~)f2)(x0) and using (4), (5), and (49) and Proposition 24, we get that for any x∈Q and N=η+θ(63)Tb-bQ~f2x-Tb-bQ~f2x=∫Rn∖Q~Kx,y-Kx0,yby-bQ~fydy≤∑k=1∞∫Rn∖Q~x-x0δx-y+x0-yn+δfyϕx-y+x0-ynNby-bQ~dy≤∑k=1∞4-δ/n4k+1Qϕ4k+1QN∫4k+1Q∖4kQby-bQ~fydy≤∑k=1∞4-δ/nϕ4k+1QNbexp⁡L,4k+1QfLlog⁡L,4k+1Q≤∑k=1∞k4-δ/nϕ4k+1QNϕ4k+1QϕbBMOθϕfLlog⁡L,4k+1Q≤CbBMOθϕMLlog⁡L,ϕ,ηfx. Thus I3≤CbBMOθϕMLlog⁡L,ϕ,ηfx.
Whenr>1, since 0<ϵ<γ<1, we have (64)1ϕQηQ∫Qb,Tfxϵdx1/ϵ≤1ϕQη/ϵ1Q∫Qbx-bQ~Tfxϵdx1/ϵ+1ϕQη/ϵ1Q∫QTb-bQ~f1xϵdx1/ϵ+1ϕQη/ϵ1Q∫QTb-bQ~f2xϵdx1/ϵ≔II1+II2+II3.Choosing any q∈(1,γ/ϵ) and denoting η>θϵ/(1-ϵ)>θγϵ/(γ-ϵ), by the Hölder inequality and Proposition 24, the same estimate with I1 is obtained that (65)II1≤C1ϕQη/ϵ1Q∫Qbx-bQ~q′ϵdx1/q′ϵ1Q∫QTfxqϵdx1/qϵ≤CϕQθ-η/ϵ+η/qϵbBMOθϕMγ,ϕ,ηTfx≤CbBMOθϕMγ,ϕ,ηTfx. Also, since η>θϵ/(1-ϵ), by Kolmogorov’s inequality, (49), and Proposition 24, we have (66)II2≤C1ϕQη/ϵTb-bQ~f1L1,∞Q,dx/Q≤CbBMOθϕfLlog⁡L,QϕQθ-η/ϵ≤CbBMOθϕMLlog⁡L,ϕ,ηfx. The last term II3 remains. Choosing c=T((b-bQ~)f2)(x0) and using (3) and (49) and Proposition 24, we get that for any x∈Q and N=θ+η+2(67)II3≤1ϕQηQ∫QTb-bQ~fydy≤C1Q∫Q∫Rn∖4QKx,yby-bQ~fydydx≤C1Q∫Q∫Rn∖4Q1x-ynϕx-ynθ+η+2by-bQ~fydydx≤C∑k=1∞14k+1Qϕ4k+1Qθ+η+1∫4k+1Qby-bQ~fydy≤C∑k=1∞kϕ4k+1Qθϕ4k+1Qθ+η+2bBMOθϕfLlog⁡L,4k+1Q≤C∑k=1∞kϕ4k+1Q2bBMOθϕMLlog⁡L,ϕ,ηfx≤CbBMOθϕMLlog⁡L,ϕ,ηfx. We complete the proof.Proof of Theorem10.
By Corollary22, Lemma 27, Lemma 23, and Theorem 6, we have (68)b,TfLpω≤CbBMOθϕMϕ,γ,η∆TfLpω+MLlog⁡L,ϕ,ηfLpω≤CbBMOθϕTfLpω+Mϕ,η/2Mϕ,η/2fLpω≤CbBMOθϕfLpω.Lemma 28.
Letω∈A1θ(ϕ) and η≥2θ. There exists a positive C such that, for any function f and λ>0,(69)ωx∈Rn:MLlog⁡L,ϕ,ηfx>λ≤C∫RnΦfxλωxdx.Proof.
LetF be any compact subset in x∈Rn:MLlog⁡L,ϕ,ηfx>λ. For x∈F, there exists a cube Q with x∈Q such that(70)fΦ,ϕ,η,Q=ϕQ-ηfΦ,Q>λ.by Vitali covering lemma, we can choose a finite family of disjoint cubes {Qj} such that F⊂∪j3Qj and Qj satisfies (70). Thus (71)ϕQ2θQj≤ϕQηQj≤∫QjΦfxλdx. From this, by (iv) in Proposition 15 with p=1 and E=Q, we obtain that (72)ω3Qj≤CϕQjωQj≤CωQjϕQjQj∫QjΦfxλdx≤Cinfx∈Qj⁡ωx∫QjΦfxλdx≤C∫QjΦfxλωxdx. Thus, (69) holds.Lemma 29.
Letb∈BMOθ(ϕ), η>2θ, and ω∈A1θ(ϕ). Then there exists a positive constant C such that for any smooth function f with compact support (73)supt>0⁡1Φt⁡ωx∈Rn:b,Tfx>λ≤CΦbBMOθϕsupt>0⁡1Φtωx∈Rn:MLlog⁡L,ϕ,ηfx>λ.Proof.
Similar to the proof of Lemma1.5 in [12] or Lemma 4.4 in [1], we can get the desired result by Proposition 16, Lemma 27, and Lemma 23. We omit the details here.Proof of Theorem11.
By homogeneity, we need only to show that(74)ωx∈Rn:b,Tf>1≤C∫RnΦfxωxdx. Now, since Φ is submultiplicative, we have by Lemma 28 and Lemma 29(75)ωx∈Rn:b,Tf>1≤Csupt>0⁡1Φ1/t⁡ωx∈Rn:b,Tfx>t≤CΦbBMOθϕsupt>0⁡1Φ1/tωx∈Rn:MLlog⁡L,ϕ,ηfx>t≤C∫RnΦfxωxdx.



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*Source: 6769293-2018-01-03.xml*