# Morrey Spaces for Nonhomogeneous Metric Measure Spaces

Authors: Cao Yonghui; Zhou Jiang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The authors give a definition of Morrey spaces for nonhomogeneous metric measure spaces and investigate the boundedness of some classical operators including maximal operator, fractional integral operator, and Marcinkiewicz integral operators.


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## Body

## 1. Introduction

During the past fifteen years, many results from real and harmonic analysis on the classical Euclidean spaces have been extended to the spaces with nondoubling measures only satisfying the polynomial growth condition (see [1–9]). The Radon measure μ on ℝd is said to only satisfy the polynomial growth condition, if there exists a positive constant c such that for all x∈ℝd and r>0, μ(B(x,r))≤crn, where n is some fixed number in (0,d] and B(x,r)={y∈ℝd:|x-y|<r}. The analysis associated with such nondoubling measures μ is proved to play a striking role in solving the long-standing open Painlevé's problem by Tolsa [6]. Obviously, the nondoubling measure μ with the polynomial growth condition may not satisfy the well-known doubling condition, which is a key assumption in harmonic analysis on spaces of homogeneous type. To unify both spaces of homogeneous type and due to the fact that the metric spaces endow with measures only satisfying the polynomial growth condition, Hytönen [10] introduced a new class of metric measure spaces satisfying both the so-called geometrically doubling and the upper doubling conditions (see Definition 3), which are called nonhomogeneous spaces. Recently, many classical results have been proved still valid if the underlying spaces are replaced by the nonhomogeneous spaces of Hytönen (see [11–17]).In this paper, we give a natural definition of Morrey spaces associated with the nonhomogeneous spaces of Hytönen and investigate the boundedness of some classical operators including maximal operator, fractional integral operator and Marcinkiewicz integrals operator. To state the main results of this paper, we first recall some necessary notion, and notations. The following notions of geometrically doubling and upper doubling metric measure spaces were originally introduced by Hytönen [10].Definition 1.
A metric space(𝒳,d) is said to be geometrically doubling if there exists some N0∈ℕ such that, for any ball B(x,r)⊂𝒳, there exists a finite ball covering {B(xi,r/2)}i of B(x,r) such that the cardinality of this covering is at most N0.Remark 2.
Let(𝒳,d) be a metric space. In [10], Hytönen showed that the following statements are mutually equivalent.(1)
(𝒳,d) is geometrically doubling.(2)
For anyɛ∈(0,1) and any ball B(x,r)⊂𝒳, there exists a finite ball covering {B(xi,ɛr)}i of B(x,r) such that the cardinality of this covering is at most N0ɛ-n, where n=log2N0.(3)
For anyɛ∈(0,1) and any ball B(x,r)⊂𝒳 contains at most N0ɛ-n centers {xi}i of disjoint balls {B(xi,ɛr)}i.(4)
There existsM∈ℕ such that any ball B(x,r)⊂𝒳 contains at most M centers {xi}i of disjoint balls {B(xi,r/4)}i=1M.Definition 3.
A metric measure space(𝒳,d,μ) is said to be upper doubling if μ is a Borel measure on 𝒳 and there exist a dominating function λ:𝒳×(0,∞)→(0,∞) and a positive constant cλ such that, for each x∈𝒳, r→λ(x,r) is nondecreasing and
(1)μ(B(x,r))≤λ(x,r)≤cλλ(x,r2)∀x∈𝒳,r>0.It was proved in [14] that there exists a dominating function λ~ related to λ satisfying the property that there exists a positive constant cλ~ such that λ~≤λ, cλ~≤cλ, and, for all x,y∈𝒳, r>0 with d(x,y)≤r, λ~(x,r)≤cλ~λ~(y,r). Based on this, in this paper, we always assume that the dominating function λ also satisfies it.The following coefficientsδ(B,S) for all ball B and S were introduced in [10] as analogues of Tolsa’s number KQ,R in [5].Definition 4.
For all ballsB⊂S, let
(2)δ(B,S)=∫(2S-B)‍dμ(x)λ(cB,d(x,cB)),
where, as in the above mentioned, and in what follows, for a ball B=B(cB,rB) and ρ>0, ρB=B(cB,ρrB).Definition 5.
Letα,β∈(0,∞). A ball B⊂𝒳 is called (α,β)-doubling if μ(αB)≤βμ(B).It was proved in [10] that if a metric measure space (𝒳,d,μ) is upper doubling and α,β∈(0,∞) satisfying β>cλlog2α=αv, then, for any ball B, there exists some j∈ℕ∪{0} such that αjB is (α,β)-doubling. Moreover, let (𝒳,d,μ) be geometrically doubling, β>αn with n=log2N0 and μ a Borel measure on 𝒳 which is finite on bounded sets. Hytönen [10] also showed that, for μ-almost every x∈𝒳, there exist arbitrary small (α,β)-doubling balls centered at x. Furthermore, the radii of these balls may be chosen to be from α-jB for j∈ℕ and any preassigned number r>0. Throughout this paper, for any α∈(1,∞) and ball B, the smallest (α,βα)-doubling ball of the form αjB with j∈ℕ is denoted by B~α, where
(3)βα=max{α3n,α3v}+30n+30v.In what follows, by a doubling ball we mean a(6,β6)-doubling ball and B~6 is simply denoted by B~.Letk>1 and 1≤q≤p<∞. We define the Morrey space Mqp(k,μ) associated with the nonhomogeneous spaces of Hytönen. This is an analogy of [18–20].Definition 6.
Letk>1 and 1≤q≤p<∞, as
(4)Mqp(k,μ)={f∈Llocq:∥f∥Mqp(k,μ)<∞},
where
(5)∥f∥Mqp(k,μ)=supBμ(kB)1/p-1/q(∫B‍|f|qdμ)1/q.Clearly we haveLp(μ)=Mpp(k,μ) and Mq1p⊂Mq2p, 1≤q2≤q1≤p. If the underlying spaces are replaced by the nonhomogeneous spaces of Tolsa or Euclidean spaces, the definition of Morrey spaces can been seen in [18]. We will prove in Section 2 that the Morrey space is independent of choice of k.In [21], Chiarenza and Frasca showed that the Hardy-Littlewood maximal operator is bounded on the Morrey space. By establishing a pointwise estimate of fractional integrals in terms of the maximal function, they also showed the boundedness of fractional integral operator on Morrey space. If the underlying spaces are replaced by the nonhomogeneous spaces of Tolsa, Sawano and Tanaka also obtained these results in [18]. When the underlying spaces are the nonhomogeneous spaces of Hytönen, these operators have been discussed in Lebesgue space and RBMO space (see [22, 23]).Main theorems of this paper are stated in each section. The definition of Morrey space and its equivalent definition are shown in Section2. Section 3 is devoted to the study of maximal operator and fractional maximal operator. Section 4 deals with the fractional integral operator for the nonhomogeneous spaces of Hytönen. In Section 5, we investigate the behavior of the Marcinkiewicz integrals operator. In what follows the letter c will be used to denote constants that may change from one occurrence to another.


## 2. Morrey Space and Its Equivalent Definition

We firstly prove that the definition of Morrey space is independent of the choice of the parameterk (see [18, Proposition 1.1]).Theorem 7.
Letk,s>1; then Mqp(k,μ)≈Mqp(s,μ).Proof.
This result is a special case of the results in [24, Theorem 1.2]. For the sake of convenience, we provide the details. Let k≤s. By the definition of Morrey space, we have
(6)∥f∥Mqp(s,μ)=supBμ(sB)1/p-1/q(∫B‍|f|dμ)1/q≤supBμ(kB)1/p-1/q(∫B‍|f|dμ)1/q=∥f∥Mqp(k,μ),
where 1/p-1/q<0. So the inclusion Mqp(k,μ)⊂Mqp(s,μ) is obvious.
Letf∈Mqp(s,μ) and ball B⊂𝒳. Exploiting Remark 2(2), where ɛ=(k-1)/s, we have that there exists ball B1,B2,…,BN with the same radius r=ɛrB such that
(7)B⊂∪i=1NBi,sBi⊂kB(i=1,2,…,N),N≤N0(sk-1)n.
Using this covering, we obtain
(8)μ(kB)1/p-1/q(∫B‍|f|qdμ)1/q≤∑i=1N‍μ(kB)1/p-1/q(∫Bi‍|f|qdμ)1/q≤∑i=1N‍μ(sBi)1/p-1/q(∫Bi‍|f|qdμ)1/q≤N0(sk-1)n∥f∥Mqp(s,μ).
That is, ∥f∥Mqp(k,μ)≤c∥f∥Mqp(s,μ). We complete the proof of the theorem.With this theorem in mind, we sometimes omit parameterk in Mqp(k,μ).LetQ={B⊂𝒳:B is (6,β6)-doubling ball}. Now we give an equivalent definition of Morrey space.Definition 8.
Let1≤q≤p<∞; as
(9)Mqp(d)={f∈Llocq:∥f∥Mqp(d)<∞},
where
(10)∥f∥Mqp(d)=supB∈Qμ(B)1/p-1/q(∫B‍|f|qdμ)1/q.
This definition and Theorem 9 are analogy of [20].Theorem 9.
Let1≤q<p<∞ and β6>(6n)qp/(p-q); then Mqp(d)≈Mqp(μ).Proof.
We only need to prove that∥f∥Mqp(μ)≤c∥f∥Mqp(d).
For every ballB0=B(cB,r0) and x∈B0, let B(x,6-ixr0) be the largest doubling ball centered at x, having radius 6-ixr0, ix∈ℕ. So B(cB,r0)⊂∪x∈B0B(x,6-ixr0). By Besicovitch covering lemma, there is a subcollection A={B(xi,6-jxir0)}i=1∞ that covers B(cB,r0) so that no point belongs to more than c𝒳 of {B(xi,6-jxir0)}i=1∞, where c𝒳 only depends on space 𝒳. We write Ai={B∈A:rB=6-ir0}. Using Remark 2(2), we know, cardinal number of set Ai≤N06in. For all B(x,6-ir0)∈Ai, we have
(11)μ(6B0)≥μ(B(x,r0))≥β6iμ(B(x,6-ir0)).
So
(12)μ(6B0)1/p-1/q(∫B0‍|f|qdμ)1/q≤∑i=1∞‍∑B∈Ai‍μ(6B0)1/p-1/q(∫B‍|f|qdμ)1/q≤∑i=1∞‍∑B∈Ai‍(β6i)1/p-1/q(μ(B(x,6-ir0)))1/p-1/q≤∑i=1∞‍∑B∈Ai‍×(∫B(x,6-ir0)‍|f|qdμ)1/q≤c∥f∥Mqp(d)∑i=1∞‍N06in(β6i)1/p-1/q≤c∥f∥Mqp(d).


## 3. Maximal Inequalities

In this section we will investigate some maximal inequalities. Now we give the definitions of some maximal operators.Definition 10.
Letρ>0, r>1, α∈(0,1), as
(13)Mρf(x)=supx∈B1μ(ρB)∫B‍|f|dμ,Mραf(x)=supx∈B1μ(ρB)1-α∫B‍|f|dμ,Mr,ρf(x)=supx∈B[1μ(ρB)∫B‍|f|rdμ]1/r,Mr,ραf(x)=supx∈B[1μ(ρB)1-αr∫B‍|f|rdμ]1/r.In [11, 22, 25–27], the boundedness of these maximal operators has been proven in Lebesgue spaces.Lemma 11.
Letp>1, ρ>0. Then the maximal operators Mρ and Mr,ρ are bounded on Lp(μ) space.Lemma 12.
Letα∈(0,1), 1<r<p<1/α, ρ≥5, and 1/q=1/p-α. Then the maximal operator Mr,ρα is bounded from Lp(μ) space to Lq(μ) space.Remark 13.
Whenr=1, Lemma 11 also is right.
Now we extend these results to the Morrey spaces.Theorem 14.
Ifρ>1 and 1<r<q≤p<∞, then the maximal operators Mρ and Mr,ρ are bounded on Mqp(μ) space.Proof.
The proof of the boundedness ofMρ has been obtained in [24, 28]. We only prove the boundedness of Mr,ρ. For simplicity, we take ρ=2. Let B0=B(x0,r0) and f=f1+f2, where f1(x)=f(x)χ9B0(x). Then for every y∈B0 we have
(14)Mr,ρf(y)≤Mr,ρf1(y)+Mr,ρf2(y).
From the definitions of Mr,ρ and f2 it follows that
(15)Mr,ρf2(y)≤supy∈B,rB≥8r0[1μ(2B)∫B‍|f|rdμ]1/r.
For y∈B0∩B, rB≥8r0, the simple calculus yields B0⊂(3/2)B. Thus we have
(16)Mr,ρf2(y)≤supy∈B0⊂B[1μ((4/3)B)∫B‍|f|rdμ]1/r.
It follows that
(17)μ(12B0)1/p-1/q(∫B0‍|Mr,ρf|qdμ)1/q≤μ(12B0)1/p-1/q(∫B0‍|Mr,ρf1|qdμ)1/q+μ(12B0)1/p-1/q(∫B0‍|Mr,ρf2|qdμ)1/q≤μ(12B0)1/p-1/q(∫𝒳‍|Mr,ρf1|qdμ)1/q+μ(B0)1/p-1/q(∫B0‍|Mr,ρf2|qdμ)1/q≤cμ(12B0)1/p-1/q(∫𝒳‍|f1|qdμ)1/q+μ(B0)1/psupy∈B0⊂B(1μ((4/3)B)∫B‍|f|rdμ)1/r≤cμ(12B0)1/p-1/q(∫9B0‍|f|qdμ)1/q+μ(B0)1/psupy∈B0⊂Bμ(43B)-1/rμ(B)1/r-1/q(∫B‍|f|qdμ)1/q≤cμ(439B0)1/p-1/q(∫9B0‍|f|qdμ)1/q+μ(B0)1/psupy∈B0⊂Bμ(43B)-1/rμ(B)1/r-1/q(∫B‍|f|qdμ)1/q≤c∥f∥Mqp(4/3,μ)+∥f∥Mqp(4/3,μ)supy∈B0⊂Bμ(B)1/p+1/r-1/qμ(43B)1/q-1/p-1/r≤c∥f∥Mqp(4/3,μ).We obtain the conclusion of the theorem.Lemma 15.
Ifα∈(0,1), 1≤r<v≤u<∞, r<1/α, and 1<u<1/α, then
(18)|Mr,ραf(x)|≤c∥f∥Mvu(μ)uα|Mr,ρf(x)|1-uα.Proof.
This Proof is an analogy of [18, 29]. For every x∈𝒳, we write lx1/u=∥f∥Mvu(μ)/Mr,ρf(x). So
(19)|Mr,ραf(x)|≤supx∈B,μ(ρB)≤lx[1μ(ρB)1-αr∫B‍|f|rdμ]1/r+supx∈B,μ(ρB)>ix[1μ(ρB)1-αr∫B‍|f|rdμ]1/r=I+II.
ForI, we have
(20)I≤supx∈B,μ(ρB)≤lxμ(ρB)αμ(ρB)-1/r[∫B‍|f|rdμ]1/r≤lxαMr,ρf(x)=∥f∥Mvu(μ)uα|Mr,ρf(x)|1-uα.
If μ(ρB)>lx, there exists a i∈ℕ such that 2i-1lx≤μ(ρB)≤2ilx. It follows that
(21)II≤supx∈B,μ(ρB)≥lx(2i-1lx)α-1/r(∫B‍|f|rdμ)1/r≤supx∈B,μ(ρB)≥lx(2i-1lx)α-1/rμ(B)1/r-1/v(∫B‍|f|vdμ)1/v≤c∥f∥Mvu(μ)supx∈B,μ(ρB)≥lx(2i-1lx)α-1/rμ(ρB)1/r-1/u≤c∥f∥Mvu(μ)supi∈ℕ(2i-1lx)α-1/r(2ilx)1/r-1/u≤c∥f∥Mvu(μ)supi∈ℕ(2i)α-1/ulxα-1/u≤c∥f∥Mvu(μ)uα|Mr,ρf(x)|1-uα.
We complete the proof of Lemma 15.Using Lemma15 and Theorem 14, we have the following theorem.Theorem 16.
If1<s≤t<∞, 1<r<u≤v<1/α<∞, α=1/u-1/s, and s/t=u/v, then operator Mr,ρα is bounded from Mvu(μ) to Mts(μ).


## 4. Fractional Integral Operator

In this section, we prove the boundedness of fractional integral operator on Morrey space. The definition of fractional integral operator can be seen in [22]. The investigation of fractional integrals on quasimetric measure spaces with nondoubling measure (nonhomogeneous spaces) in Lebesgue spaces was researched in [30, chapter 6].Definition 17.
Let0<α<1, for all f∈L∞(μ) with bounded support, as
(22)Iαf(x)=∫𝒳‍f(y)λ(y,d(x,y))1-αdμ(y).In what follows, we assume that the dominating functionλ satisfies
(23)λ(x,ar)=amλ(x,r)∀x∈𝒳,a,r∈(0,∞),
where λ is the dominating function of the measure of μ in Definition 3. The condition about λ was first introduced by Bui and Duong in [11] to study the boundedness of commutators of Calderón-Zygmund operators. In [22], the authors obtain the boundedness of Iα. The boundedness of fractional integral operators of other type can be seen in [31, 32].Lemma 18.
Letα∈(0,1), 1<p<1/α, and 1/q=1/p-α. Then Iα is bounded from Lp(μ) space to Lq(μ) space.Lemma 19.
Letα∈(0,1), 1<q≤p<1/α, and 1/t=1/p-α. Then
(24)|Iαf(x)|≤c∥f∥Mqp(μ)1-p/t(M6f(x))p/t.Proof.
Lets∈(0,∞). We write
(25)|Iαf(x)|≤∫B(x,s)‍|f(y)|λ(y,d(x,y))1-αdμ(y)+∫𝒳-B(x,s)‍|f(y)|λ(y,d(x,y))1-αdμ(y)=I+II.
For I, we have
(26)I≤∑j=0∞‍∫B(x,6-js)-B(x,6-j-1s)‍|f(y)|λ(y,d(x,y))1-αdμ(y)≤∑j=0∞‍1λ(x,6-j-1s)1-α∫B(x,6-js)‍|f(y)|dμ(y)≤∑j=0∞‍μ(B(x,6-j+1s))λ(x,6-j-1s)1-α1μ(B(x,6-j+1s))∑j=0∞‍×∫B(x,6-js)‍|f(y)|dμ(y)≤M6f(x)∑j=0∞‍λ(x,6-j+1s)λ(x,6-j-1s)1-α≤M6f(x)∑j=0∞‍(6-j+1s)mλ(x,1)(6-j-1s)m(1-α)λ(x,1)1-α≤cM6f(x)∑j=0∞‍(6mα)-jλ(x,1)αsmα≤cM6f(x)λ(x,1)αsmα.
Similarly, we have
(27)II≤∑j=1∞‍∫B(x,6js)-B(x,6j-1s)‍|f(y)|λ(y,d(x,y))1-αdμ(y)≤∑j=1∞‍1λ(x,6j-1s)1-α∫B(x,6js)‍|f(y)|dμ(y)≤∑j=1∞‍μ(B(x,6js))1-1/qλ(x,6j-1s)1-αμ(B(x,6j+1s))1/p-1/qμ(B(x,6j+1s))1/p-1/q∑j=1∞‍×(∫B(x,6js)‍|f(y)|qdμ(y))1/q≤∥f∥Mqp(μ)∑j=1∞‍λ(x,6j+1s)1-1/pλ(x,6j-1s)1-α≤c∥f∥Mqp(μ)λ(x,1)α-1/p(sm)α-1/p.
For everyx∈𝒳, we take s that satisfies λ(x,1)sm=(∥f∥Mqp(μ)/M6f(x))p. Then
(28)I≤c∥f∥Mqp(μ)pαM6f(x)1-pα≤c∥f∥Mqp(μ)1-p/tM6f(x)p/t,II≤c∥f∥Mqp(μ)1-p/tM6f(x)p/t.
So we have
(29)|Iαf(x)|≤c∥f∥Mqp(μ)1-p/t(M6f(x))p/t.Using this lemma and the boundedness of maximal operator, we obtain the following result.The following proof of Theorem20 is similar to that of [33].Theorem 20.
Let1<q≤p<∞, 1<t≤s<∞, α∈(0,1), and 1/s=1/p-α, s/t=p/q. Then Iα is bounded from Mqp(μ) space to Mts(μ) space.Proof.
For all ballB(x,r), we have
(30)μ(2B)t/s-1∫B‍|Iαf|tdμ≤cμ(2B)t/s-1∫B‍∥f∥Mqp(μ)t-tp/s(M6f(x))tp/sdμ≤cμ(2B)t/s-1∥f∥Mqp(μ)t-q∫B‍(M6f(x))qdμ≤c∥f∥Mqp(μ)t.
Thus we have proved the theorem.


## 5. Marcinkiewicz Integral Operator

Firstly, we introduce the definition of Marcinkiewicz integral operator (see [23]).Definition 21.
LetK be a locally integrable function on (𝒳×𝒳-{(x,x):x∈𝒳}). Assume that there exists a positive constant c such that, for all x,y,z∈𝒳 with x≠y,
(31)|K(x,y)|≤cd(x,y)λ(x,d(x,y)),∫d(x,y)≥2d(y,z)[|K(x,y)-K(x,z)|+|K(y,x)-K(z,x)|]∫d(x,y)≥2d(y,z)×1d(x,y)dμ(x)≤c.
The Marcinkiewicz integral ℳ(f) associated with the above kernel K is defined by setting
(32)ℳ(f)(x)=[∫0∞‍|∫d(x,y)<t‍K(x,y)f(y)dμ(y)|2dtt3]1/2∀x∈𝒳.The boundedness onLp(μ) has been proved in [23].Lemma 22.
Suppose thatℳ is bounded on Lp0(μ) space for some p0∈(1,∞). Then ℳ is bounded on Lp(μ) spaces for all p∈(1,∞).Now we extend this result to the Morrey spacesMqp(μ).Theorem 23.
Let1<p≤q<∞. If ℳ is bounded on Lp0(μ) space for some p0∈(1,∞), then ℳ is bounded on Mqp(μ) space.Proof.
For every ballB=B(x0,r), f∈Mqp(μ), let f(x)=f1(x)+f2(x), where f1(x)=f(x)χ2B(x).
We can estimate(33)μ(4B)1/p-1/q(∫B‍|ℳ(f)|qdμ)1/q≤μ(4B)1/p-1/q(∫B‍|ℳ(f1)|qdμ)1/q+μ(4B)1/p-1/q(∫B‍|ℳ(f2)|qdμ)1/q≤I+II.
For the first term I, we have
(34)I≤μ(4B)1/p-1/q(∫𝒳‍|f1|qdμ)1/q≤μ(4B)1/p-1/q(∫2B‍|f|qdμ)1/q≤∥f∥Mqp(μ).
For II, we firstly estimate ℳ(f2)(x), as
(35)ℳ(f2)(x)=[∫0∞‍|∫d(x,y)<t‍K(x,y)f2(y)dμ(y)|2dtt3]1/2≤[∫0d(x0,y)+2r‍|∫d(x,y)<t‍K(x,y)f2(y)dμ(y)|2dtt3]1/2+[∫d(x0,y)+2r∞‍|∫d(x,y)<t‍K(x,y)f2(y)dμ(y)|2dtt3]1/2≤II1+II2.
For any ball B(x0,r), y∈(kB)c, k≥2, and x∈B, we have
(5)λ(x0,d(y,x0))~λ(y,d(y,x0))~λ(x,d(x,y)),II1≤∫𝒳‍K(x,y)|f2(y)|[∫d(x,y)d(x0,y)+2r‍1t3dt]1/2dμ(y)≤∫𝒳‍K(x,y)|f2(y)|[1(d(x0,y)+2r)2-1d(x,y)2]1/2dμ(y)≤c∫𝒳‍K(x,y)|f2(y)|[r(d(x0,y))3]1/2dμ(y)≤c∫𝒳‍d(x,y)λ(x,d(x,y))|f2(y)|[r(d(x0,y))3]1/2dμ(y)≤c∫𝒳-2B‍r1/2d(x0,y)1/2λ(x0,d(x0,y))|f2(y)|dμ(y)≤cr1/2∑i=2∞‍∫B(x0,2ir)-B(x0,2i-1r)‍|f(y)|d(x0,y)1/2λ(x0,d(x0,y))dμ(y)≤cr1/2∑i=2∞‍1λ(x0,2i-1r)(2i-1r)1/2∫B(x0,2ir)‍|f(y)|dμ(y)≤c∥f∥Mqp(μ)∑i=2∞‍μ(B(x0,2i+1r))1-1/pλ(x0,2i-1r)≤c∥f∥Mqp(μ)∑i=2∞‍(2i+1r)m-m/pλ(x0,1)1-1/p(2i-1r)mλ(x0,1)≤c∥f∥Mqp(μ)rm(-1/p)λ(x,1)-1/p≤c∥f∥Mqp(μ)λ(x0,r)-1/p.
Similarly, we obtain
(37)II2≤c∫𝒳-2B‍1λ(x0,d(x0,y))|f(y)|dμ(y)≤c∑i=2∞‍∫B(x0,2ir)-B(x0,2i-1r)‍|f(y)|λ(x0,d(x0,y))dμ(y)≤c∑i=2∞‍1λ(x0,2i-1r)∫B(x0,2ir)‍|f(y)|dμ(y)≤c∥f∥Mqp(μ)∑i=2∞‍μ(B(x0,2i+1r))1-1/pλ(x0,2i-1r)≤c∥f∥Mqp(μ)λ(x0,r)-1/p.
That is to say,ℳ(f2)(x)≤cλ(x0,r)-1/p∥f∥Mqp(μ) for all ball B(x0,r) and x∈B(x0,r).
Using it we have(38)II≤μ(4B)1/p-1/q(∫B‍|ℳ(f2)|qdμ)1/q≤cμ(4B)1/p-1/q(∫B‍|λ(x0,r)-1/p∥f∥Mqp(μ)|qdμ)1/q≤c∥f∥Mqp(μ)μ(4B)1/p-1/qλ(x0,r)-1/pμ(B)1/q≤c∥f∥Mqp(μ)μ(B)1/pλ(x0,r)-1/p≤c∥f∥Mqp(μ).
The proof of Theorem 23 is completed.



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*Source: 196459-2013-09-23.xml*