# Sobolev Embeddings for Generalized Riesz Potentials of Functions in Morrey SpacesL(1,φ)(G) over Nondoubling Measure Spaces

Authors: Yoshihiro Sawano; Tetsu Shimomura
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

Our aim in this paper is to deal with the Sobolev embeddings for generalized Riesz potentials of functions in Morrey spacesL(1,φ)(G) over nondoubling measure spaces.


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## Body

## 1. Introduction

In this paper, we show that many endpoint results about the Adams theorem still hold in the nondoubling setting and that the integral kernel can be generalized to a large extent. In [1], in the setting of the Lebesgue measure, for 0<α<n, recall that Adams considered and proved the boundedness of the fractional integral operator Iα given by
(1)Iαf(x)≔∫ℝn‍f(y)|x-y|n-αdy.
The operator Iα is also called the fractional integral operator or the Riesz potential. We denote by B(z,r) the ball {x∈ℝn:|x-z|<r} with center z and of radius r>0, and by |B(z,r)| its Lebesgue measure, that is, |B(z,r)|=ωnrn, where ωn is the volume of the unit ball in ℝn. Let G be a bounded open subset of ℝn. We denote its diameter by dG;
(2)dG=sup{|x-y|:x,y∈G}.Foru∈L1(G), we define the integral mean over B(z,r) by
(3)uB(z,r)=⨍B(z,r)u(x)dx≔1|B(z,r)|∫G∩B(z,r)‍u(x)dx.
Let 1≤p<∞. If φ is a positive function on the interval (0,∞) satisfying the doubling condition (see (23)), then we define the Morrey space L(p,φ)(G) to be the family of all f∈Llocp(G) for which there is a positive constant C such that
(4)⨍B(z,r)|f(x)|pdx≤Cpφ(r)wheneverz∈G,0<r≤dG.
The norm of f∈L(p,φ)(G) is defined by the infimum of the constants C satisfying the inequality above. When φ(r)≡r-λ(r>0), L(p,φ)(G) is denoted by Lp,λ(G).A direct consequence of this notation is that(5)Lp,λp(G)⊃Ln/λ(G)
for 0<λ≤n and p∈[1,n/λ).Some prefer to use the notation(6)∥f∥Mp,w(·)=supr>0w(r)∥f∥Lp(B(z,r))
with
(7)w(r)=(ωnrn)-1/pφ(r)-1/p
references [2–5].Much about the casep>1 is known. Recall that the Adams theorem about the boundedness of fractional integral operators [1, Theorem 3.1] asserts that
(8)∥Iαf∥Lq,λ≤C∥f∥Lp,λ,
provided the parameters p, q, λ satisfy
(9)0<λ≤n,1<p<q<∞,1q=1p-αλ.
See also research papers [2–4, 6–16] and a survey [5].Meanwhile, only a few results are known for the casep=1. Trudinger [17, Theorem 1] proved that if f∈L1,1(G)=L1(G) then exp(a|I1f|)∈L1(G) for some constant a>0; this implies that the operator I1 is bounded from L1,1(G) to exp(L1)(G). See also Serrin [18] for an alternative proof. Recently, the boundedness of Riesz potentials from L(1,φ)(G) to Orlicz-Morrey spaces was shown in [19]. This result extends [20, 21]. One of the reasons why the case when p=1 is difficult is the failure of the boundedness of the Hardy-Littlewood maximal operator M. In connection with this failure, we do not have Littlewood-Paley characterization. Due to these two difficulties, the case when p=1 is hard to analyze. However, from the point of PDEs, we are faced with analyzing the quantity
(10)limr↓0(supx∈Rn∫B(x,r)‍|V(y)||x-y|n-αdy)
in connection of the Kato condition, where V is the potential operator of the operator -Δ+V. See [22, Section 2], for example. Consequently, despite the difficulty arising from harmonic analysis, the case when p=1 occurs naturally. As another evidence that the case when p=1 is of importance, we recall that the space L1,λ(ℝn) appears naturally in the following sharp maximal inequalities [23, Theorem 4.7], [24, Theorem 1.3], and [25, Theorem 1.2]: let 1<p<∞ and λ∈(0,n]. Then, there exists a constant C>0 such that
(11)C-1(∥M♯f∥Lp,λ+∥f∥L1,λ)≤∥f∥Lp,λ≤C(∥M♯f∥Lp,λ+∥f∥L1,λ)
for any measurable function f, where
(12)M♯f(x)=supy∈ℝn,r>01|B(y,r)|×∫B(y,r)|f(z)-1|B(y,r)|∫B(y,r)‍f(w)dw|dz
is the sharp maximal operator due to Fefferman and Stein [26]. A disadvantage of using the Littlewood-Paley theory is that we lose the integrability of functions a little when we consider the inequality
(13)supj∈ℤ∥Sjf∥L1,λ≤C∥f∥L1,λ,
where {Sj}j=-∞∞ is a Littlewood-Paley patch. By choosing a smooth function φ∈C∞(ℝn) such that χB(0,4)∖B(0,2)≤φ≤χB(0,8)∖B(0,1), recall that we can define the jth Littlewood-Paley patch by
(14)Sjf(x)≔ℱ-1[φ(2-j·)·ℱf](x)
for f∈𝒮′(ℝn). Note that (13) is a direct consequence of the translation invariance of the space L1,λ(ℝn). But this loss caused by (13) is quite big. Note that
(15)∥f∥L1,λ≤Csupj∈ℤ∥Sjf∥L1,λ
fails. See the appendix for a proof. When p>1, an approach using the Littlewood-Paley patch is taken effectively [27]. Indeed,
(16)C-1∥f∥Lp,λ≤∥(∑j=-∞∞‍|Sjf|2)1/2∥Lp,λ≤C∥f∥Lp,λ
for all f∈Lp,λ(ℝn). However, for the case when p=1, due to the fact that the estimate (13) is essential when we consider the Littlewood-Paley patch, we prefer to avoid the Littlewood-Paley patch. See [28–43] for a huge amount of culmination of this approach.Instead of using the Littlewood-Paley patch, we still have a good approach for the case whenp=1. Just make a closer look at the integral kernel. Our method being simple enough, there is no need to stick to the geometric structure of ℝn. Our result relies completely only upon the positivity of the integral kernel. So, here and below, we work on a separable metric space X equipped with a nonnegative Radon measure μ, where we do not postulate any other condition on μ. By B(x,r), we denote the open ball centered at x of radius r>0. While, given a point p1 and p2 in ℝn, we write |p1-p2| for the distance of the points p1 and p2, and we write d(x,y) for the distance of the points x and y in X. We assume that μ({x})=0 and that 0<μ(B(x,r))<∞ for x∈X and r>0 for simplicity. In the present paper, we do not postulate on μ the “so-called” doubling condition. Recall that a Radon measure μ is said to be doubling, if there exists a constant C>0 such that
(17)μ(B(x,2r))≤Cμ(B(x,r))
for all x∈supp(μ)(=X) and r>0. Otherwise μ is said to be nondoubling. In connection with the 5r-covering lemma, the doubling condition had been a key condition in harmonic analysis.Our aim in this paper is to show that, for the casep=1, the operator Iα and its generalization Iρ are bounded from Morrey spaces L(1,φ) to Orlicz-Morrey spaces, or, to generalized Hölder spaces, whose definitions will be given in the next section, in the nondoubling setting. Our result extends the results in [17–21]. The definition of Iρ is the following: let ρ be a function from (0,∞) to itself and satisfy
(18)∫0r‍ρ(t)tdt<+∞
for all sufficiently small r>0. We do not have to postulate the doubling condition on ρ. See Remark 3 for an example which fails the doubling condition. We define
(19)Iρf(x)=∫G‍ρ(d(x,y))μ(B(x,4d(x,y)))f(y)dμ(y),
where f∈L1(G). Instead of using
(20)Iρ†f(x)=∫G‍ρ(d(x,y))μ(B(x,d(x,y)))f(y)dμ(y),
we discuss Iρ defined above. This modification will be necessary in Lemma 9 for example. An example in [44, Section 2] shows that Iρ† is less likely to be bounded in general, although there does not exist a proof. We refer to [45] for an attempt of definining fractional integral operators by using the underlying measure μ.Note that (18) is necessary in order that the image by Iρ of χB(x,r), the indicator functions of the balls, belongs to Lp,φ(G) at least when μ is the Lebesgue measure. Indeed, if
(21)∫0r‍ρ(t)tdt=∞
for any sufficiently small r>0. Then, for y∈B(x,r/2) such that B(x,r)⊂G, we have
(22)IρχB(x,r)(y)=∫B(x,r)‍ρ(|y-z|)|B(y,4|y-z|)|dz≥∫B(y,r/2)‍ρ(|y-z|)|B(y,4|y-z|)|dz=C∫0r/2‍ρ(t)tdt=∞
by using the spherical coordinate.We organize the remaining part of the present paper as follows. In Section2, we set up some notations. Section 3 is devoted to stating our main results fully based on the notations in Section 2. Some auxiliary lemmas are collected in Section 4. Finally, theorems in the present paper are proven in Section 5.


## 2. Notation and Terminologies

Let𝒢 be the set of all continuous functions from (0,∞) to itself with the doubling condition, that is, there exists a constant cφ≥1 such that
(23)1cφ≤φ(r)φ(s)≤cφforr,s>0with12≤rs≤2.
We call the smallest number cφ satisfying (23) the doubling constant of φ. Note that in view of [46, page 445] and [47, (1.2)], the doubling condition on φ is a natural one. For φ∈𝒢, we define the Morrey space L(1,φ)(G) as follows:
(24)L(1,φ)(G)≔{f∈Lloc1(G):∥f∥L(1,φ)(G)<∞}
with the norm
(25)∥f∥L(1,φ)(G)=supz∈G,0<r≤dG1φ(r)·1μ(B(z,2r))∫B(z,r)‍|f(x)|dμ(x).
Then, a routine argument shows that L(1,φ)(G) is a Banach space. Due to the fact that ℝn is a geometrically doubling space, we can prove that
(26)C-1∥f∥L(1,φ)(G)≤supz∈G,0<r≤dG1φ(r)·1μ(B(z,kr))∫B(z,r)‍|f(x)|dμ(x)≤C∥f∥L(1,φ)(G)
for all k>1. See [48, Proposition 1.1] for a technique used to prove this inequality. Note here that if φ1, φ2∈𝒢 and φ1/φ2 is bounded above on (0,dG), then
(27)L(1,φ1)(G)⊂L(1,φ2)(G),
in particular, if there exists a constant C≥1 such that C-1φ1(r)≤φ2(r)≤Cφ1(r) for all r>0, then
(28)L(1,φ1)(G)=L(1,φ2)(G)
with equivalent norms. A ball testing shows the following.Proposition 1.
The functionφ1/φ2 is bounded above on (0,dG) if L(1,φ1)(G)⊂L(1,φ2)(G) when μ=dx.Here and below, we writeA≲B to indicate that there exists a constant C independent of Morrey functions such that A≤CB. The symbol A~B stands for A≲B≲A.Proof.
According to [49, Proposition A], for any ball B(x0,r) contained in G, we have
(29)∥χB(x0,r)∥L(1,φ1)(G)~1φ1(r),∥χB(x0,r)∥L(1,φ2)(G)~1φ2(r).
If L(1,φ1)(G)⊂L(1,φ2)(G), in the sense of sets, then by the closed graph theorem and the doubling condition on φ1 and φ2, we conclude
(30)∥f∥L(1,φ2)(G)≤C∥f∥L(1,φ1)(G).
If we combine (29) and (30), then we obtain that φ1/φ2 is bounded above on (0,dG).Let us consider the family𝒴 of all continuous, increasing, convex, and bijective functions from [0,∞) to itself. For Φ∈𝒴, the Orlicz space LΦ(G) is defined by
(31)LΦ(G)≔{f∈Lloc1(G):∥f∥LΦ(G)<∞},
where
(32)∥f∥LΦ(G)≔inf{λ>0:∫G‍Φ(|f(x)|λ)dμ(x)≤1}.
If Φ1, Φ2∈𝒴 are equivalent in the sense that there exists a constant C≥1 with
(33)Φ1(C-1r)≤Φ2(r)≤Φ1(Cr)
for all r>0, then we see easily that
(34)LΦ1(G)=LΦ2(G)
with equivalent norms. If
(35)Φ(r)=exp(rp),exp(exp(rp)),rp(logr)λorrp(logr)q(log(logr))λ(r>0)
for large r>0, then LΦ(G) will be denoted by
(36)exp(Lp)(G),expexp(Lp)(G),Lp(logL)λ(G)orLp(logL)q(loglogL)λ(G),
respectively.ForΦ∈𝒴 and φ∈𝒢, the Orlicz-Morrey space L(Φ,φ)(G) is defined by
(37)L(Φ,φ)(G)∶={f∈Lloc1(G):∥f∥L(Φ,φ)(G)<∞},
where
(38)∥f∥L(Φ,φ)(G)∶=supz∈G,0<r≤dGinf{(|f(x)|λ)λ>0:1μ(B(z,2r))∶=supz∈G,0<r≤dGinfm×∫B(z,r)‍Φ(|f(x)|λ)dμ(x)≤φ(r)}
(see [50, 51]). Then, again it is routine to prove that ∥·∥L(Φ,φ)(G) is a norm and that L(Φ,φ)(G) is a Banach space. Note that the space LΦ is a special case of Orlicz-Morrey spaces when μ=dx.Forφ∈𝒢 such that φ is bounded, the generalized Hölder space is defined by
(39)Λφ(G)={f:∥f∥Λφ(G)<∞},
where
(40)∥f∥Λφ(G)=supx,y∈G,x≠y|f(x)-f(y)|φ(2d(x,y)).
Then, ∥f∥Λφ(G) is a norm modulo constants and thereby Λφ(G) is a Banach space. Since φ is bounded, every f∈Λφ(G) is bounded. If φ(r)→0 as r↓0, then every f∈Λφ(G) is continuous. For details, we refer to [52].


## 3. Main Results

In this section, we state our main theorems, whose proofs are given in Section5.Throughout this paper, letG be a bounded open set in X and denote by cφ, the doubling constant of φ∈𝒢.Let us begin with the following result, which is the one of Gunawan type [9].Theorem 2.
Letρ:(0,∞)→(0,∞) be a measurable function such that there exist k1, k2, Cρ such that
(41)0<16k1≤1≤k2<∞,supr/2≤s≤rρ(s)≤Cρ∫k1rk2r‍ρ(s)sds(r>0).
Let φ∈𝒢, and define
(42)ψ(r)≔(∫04k2r‍ρ(t)tdt)φ(r)+∫2k1r4k2dG‍ρ(t)φ(t)tdt
for 0<r≤dG. Then, there exists a constant C>0 such that
(43)1μ(B(z,4r))∫B(z,r)‍|Iρf(x)|dμ(x)≤Cψ(r)∥f∥L(1,φ)(G)
for z∈G, 0<r≤dG and f∈L(1,φ)(G), where C>0 is a constant depending only on Cρ, cφ, k1, and k2.Remark 3.
(1) Here it is not significant for us to choose 16; it counts that any number will do as long as it is small enough.
(2) The number 4 in the right-hand side seems to be essential. According to [44, Section 2], it can happen that the norms
(44)supz∈G,0<r≤dG(rνμ(B(z,4r))∫B(z,r)‍|f(w)|pdμ(w))1/p,supz∈G,0<r≤dG(rνμ(B(z,2r))∫B(z,r)‍|f(w)|pdμ(w))1/p
are not equivalent for 1≤p<∞.
(3) In view of [53, Lemma 2.5], we see that (1-Δ)-α/2 falls under the scope of Theorem 2. Indeed, Nagayasu and Wadade showed that the kernel ρ which corresponds to (1-Δ)-α/2 satisfies
(45)ρ(r)~rα(0<r<1),ρ(r)≲e-r(r≥1).
This means that we have (41) with k1=1/16 and k2=1. Note that ρ∈𝒢 implies (41). See also [54, Remark 2.2].Remark 4.
Theorem2 is proved in [19] when G=ℝn and ρ∈𝒢. See also [21].We now state a result for Orlicz-Morrey spaces.Theorem 5.
Letρ,ρ~:(0,∞)→(0,∞) be measurable functions such that there exist k1, k2, Cρ such that 0<16k1≤1≤k2<∞ and that
(46)supr/2≤s≤rρ(s)≤Cρ∫k1rk2r‍ρ(s)sds,supr/2≤s≤rρ~(s)≤Cρ~∫k1rk2r‍ρ~(s)sds(r>0).
Let φ∈𝒢. Assume
(47)∫01‍ρ(t)φ(t)tdt=∞
and that ρ~/ρ is continuous and decreasing.
Define(48)ψ1(r)≔∫2k1r4k2dG‍ρ(t)φ(t)tdt,κ(r)≔ψ1(r)ρ~(4k2r)ρ(4k2r),ψ(r)≔(∫02k2r‍ρ~(t)tdt)φ(r)+∫2k1r4k2dG‍ρ~(t)φ(t)tdt
for 0<r≤dG. If Φ∈𝒴 satisfies
(49)CG=sup{(ψ1∘κ-1)(s)Φ-1(s):κ(dG)≤s<∞}<∞,
then there exists a constant A>0 such that
(50)1μ(B(z,4r))∫B(z,r)‍Φ(|Iρf(x)|A∥f∥L(1,φ)(G))dμ(x)≤ψ(r)
for z∈G, 0<r≤dG and f∈L(1,φ)(G), where A>0 is a constant depending only on Cρ, Cρ~, cφ, k1, k2, and CG.Remark 6.
Note thatκ is bijective from (0,dG] to [κ(dG),∞) by the assumptions in the theorem. Indeed, by the definition of κ above, κ is a decreasing function. In addition, limr↓0κ(r)=∞, showing that κ:(0,dG]→[κ(dG),∞) is bijective.Finally, we shall show a result of Gunawan type about continuity.Theorem 7.
Letρ:(0,∞)→(0,∞) be a measurable function such that there exist k1, k2, Cρ such that
(51)0<16k1≤1≤k2<∞,supr/2≤s≤rρ(s)≤Cρ∫k1rk2r‍ρ(s)sds(r>0).
Let φ∈𝒢. Assume the following condition on ρ. There are 0<θ≤1 and Cρ′>0 such that
(52)|ρ(d(x,y))μ(B(x,4d(x,y)))-ρ(d(z,y))μ(B(z,4d(z,y)))|≤Cρ′(d(x,z)d(x,y))θρ(d(x,y))μ(B(x,4d(x,y)))
whenever d(x,z)≤d(x,y)/2. Assume in addition the Dini condition
(53)∫01‍ρ(t)φ(t)tdt<∞.
If
(54)ψ(r)=∫03k2r‍ρ(t)φ(t)tdt+rθ∫2k1r4k2dG‍ρ(t)φ(t)t1+θdtρ(t)φ(t)tdtt+rθ∫2k1r4k2dG‍ρ(t)φ(t)t1+θfor0<r≤dG,
then Iρ is bounded from L(1,φ)(G) to Λψ(G). More precisely,
(55)∥Iρf∥Λψ(G)≤C∥f∥L(1,φ)(G),
where C>0 is a constant depending only on Cρ, Cρ′, cφ, k1, k2, and θ.Note that if∫01‍(ρ(t)φ(t)/t)dt<∞ and 0<θ≤1, then
(56)r∈(0,dG]⟼rθ∫2k1r4k2dG‍ρ(t)φ(t)t1+θdt∈[0,∞)
is bounded.


## 4. Preliminary Lemmas

Lemma 8.
Letρ:(0,∞)→(0,∞) be a measurable function such that there exist k1, k2, Cρ such that
(57)0<16k1≤1≤k2<∞,supr/2≤s≤rρ(s)≤Cρ∫k1rk2r‍ρ(s)sds(r>0).
Let φ∈𝒢. Then
(58)∫B(x,r)‍ρ(d(x,y))μ(B(x,4d(x,y)))|f(y)|dμ(y)≤C(∫02k2r‍ρ(t)φ(t)tdt)∥f∥L(1,φ)(G),
where C>0 is a constant depending only on Cρ, cφ, k1, and k2.
Moreover, ifk≥0, then
(59)∫B(x,dG)∖B(x,r)‍ρ(d(x,y))μ(B(x,4d(x,y)))d(x,y)k|f(y)|dμ(y)≤C(∫2k1r4k2dG‍ρ(t)φ(t)t1+kdt)∥f∥L(1,φ)(G),
where C>0 is a constant depending only on Cρ, cφ, k1, k2, and k.Proof.
Ify∈B(x,2jr)∖B(x,2j-1r) and j∈ℤ, then a geometric observation shows
(60)ρ(d(x,y))μ(B(x,4d(x,y)))d(x,y)k≤1μ(B(x,2j+1r))(2j-1r)ksup2j-1r≤s≤2jrρ(s)≤Cρμ(B(x,2j+1r))(2j-1r)k∫2jk1r2jk2r‍ρ(s)sds.
Hence,
(61)∫B(x,2jr)∖B(x,2j-1r)‍ρ(d(x,y))μ(B(x,4d(x,y)))d(x,y)k|f(y)|dμ(y)≤Cρ(2j-1r)k∫2jk1r2jk2r‍ρ(s)sds×1μ(B(x,2j+1r))∫B(x,2jr)‍|f(y)|dμ(y)=Cρφ(2jr)(2j-1r)k∫2jk1r2jk2r‍ρ(s)sds×1φ(2jr)μ(B(x,2j+1r))∫B(x,2jr)‍|f(y)|dμ(y)≤Cρφ(2jr)(2j-1r)k∫2jk1r2jk2r‍ρ(s)sds×∥f∥L(1,φ)(G).
Set d≔[1+log2(k2/k1)]. Then, by virtue of the doubling condition on φ, we have
(62)φ(2jr)(2j-1r)k∫2jk1r2jk2r‍ρ(s)sds≤2kφ(2jr)(2jr)k∫2jk1r2j+dk1r‍ρ(s)sds=2k∑l=1d‍φ(2jr)(2jr)k∫2j+l-1k1r2j+lk1r‍ρ(s)sds≤C1(2d+1k1)k∑l=1d‍∫2j+l-1k1r2j+lk1r‍φ(s)(2j+dk1r)k·ρ(s)sds≤C1(2d+1k1)k∑l=1d‍∫2j+l-1k1r2j+lk1r‍φ(s)ρ(s)s1+kds=C1(2d+1k1)k∫2jk1r2j+dk1r‍φ(s)ρ(s)s1+kds,
where C1>0 is a constant depending only on cφ, k1, and k2.
Consequently, sinceμ({x})=0,
(63)∫B(x,r)‍ρ(d(x,y))μ(B(x,4d(x,y)))|f(y)|dμ(y)=∑j=0∞‍∫B(x,2-jr)∖B(x,2-j-1r)‍ρ(d(x,y))μ(B(x,4d(x,y)))|f(y)|dμ(y)≤∑j=0∞‍CρC1∫2-jk1r2-j+dk1r‍φ(s)ρ(s)sds×∥f∥L(1,φ)(G)≤CρC1(∫02k2r‍ρ(t)φ(t)tdt)∥f∥L(1,φ)(G),
which proves (58).
We choosej0∈ℤ, so that dG≤2j0r<2dG. Then, we have
(64)∫B(x,dG)∖B(x,r)‍ρ(d(x,y))μ(B(x,4d(x,y)))d(x,y)k|f(y)|dμ(y)≤∑j=1j0‍∫B(x,2jr)∖B(x,2j-1r)‍ρ(d(x,y))μ(B(x,4d(x,y)))d(x,y)kmmmmmmmmmmmk×|f(y)|dμ(y)≤CρC1(2d+1k1)k∑j=1j0‍∫2jk1r2j+dk1r‍φ(s)ρ(s)s1+kds×∥f∥L(1,φ)(G)≤CρC1(2d+1k1)k(∫2k1r4k2dG‍ρ(t)φ(t)t1+kdt)∥f∥L(1,φ)(G).
Thus, since k, k1, d being constants, (59) follows.Lemma 9.
Letρ:(0,∞)→(0,∞) be a measurable function such that there exist k1, k2, Cρ such that
(65)0<16k1≤1≤k2<∞,
and that
(66)supr/2≤s≤rρ(s)≤Cρ∫k1rk2r‍ρ(s)sds(r>0).
Then, for all f∈L(1,φ)(G),
(67)1μ(B(z,2r))×∫B(z,r)‍(∫B(z,r)‍ρ(d(x,y))μ(B(x,4d(x,y)))|f(y)|dμ(y))dμ(x)≤Cφ(r)(∫02k2r‍ρ(t)tdt)∥f∥L(1,φ)(G).Proof.
By Fubini's theorem and the dyadic decomposition of the ball, we have(68)∫B(z,r)‍(∫B(z,r)‍ρ(d(x,y))μ(B(x,4d(x,y)))|f(y)|dμ(y))dμ(x)=∫B(z,r)‍|f(y)|=mmm×(∫B(z,r)‍ρ(d(x,y))μ(B(x,4d(x,y)))dμ(x))dμ(y)≤∫B(z,r)‍|f(y)|=mmm×(∑j=0∞‍∫B(y,2-j+1r)∖B(y,2-jr)sup2-jr≤s≤2-j+1rρ(s)μ(B(x,2-j+2r))=mmmmmmmmmmmmmmm×dμ(x)∑j=0∞‍∫B(y,2-j+1r)∖B(y,2-jr)sup2-jr≤s≤2-j+1rρ(s)μ(B(x,2-j+2r)))dμ(y)≤∫B(z,r)‍|f(y)|=mmm×(∑j=0∞‍∫B(y,2-j+1r))sup2-jr≤s≤2-j+1rρ(s)μ(B(y,2-j+1r))=mmmmmmmmmmmm×dμ(x)∑j=0∞‍∫B(y,2-j+1r)∖B(y,2-jr)sup2-jr≤s≤2-j+1rρ(s)μ(B(x,2-j+2r)))dμ(y).
Since ρ satisfies (66), we have
(69)∫B(z,r)‍(∫B(z,r)‍ρ(d(x,y))μ(B(x,4d(x,y)))|f(y)|dμ(y))dμ(x)≤Cρ∫B(z,r)‍|f(y)|(∑j=0∞‍∫2-j+1k1r2-j+1k2r‍ρ(s)sdt)dμ(y)≤Cρ(∫02k2r‍ρ(t)tdt)∫B(z,r)‍|f(y)|dμ(y)≤Cρφ(r)μ(B(z,2r))(∫02k2r‍ρ(t)tdt)∥f∥L(1,φ)(G),
as required.


## 5. Proofs of the Theorems

We are now ready to prove our theorems.Proof of Theorem2.
Letz∈G and r∈(0,dG]. By the positivity of the kernel, we may assume that f≥0. We write
(70)1μ(B(z,4r))∫B(z,r)‍Iρf(x)dμ(x)≤1μ(B(z,4r))×∫B(z,r)‍(∫B(x,r)‍ρ(d(x,y))μ(B(x,4d(x,y)))mmmmmmmmmm×f(y)dμ(y)∫B(x,r)‍)dμ(x)+1μ(B(z,4r))×∫B(z,r)‍(∫B(x,dG)∖B(x,r)‍ρ(d(x,y))μ(B(x,4d(x,y)))mmmmmmmmmmmmm×f(y)dμ(y)∫B(x,dG)∖B(x,r)‍)dμ(x)≤1μ(B(z,4r))×∫B(z,2r)‍(∫B(z,2r)‍ρ(d(x,y))μ(B(x,4d(x,y)))mmmmmmmlmmm×f(y)dμ(y)∫B(z,2r)‍)dμ(x)+1μ(B(z,4r))×∫B(z,r)‍(∫B(x,dG)∖B(x,r)‍ρ(d(x,y))μ(B(x,4d(x,y)))mmmmmmmmmmmlmm×f(y)dμ(y)∫B(x,dG)∖B(x,r)‍)dμ(x)=I1+I2
for z∈G and 0<r≤dG. By Lemma 9, we have
(71)I1≤C1φ(r)(∫04k2r‍ρ(t)tdt)∥f∥L(1,φ)(G)≤C1ψ(r)∥f∥L(1,φ)(G).
Meanwhile, by Lemma 8 we have
(72)I2≤C2(∫2k1r4k2dG‍ρ(t)φ(t)tdt)∥f∥L(1,φ)(G)≤C2ψ(r)∥f∥L(1,φ)(G).
Hence, it follows from (71) and (72) that
(73)1μ(B(z,4r))∫B(z,r)‍Iρf(x)dμ(x)≤Cψ(r)∥f∥L(1,φ)(G),
where C>0 depends only on Cρ, cφ, k1, and k2.Proof of Theorem5.
By Theorem2, we have
(74)1μ(B(z,4r))∫B(z,r)‍|Iρ~f(x)|dμ(x)≤C1ψ(r)∥f∥L(1,φ)(G)
for z∈G and 0<r≤dG.
Letg≔|f|/∥f∥L(1,φ)(G). For x∈G and 0<δ≤dG, since ρ~/ρ is decreasing, we have by Lemma 8(75)Iρg(x)=∫B(x,δ)‍ρ(d(x,y))μ(B(x,4d(x,y)))g(y)dμ(y)+∫B(x,dG)∖B(x,δ)‍ρ(d(x,y))μ(B(x,4d(x,y)))g(y)dμ(y)≤ρ(δ)ρ~(δ)∫B(x,δ)‍ρ~(d(x,y))μ(B(x,4d(x,y)))g(y)dμ(y)+C2∫2k1δ4k2dG‍ρ(t)φ(t)tdt≤ρ(δ)ρ~(δ)Iρ~g(x)+C2ψ1(δ)≤ρ(4k2δ)ρ~(4k2δ)Iρ~g(x)+C2ψ1(δ).
Hence, in view of the definition of κ, we have
(76)Iρg(x)≤ψ1(δ)κ(δ)Iρ~g(x)+C2ψ1(δ).
Now let
(77)δ≔{κ-1(Iρ~g(x))whenIρ~g(x)≥κ(dG),dGwhenIρ~g(x)<κ(dG).
Observe that
(78)ψ1(δ)={ψ1(κ-1(Iρ~g(x)))whenIρ~g(x)≥κ(dG),ψ1(dG)whenIρ~g(x)<κ(dG),
by definition.
We claim that(79)ψ1(δ)κ(δ)Iρ~g(x)≤{ψ1(κ-1(Iρ~g(x)))whenIρ~g(x)≥κ(dG),ψ1(dG)whenIρ~g(x)<κ(dG).
Indeed, when Iρ~g(x)<κ(dG), we have δ=dG. Hence,
(80)ψ1(δ)κ(δ)Iρ~g(x)=ψ1(dG)×1κ(dG)Iρ~g(x)≤ψ1(dG).
When Iρ~g(x)≥κ(dG), we have δ=κ-1(Iρ~g(x)). Hence,
(81)ψ1(δ)κ(δ)Iρ~g(x)=ψ1(κ-1(Iρ~g(x)))Iρ~g(x)Iρ~g(x)=ψ1(κ-1(Iρ~g(x))).
Consequently our claim (79) is justified.
It follows from (76) and (79) that
(82)Iρg(x)≤(1+C2)max{ψ1(κ-1(Iρ~g(x))),ψ1(dG)}.
By (49), we obtain
(83)(ψ1∘κ-1)(s)≤CGΦ-1(s)forκ(dG)≤s<∞.
Hence, taking A≔CG(C1+1)(1+C2), we establish
(84)|Iρf(x)|A∥f∥L(1,φ)(G)≤Iρg(x)A≤max{ψ1(κ-1(Iρ~g(x))),ψ1(dG)}CG(C1+1)=max{ψ1(κ-1(Iρ~g(x))),ψ1(κ-1(κ(dG)))}CG(C1+1)≤max{Φ-1(Iρ~g(x)),Φ-1(κ(dG))}C1+1.
Since ρ~/ρ is decreasing and
(85)(ρ~(4k2dG)/ρ(4k2dG))ψ1(dG)=κ(dG),
we see that
(86)ψ(r)≥∫2k1dG4k2dG‍ρ~(t)φ(t)tdt≥ρ~(4k2dG)ρ(4k2dG)∫2k1dG4k2dG‍ρ(t)φ(t)tdt=κ(dG)
for all 0<r≤dG. Hence, with the aid of (74), we have
(87)1μ(B(z,4r))∫B(z,r)‍Φ(|Iρf(x)|A∥f∥L(1,φ)(G))dμ(x)≤1C1+1×1μ(B(z,4r))∫B(z,r)‍max{Iρ~g(x),κ(dG)}dμ(x)≤1C1+1×(1μ(B(z,4r))∫B(z,r)‍Iρ~g(x)dμ(x)mmmmmm+1μ(B(z,4r))∫B(z,r)‍κ(dG)dμ(x))≤1C1+1(C1ψ(r)+ψ(r))=ψ(r),
which proves (50).Proof of Theorem7.
Write(88)Iρf(x)-Iρf(z)=∫B(x,2d(x,z))‍ρ(d(x,y))μ(B(x,4d(x,y)))f(y)dμ(y)-∫B(x,2d(x,z))‍ρ(d(z,y))μ(B(z,4d(z,y)))f(y)dμ(y)+∫G∖B(x,2d(x,z))‍(ρ(d(x,y))μ(B(x,4d(x,y)))mmmmmmmmmmmm-ρ(d(z,y))μ(B(z,4d(z,y))))mmmmmmmmmm×f(y)dμ(y).
By (58), we have
(89)∫B(x,2d(x,z))‍ρ(d(x,y))μ(B(x,4d(x,y)))|f(y)|dμ(y)≤C1ψ(2d(x,z))∥f∥L(1,φ)(G),(90)∫B(x,2d(x,z))‍ρ(d(z,y))μ(B(z,4d(z,y)))|f(y)|dμ(y)≤∫B(z,3d(x,z))‍ρ(d(z,y))μ(B(z,4d(z,y)))|f(y)|dμ(y)≤C1′ψ(2d(x,z))∥f∥L(1,φ)(G)
for x,z∈G. On the other hand, we have by (52) and (59)
(91)∫G∖B(x,2d(x,z))‍|ρ(d(x,y))μ(B(x,4d(x,y)))-ρ(d(z,y))μ(B(z,4d(z,y)))|mmmmmm×|f(y)|dμ(y)≤Cρ′d(x,z)θ∫G∖B(x,2d(x,z))‍ρ(d(x,y))μ(B(x,4d(x,y)))d(x,y)θmmmmmmmmmmmmmml×|f(y)|dμ(y)≤C2d(x,z)θ(∫4k1d(x,z)4k2dG‍ρ(t)φ(t)t1+θdt)∥f∥L(1,φ)(G)≤C2ψ(2d(x,z))∥f∥L(1,φ)(G).
Now from (89), (90), and (91), we establish
(92)|Iρf(x)-Iρf(z)|≤Cψ(2d(x,z))∥f∥L(1,φ)(G)
for x,z∈G, as required.



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*Source: 984259-2013-04-11.xml*