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

Abstract

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

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 𝐼_𝛼 given by 𝐼_𝛼𝑓(𝑥)∶=_ℝ𝑛𝑓𝑦𝑥−𝑦^𝑛−𝛼𝑑𝑦. (1) The operator 𝐼_𝛼 is also called the fractional integral operator or the Riesz potential. We denote by 𝐵(𝑧,𝑟) the ball {𝑥∈ℝ^𝑛∶|𝑥−𝑧|<𝑟} with center 𝑧 and of radius 𝑟>0, and by |𝐵(𝑧,𝑟)| its Lebesgue measure, that is, |𝐵(𝑧,𝑟)|=𝜔_𝑛𝑟^𝑛, where 𝜔_𝑛 is the volume of the unit ball in ℝ^𝑛. Let 𝐺 be a bounded open subset of ℝ^𝑛. We denote its diameter by 𝑑_𝐺; 𝑑_𝐺=sup𝑥−𝑦∶𝑥,𝑦∈𝐺. (2)

For 𝑢∈𝐿¹(𝐺), we define the integral mean over 𝐵(𝑧,𝑟) by 𝑢_{𝐵(𝑧,𝑟)}=_{𝐵(𝑧,𝑟)}𝑢(𝑥)1𝑑𝑥∶=|𝐵(𝑧,𝑟)|_{𝐺∩𝐵(𝑧,𝑟)}𝑢(𝑥)𝑑𝑥. (3) 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∈𝐿^𝑝_loc(𝐺) for which there is a positive constant 𝐶 such that _{𝐵(𝑧,𝑟)}𝑓(𝑥)^𝑝𝑑𝑥≤𝐶^𝑝𝜑(𝑟)whenever𝑧∈𝐺,0<𝑟≤𝑑_𝐺. (4) 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 𝐿_𝑝,𝜆(𝐺).

A direct consequence of this notation is that 𝐿_{𝑝,𝜆𝑝}(𝐺)⊃𝐿^𝑛/𝜆(𝐺) (5) for 0<𝜆≤𝑛 and 𝑝∈[1,𝑛/𝜆).

Some prefer to use the notation 𝑓_{𝑀𝑝,𝑤(⋅)}=sup_𝑟>0𝑤(𝑟)𝑓_{𝐿𝑝(𝐵(𝑧,𝑟))} (6) with 𝑤(𝑟)=𝜔_𝑛𝑟^{𝑛−1/𝑝}𝜑(𝑟)^−1/𝑝 (7) references [2–5].

Much about the case 𝑝>1 is known. Recall that the Adams theorem about the boundedness of fractional integral operators [1, Theorem 3.1] asserts that 𝐼_𝛼𝑓_{𝐿𝑞,𝜆}≤𝐶𝑓_{𝐿𝑝,𝜆}, (8) provided the parameters 𝑝, 𝑞, 𝜆 satisfy 10<𝜆≤𝑛,1<𝑝<𝑞<∞,𝑞=1𝑝−𝛼𝜆. (9) See also research papers [2–4, 6–16] and a survey [5].

Meanwhile, only a few results are known for the case 𝑝=1. Trudinger [17, Theorem 1] proved that if 𝑓∈𝐿_1,1(𝐺)=𝐿¹(𝐺) then exp(𝑎|𝐼₁𝑓|)∈𝐿¹(𝐺) for some constant 𝑎>0; this implies that the operator 𝐼₁ is bounded from 𝐿_1,1(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 lim_𝑟↓0sup_{𝑥∈𝐑𝑛𝐵(𝑥,𝑟)}𝑉𝑦𝑥−𝑦^𝑛−𝛼𝑑𝑦 (10) 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 𝐿_1,𝜆(ℝ^𝑛) 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 𝐶⁻¹𝑀^♯𝑓_{𝐿𝑝,𝜆}+𝑓_𝐿1,𝜆≤𝑓_{𝐿𝑝,𝜆}≤𝐶𝑀^♯𝑓_{𝐿𝑝,𝜆}+𝑓_𝐿1,𝜆 (11) for any measurable function 𝑓, where 𝑀^♯𝑓(𝑥)=sup_{𝑦∈ℝ𝑛,𝑟>0}1𝐵×𝑦,𝑟_{𝐵𝑦,𝑟}𝑓(𝑧)−1𝐵𝑦,𝑟_{𝐵(𝑦,𝑟)}𝑓(𝑤)𝑑𝑤𝑑𝑧 (12) 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 sup_𝑗∈ℤ𝑆_𝑗𝑓_𝐿1,𝜆≤𝐶𝑓_𝐿1,𝜆, (13) where {𝑆_𝑗}^∞_𝑗=−∞ is a Littlewood-Paley patch. By choosing a smooth function 𝜑∈𝐶^∞(ℝ^𝑛) such that 𝜒_{𝐵(0,4)⧵𝐵(0,2)}≤𝜑≤𝜒_{𝐵(0,8)⧵𝐵(0,1)}, recall that we can define the 𝑗th Littlewood-Paley patch by 𝑆_𝑗𝑓(𝑥)∶=ℱ⁻¹𝜑2^−𝑗⋅⋅ℱ𝑓(𝑥) (14) for 𝑓∈𝒮^′(ℝ^𝑛). Note that (13) is a direct consequence of the translation invariance of the space 𝐿_1,𝜆(ℝ^𝑛). But this loss caused by (13) is quite big. Note that 𝑓_𝐿1,𝜆≤𝐶sup_𝑗∈ℤ𝑆_𝑗𝑓_𝐿1,𝜆 (15) fails. See the appendix for a proof. When 𝑝>1, an approach using the Littlewood-Paley patch is taken effectively [27]. Indeed, 𝐶⁻¹𝑓_{𝐿𝑝,𝜆}≤^∞_𝑗=−∞𝑆_𝑗𝑓^21/2_{𝐿𝑝,𝜆}≤𝐶𝑓_{𝐿𝑝,𝜆} (16) for all 𝑓∈𝐿_𝑝,𝜆(ℝ^𝑛). 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 when 𝑝=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 ℝ^𝑛. 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 ℝ^𝑛, 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 𝜇(𝐵(𝑥,2𝑟))≤𝐶𝜇(𝐵(𝑥,𝑟)) (17) 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 case 𝑝=1, the operator 𝐼_𝛼 and its generalization 𝐼_𝜌 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 𝐼_𝜌 is the following: let 𝜌 be a function from (0,∞) to itself and satisfy ^𝑟₀𝜌(𝑡)𝑡𝑑𝑡<+∞ (18) 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 𝐼_𝜌𝑓(𝑥)=_𝐺𝜌𝑑𝑥,𝑦𝜇𝐵𝑓𝑦𝑦,𝑥,4𝑑𝑥,𝑦𝑑𝜇 (19) where 𝑓∈𝐿¹(𝐺). Instead of using 𝐼^†_𝜌𝑓(𝑥)=_𝐺𝜌𝑑𝑥,𝑦𝜇𝐵𝑓𝑦𝑦,𝑥,𝑑𝑥,𝑦𝑑𝜇 (20) we discuss 𝐼_𝜌 defined above. This modification will be necessary in Lemma 9 for example. An example in [44, Section 2] shows that 𝐼^†_𝜌 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 𝐼_𝜌 of 𝜒_{𝐵(𝑥,𝑟)}, the indicator functions of the balls, belongs to 𝐿^𝑝,𝜑(𝐺) at least when 𝜇 is the Lebesgue measure. Indeed, if ^𝑟₀𝜌(𝑡)𝑡𝑑𝑡=∞ (21) for any sufficiently small 𝑟>0. Then, for 𝑦∈𝐵(𝑥,𝑟/2) such that 𝐵(𝑥,𝑟)⊂𝐺, we have 𝐼_𝜌𝜒_{𝐵(𝑥,𝑟)}𝑦=_{𝐵(𝑥,𝑟)}𝜌𝑦−𝑧𝐵≥𝑦,4𝑦−𝑧𝑑𝑧_{𝐵(𝑦,𝑟/2)}𝜌𝑦−𝑧𝐵y,4𝑦−𝑧𝑑𝑧=𝐶₀^𝑟/2𝜌(𝑡)𝑡𝑑𝑡=∞ (22) by using the spherical coordinate.

We organize the remaining part of the present paper as follows. In Section 2, 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

Let 𝒢 be the set of all continuous functions from (0,∞) to itself with the doubling condition, that is, there exists a constant 𝑐_𝜑≥1 such that 1𝑐_𝜑≤𝜑(𝑟)𝜑(𝑠)≤𝑐_𝜑for𝑟,𝑠>0with12≤𝑟𝑠≤2. (23) We call the smallest number 𝑐_𝜑 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: 𝐿^(1,𝜑)(𝐺)∶=𝑓∈𝐿¹_loc(𝐺)∶𝑓_{𝐿(1,𝜑)(𝐺)}<∞ (24) with the norm 𝑓_{𝐿(1,𝜑)(𝐺)}=sup_{𝑧∈𝐺,0<𝑟≤𝑑𝐺}1𝜑(𝑟)⋅1𝜇(𝐵(𝑧,2𝑟))_{𝐵(𝑧,𝑟)}𝑓(𝑥)𝑑𝜇(𝑥). (25) Then, a routine argument shows that 𝐿^(1,𝜑)(𝐺) is a Banach space. Due to the fact that ℝ^𝑛 is a geometrically doubling space, we can prove that 𝐶⁻¹𝑓_{𝐿(1,𝜑)(𝐺)}≤sup_{𝑧∈𝐺,0<𝑟≤𝑑𝐺}1𝜑(𝑟)⋅1𝜇(𝐵(𝑧,𝑘𝑟))_{𝐵(𝑧,𝑟)}𝑓(𝑥(𝑥)𝑓)𝑑𝜇≤𝐶_{𝐿(1,𝜑)(𝐺)} (26) 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,𝑑_𝐺), then 𝐿^(1,𝜑1)(𝐺)⊂𝐿^(1,𝜑2)(𝐺), (27) in particular, if there exists a constant 𝐶≥1 such that 𝐶⁻¹𝜑₁(𝑟)≤𝜑₂(𝑟)≤𝐶𝜑₁(𝑟) for all 𝑟>0, then 𝐿^(1,𝜑1)(𝐺)=𝐿^(1,𝜑2)(𝐺) (28) with equivalent norms. A ball testing shows the following.

Proposition 1. The function 𝜑₁/𝜑₂ is bounded above on (0,𝑑_𝐺) if 𝐿^(1,𝜑1)(𝐺)⊂𝐿^(1,𝜑2)(𝐺) when 𝜇=𝑑𝑥.

Here and below, we write 𝐴≲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 𝜒_{𝐵(𝑥0,𝑟)𝐿1)(1,𝜑(𝐺)}∼1𝜑₁(𝑟),𝜒_{𝐵(𝑥0,𝑟)𝐿2)(1,𝜑(𝐺)}∼1𝜑₂(𝑟). (29) 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 𝑓_{𝐿2)(1,𝜑(𝐺)}≤𝐶𝑓_{𝐿1)(1,𝜑(𝐺)}. (30) If we combine (29) and (30), then we obtain that 𝜑₁/𝜑₂ is bounded above on (0,𝑑_𝐺).

Let us consider the family 𝒴 of all continuous, increasing, convex, and bijective functions from [0,∞) to itself. For Φ∈𝒴, the Orlicz space 𝐿^Φ(𝐺) is defined by 𝐿^Φ(𝐺)∶=𝑓∈𝐿¹_loc(𝐺)∶𝑓_𝐿Φ(𝐺)<∞, (31) where 𝑓_𝐿Φ(𝐺)∶=inf𝜆>0∶_𝐺Φ𝑓(𝑥)𝜆𝑑𝜇(𝑥)≤1. (32) If Φ₁, Φ₂∈𝒴 are equivalent in the sense that there exists a constant 𝐶≥1 with Φ₁𝐶⁻¹𝑟≤Φ₂(𝑟)≤Φ₁(𝐶𝑟) (33) for all 𝑟>0, then we see easily that 𝐿^Φ1(𝐺)=𝐿^Φ2(𝐺) (34) with equivalent norms. If Φ(𝑟)=exp𝑟^𝑝,expexp𝑟^𝑝,𝑟^𝑝log𝑟^𝜆or𝑟^𝑝log𝑟^𝑞loglog𝑟^𝜆(𝑟>0) (35) for large 𝑟>0, then 𝐿^Φ(𝐺) will be denoted by exp𝐿^𝑝(𝐺),expexp𝐿^𝑝(𝐺),𝐿^𝑝log𝐿^𝜆(𝐺)or𝐿^𝑝log𝐿^𝑞loglog𝐿^𝜆(𝐺), (36) respectively.

For Φ∈𝒴 and 𝜑∈𝒢, the Orlicz-Morrey space 𝐿^(Φ,𝜑)(𝐺) is defined by 𝐿^(Φ,𝜑)(𝐺)∶=𝑓∈𝐿¹_loc(𝐺)∶𝑓_{𝐿(Φ,𝜑)(𝐺)}<∞, (37) where 𝑓_{𝐿(Φ,𝜑)(𝐺)}∶=sup_{𝑧∈𝐺,0<𝑟≤𝑑𝐺}1inf𝜆>0∶𝜇(𝐵(×𝑧,2𝑟))_{𝐵(𝑧,𝑟)}Φ𝑓(𝑥)𝜆𝑑𝜇(𝑥)≤𝜑(𝑟) (38) (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 𝐿^Φ is a special case of Orlicz-Morrey spaces when 𝜇=𝑑𝑥.

For 𝜑∈𝒢 such that 𝜑 is bounded, the generalized Hölder space is defined by Λ_𝜑(𝐺)=𝑓∶𝑓_Λ𝜑(𝐺)<∞, (39) where 𝑓_Λ𝜑(𝐺)=sup_{𝑥,𝑦∈𝐺,𝑥≠𝑦}𝑓(𝑥)−𝑓𝑦𝜑.2𝑑𝑥,𝑦 (40) Then, ‖𝑓‖_Λ𝜑(𝐺) is a norm modulo constants and thereby Λ_𝜑(𝐺) is a Banach space. Since 𝜑 is bounded, every 𝑓∈Λ_𝜑(𝐺) is bounded. If 𝜑(𝑟)→0 as 𝑟↓0, then every 𝑓∈Λ_𝜑(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 Section 5.

Throughout this paper, let 𝐺 be a bounded open set in 𝑋 and denote by 𝑐_𝜑, the doubling constant of 𝜑∈𝒢.

Let us begin with the following result, which is the one of Gunawan type [9].

Theorem 2. Let 𝜌∶(0,∞)→(0,∞) be a measurable function such that there exist 𝑘₁, 𝑘₂, 𝐶_𝜌 such that 0<16𝑘₁≤1≤𝑘₂<∞,sup_{𝑟/2≤𝑠≤𝑟}𝜌(𝑠)≤𝐶_𝜌^𝑘2𝑟_𝑘1𝑟𝜌(𝑠)𝑠𝑑𝑠(𝑟>0). (41) Let 𝜑∈𝒢, and define 𝜓(𝑟)∶=^4𝑘2𝑟₀𝜌(𝑡)𝑡𝑑𝑡𝜑(𝑟)+^{4𝑘2𝑑𝐺}_2𝑘1𝑟𝜌(𝑡)𝜑(𝑡)𝑡𝑑𝑡 (42) for 0<𝑟≤𝑑_𝐺. Then, there exists a constant 𝐶>0 such that 1𝜇(𝐵(𝑧,4𝑟))_{𝐵(𝑧,𝑟)}𝐼_𝜌𝑓(𝑥)𝑑𝜇(𝑥)≤𝐶𝜓(𝑟)𝑓_{𝐿(1,𝜑)(𝐺)} (43) for 𝑧∈𝐺, 0<𝑟≤𝑑_𝐺 and 𝑓∈𝐿^(1,𝜑)(𝐺), where 𝐶>0 is a constant depending only on 𝐶_𝜌, 𝑐_𝜑, 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 sup_{𝑧∈𝐺,0<𝑟≤𝑑𝐺}𝑟^𝜈𝜇(𝐵(𝑧,4𝑟))_{𝐵(𝑧,𝑟)}𝑓(𝑤)^𝑝𝑑𝜇(𝑤)^1/𝑝,sup_{𝑧∈𝐺,0<𝑟≤𝑑𝐺}𝑟^𝜈𝜇(𝐵(𝑧,2𝑟))_{𝐵(𝑧,𝑟)}𝑓(𝑤)^𝑝𝑑𝜇(𝑤)^1/𝑝 (44) 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 𝜌(𝑟)∼𝑟^𝛼(0<𝑟<1),𝜌(𝑟)≲𝑒^−𝑟(𝑟≥1). (45) 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. Theorem 2 is proved in [19] when 𝐺=ℝ^𝑛 and 𝜌∈𝒢. See also [21].

We now state a result for Orlicz-Morrey spaces.

Theorem 5. Let 𝜌,𝜌∶(0,∞)→(0,∞) be measurable functions such that there exist 𝑘₁, 𝑘₂, 𝐶_𝜌 such that 0<16𝑘₁≤1≤𝑘₂<∞ and that sup_{𝑟/2≤𝑠≤𝑟}𝜌(𝑠)≤𝐶_𝜌^𝑘2𝑟_𝑘1𝑟𝜌(𝑠)𝑠𝑑𝑠,sup_{𝑟/2≤𝑠≤𝑟}𝜌(𝑠)≤𝐶_𝜌^𝑘2𝑟_𝑘1𝑟𝜌(𝑠)𝑠𝑑𝑠(𝑟>0). (46) Let 𝜑∈𝒢. Assume ¹₀𝜌(𝑡)𝜑(𝑡)𝑡𝑑𝑡=∞ (47) and that 𝜌/𝜌 is continuous and decreasing.
Define 𝜓₁(𝑟)∶=^{4𝑘2𝑑𝐺}_2𝑘1𝑟𝜌(𝑡)𝜑(𝑡)𝑡𝜅(𝑟)𝜓𝑑𝑡,∶=¹(𝑟)𝜌4𝑘₂𝑟𝜌4𝑘₂𝑟,𝜓(𝑟)∶=^2𝑘2𝑟₀𝜌(𝑡)𝑡𝑑𝑡𝜑(𝑟)+^{4𝑘2𝑑𝐺}_2𝑘1𝑟𝜌(𝑡)𝜑(𝑡)𝑡𝑑𝑡 (48) for 0<𝑟≤𝑑_𝐺. If Φ∈𝒴 satisfies 𝐶_𝐺𝜓=sup¹∘𝜅⁻¹(𝑠)Φ⁻¹(𝑠)∶𝜅𝑑_𝐺≤𝑠<∞<∞, (49) then there exists a constant 𝐴>0 such that 1𝜇(𝐵(𝑧,4𝑟))_{𝐵(𝑧,𝑟)}Φ𝐼_𝜌𝑓(𝑥)𝐴𝑓_{𝐿(1,𝜑)(𝐺)}𝑑𝜇(𝑥)≤𝜓(𝑟) (50) for 𝑧∈𝐺, 0<𝑟≤𝑑_𝐺 and 𝑓∈L^(1,𝜑)(𝐺), where 𝐴>0 is a constant depending only on 𝐶_𝜌, 𝐶_𝜌, 𝑐_𝜑, 𝑘₁, 𝑘₂, and 𝐶_𝐺.

Remark 6. Note that 𝜅 is bijective from (0,𝑑_𝐺] to [𝜅(𝑑_𝐺),∞) by the assumptions in the theorem. Indeed, by the definition of 𝜅 above, 𝜅 is a decreasing function. In addition, lim_𝑟↓0𝜅(𝑟)=∞, showing that 𝜅∶(0,𝑑_𝐺]→[𝜅(𝑑_𝐺),∞) is bijective.

Finally, we shall show a result of Gunawan type about continuity.

Theorem 7. Let 𝜌∶(0,∞)→(0,∞) be a measurable function such that there exist 𝑘₁, 𝑘₂, 𝐶_𝜌 such that 0<16𝑘₁≤1≤𝑘₂<∞,sup_{𝑟/2≤𝑠≤𝑟}𝜌(𝑠)≤𝐶_𝜌^𝑘2𝑟_𝑘1𝑟𝜌(𝑠)𝑠𝑑𝑠(𝑟>0). (51) Let 𝜑∈𝒢. Assume the following condition on 𝜌. There are 0<𝜃≤1 and 𝐶^′_𝜌>0 such that 𝜌𝑑𝑥,𝑦𝜇𝐵−𝜌𝑑𝑥,4𝑑𝑥,𝑦𝑧,𝑦𝜇𝐵𝑧,4𝑑𝑧,𝑦≤𝐶^′_𝜌𝑑(𝑥,𝑧)𝑑(𝑥,𝑦)^𝜃𝜌𝑑𝑥,𝑦𝜇𝐵𝑥,4𝑑𝑥,𝑦 (52) whenever 𝑑(𝑥,𝑧)≤𝑑(𝑥,𝑦)/2. Assume in addition the Dini condition ¹₀𝜌(𝑡)𝜑(𝑡)𝑡𝑑𝑡<∞. (53) If 𝜓(𝑟)=^3𝑘2𝑟₀𝜌(𝑡)𝜑(𝑡)𝑡𝑑𝑡+𝑟^{𝜃4𝑘2𝑑𝐺}_2𝑘1𝑟𝜌(𝑡)𝜑(𝑡)𝑡^1+𝜃𝑑𝑡for0<𝑟≤𝑑_𝐺, (54) then 𝐼_𝜌 is bounded from 𝐿^(1,𝜑)(𝐺) to Λ_𝜓(𝐺). More precisely, 𝐼_𝜌𝑓_Λ𝜓(𝐺)≤𝐶𝑓_{𝐿(1,𝜑)(𝐺)}, (55) where 𝐶>0 is a constant depending only on 𝐶_𝜌, 𝐶^′_𝜌, 𝑐_𝜑, 𝑘₁, 𝑘₂, and 𝜃.

Note that if ∫¹₀(𝜌(𝑡)𝜑(𝑡)/𝑡)𝑑𝑡<∞ and 0<𝜃≤1, then 𝑟∈0,𝑑_𝐺⟼𝑟^{𝜃4𝑘2𝑑𝐺}_2𝑘1𝑟𝜌(𝑡)𝜑(𝑡)𝑡^1+𝜃𝑑𝑡∈[0,∞) (56) is bounded.

4. Preliminary Lemmas

Lemma 8. Let 𝜌∶(0,∞)→(0,∞) be a measurable function such that there exist 𝑘₁, 𝑘₂, 𝐶_𝜌 such that 0<16𝑘₁≤1≤𝑘₂<∞,sup_{𝑟/2≤𝑠≤𝑟}𝜌(𝑠)≤𝐶_𝜌^𝑘2𝑟_𝑘1𝑟𝜌(𝑠)𝑠𝑑𝑠(𝑟>0). (57) Let 𝜑∈𝒢. Then _{𝐵(𝑥,𝑟)}𝜌𝑑𝑥,𝑦𝜇𝐵𝑓𝑦𝑦𝑥,4𝑑𝑥,𝑦𝑑𝜇≤𝐶^2𝑘2𝑟₀𝜌(𝑡)𝜑(𝑡)𝑡𝑑𝑡𝑓_{𝐿(1,𝜑)(𝐺)}, (58) where 𝐶>0 is a constant depending only on 𝐶_𝜌, 𝑐_𝜑, 𝑘₁, and 𝑘₂.
Moreover, if 𝑘≥0, then _{𝐵(𝑥,𝑑𝐺)⧵𝐵(𝑥,𝑟)}𝜌𝑑𝑥,𝑦𝜇𝐵𝑥,4𝑑𝑥,𝑦𝑑𝑥,𝑦^𝑘𝑓𝑦𝑑𝜇𝑦≤𝐶^{4𝑘2𝑑𝐺}_2𝑘1𝑟𝜌(𝑡)𝜑(𝑡)𝑡^1+𝑘𝑑𝑡𝑓_{𝐿(1,𝜑)(𝐺)}, (59) where 𝐶>0 is a constant depending only on 𝐶_𝜌, 𝑐_𝜑, 𝑘₁, 𝑘₂, and 𝑘.

Proof. If 𝑦∈𝐵(𝑥,2^𝑗𝑟)⧵𝐵(𝑥,2^𝑗−1𝑟) and 𝑗∈ℤ, then a geometric observation shows 𝜌𝑑𝑥,𝑦𝜇𝐵𝑥,4𝑑𝑥,𝑦𝑑𝑥,𝑦^𝑘≤1𝜇𝐵𝑥,2^𝑗+1𝑟2^𝑗−1𝑟^𝑘sup_{2𝑗−1𝑟≤𝑠≤2𝑗𝑟}𝜌(𝑠)≤𝐶_𝜌𝜇𝐵𝑥,2^𝑗+1𝑟2^𝑗−1𝑟^{𝑘2𝑗𝑘2𝑟2𝑗𝑘1𝑟}𝜌(𝑠)𝑠𝑑𝑠. (60) Hence, _{𝐵(𝑥,2𝑗𝑟)⧵𝐵(𝑥,2𝑗−1𝑟)}𝜌𝑑𝑥,𝑦𝜇𝐵𝑥,4𝑑𝑥,𝑦𝑑𝑥,𝑦^𝑘𝑓𝑦𝑦≤𝐶𝑑𝜇_𝜌2^𝑗−1𝑟^{𝑘2𝑗𝑘2𝑟2𝑗𝑘1𝑟}𝜌(𝑠)𝑠×1𝑑𝑠𝜇𝐵𝑥,2^𝑗+1𝑟_𝐵(𝑥,2^𝑗_𝑟)𝑓𝑦𝑑𝜇𝑦=𝐶_𝜌𝜑2^𝑗𝑟2^𝑗−1𝑟^{𝑘2𝑗𝑘2𝑟2𝑗𝑘1𝑟}𝜌(𝑠)𝑠×1𝑑𝑠𝜑2^𝑗𝑟𝜇𝐵𝑥,2^𝑗+1𝑟_𝐵(𝑥,2^𝑗_𝑟)𝑓𝑦𝑑𝜇𝑦≤𝐶_𝜌𝜑2^𝑗𝑟2^𝑗−1𝑟^{𝑘2𝑗𝑘2𝑟2𝑗𝑘1𝑟}𝜌(𝑠)𝑠𝑑𝑠×𝑓_{𝐿(1,𝜑)(𝐺)}. (61) Set 𝑑∶=[1+log₂(𝑘₂/𝑘₁)]. Then, by virtue of the doubling condition on 𝜑, we have 𝜑2^𝑗𝑟2^𝑗−1𝑟^{𝑘2𝑗𝑘2𝑟2𝑗𝑘1𝑟}𝜌(𝑠)𝑠𝑑𝑠≤2^𝑘𝜑2^𝑗𝑟2^𝑗𝑟^{𝑘2𝑗+𝑑𝑘1𝑟2𝑗𝑘1𝑟}𝜌(𝑠)𝑠𝑑𝑠=2^𝑘𝑑_𝑙=1𝜑2^𝑗𝑟2^𝑗𝑟^{𝑘2𝑗+𝑙𝑘1𝑟2𝑗+𝑙−1𝑘1𝑟}𝜌(𝑠)𝑠𝑑𝑠≤𝐶₁2^𝑑+1𝑘₁^𝑘𝑑_𝑙=1^{2𝑗+𝑙𝑘1𝑟}_{2𝑗+𝑙−1𝑘1𝑟}𝜑(𝑠)2^𝑗+𝑑𝑘₁𝑟^𝑘⋅𝜌(𝑠)𝑠𝑑𝑠≤𝐶₁2^𝑑+1𝑘₁^𝑘𝑑_𝑙=1^{2𝑗+𝑙𝑘1𝑟}_{2𝑗+𝑙−1𝑘1𝑟}𝜑(𝑠)𝜌(𝑠)𝑠^1+𝑘𝑑𝑠=𝐶₁2^𝑑+1𝑘₁^{𝑘2𝑗+𝑑𝑘1𝑟}_{2𝑗𝑘1𝑟}𝜑(𝑠)𝜌(𝑠)𝑠^1+𝑘𝑑𝑠, (62) where 𝐶₁>0 is a constant depending only on 𝑐_𝜑, 𝑘₁, and 𝑘₂.
Consequently, since 𝜇({𝑥})=0, _{𝐵(𝑥,𝑟)}𝜌𝑑𝑥,𝑦𝜇𝐵𝑓𝑦𝑦=𝑥,4𝑑𝑥,𝑦𝑑𝜇_{∞𝑗=0𝐵(𝑥,2−𝑗𝑟)⧵𝐵(𝑥,2−𝑗−1𝑟)}𝜌𝑑𝑥,𝑦𝜇𝐵𝑓𝑦𝑦≤𝑥,4𝑑𝑥,𝑦𝑑𝜇_∞𝑗=0𝐶_𝜌𝐶₁^{2−𝑗+𝑑𝑘1𝑟}_{2−𝑗𝑘1𝑟}𝜑(𝑠)𝜌(𝑠)𝑠𝑑𝑠×𝑓_{𝐿(1,𝜑)(𝐺)}≤𝐶_𝜌𝐶₁^2𝑘2𝑟₀𝜌(𝑡)𝜑(𝑡)𝑡𝑑𝑡𝑓_{𝐿(1,𝜑)(𝐺)}, (63) which proves (58).
We choose 𝑗₀∈ℤ, so that 𝑑_𝐺≤2^𝑗0𝑟<2𝑑_𝐺. Then, we have _{𝐵(𝑥,𝑑𝐺)⧵𝐵(𝑥,𝑟)}𝜌𝑑𝑥,𝑦𝜇𝐵𝑥,4𝑑𝑥,𝑦𝑑𝑥,𝑦^𝑘𝑓𝑦𝑦≤𝑑𝜇_{𝑗0𝑗=1𝐵(𝑥,2𝑗𝑟)⧵𝐵(𝑥,2𝑗−1𝑟)}𝜌𝑑𝑥,𝑦𝜇𝐵𝑥,4𝑑𝑥,𝑦𝑑𝑥,𝑦^𝑘×𝑓𝑦𝑑𝜇𝑦≤𝐶_𝜌𝐶₁2^𝑑+1𝑘₁^𝑘𝑗0_𝑗=1^{2𝑗+𝑑𝑘1𝑟}_{2𝑗𝑘1𝑟}𝜑(𝑠)𝜌(𝑠)𝑠^1+𝑘𝑑𝑠×𝑓_{𝐿(1,𝜑)(𝐺)}≤𝐶_𝜌𝐶₁2^𝑑+1𝑘₁^{𝑘4𝑘2𝑑𝐺}_2𝑘1𝑟𝜌(𝑡)𝜑(𝑡)𝑡^1+𝑘𝑑𝑡𝑓_{𝐿(1,𝜑)(𝐺)}. (64) Thus, since 𝑘, 𝑘₁, 𝑑 being constants, (59) follows.

Lemma 9. Let 𝜌∶(0,∞)→(0,∞) be a measurable function such that there exist 𝑘₁, 𝑘₂, 𝐶_𝜌 such that 0<16𝑘₁≤1≤𝑘₂<∞, (65) and that sup_{𝑟/2≤𝑠≤𝑟}𝜌(𝑠)≤𝐶_𝜌^𝑘2𝑟_𝑘1𝑟𝜌(𝑠)𝑠𝑑𝑠(𝑟>0). (66) Then, for all 𝑓∈𝐿^(1,𝜑)(𝐺), 1𝜇(𝐵(×𝑧,2𝑟))_{𝐵(𝑧,𝑟)𝐵(𝑧,𝑟)}𝜌𝑑𝑥,𝑦𝜇𝐵𝑓𝑦𝑦(𝑥)(𝑟𝑥,4𝑑𝑥,𝑦𝑑𝜇𝑑𝜇≤𝐶𝜑)^2𝑘2𝑟₀𝜌(𝑡)𝑡𝑑𝑡𝑓_{𝐿(1,𝜑)(𝐺)}. (67)

Proof. By Fubini's theorem and the dyadic decomposition of the ball, we have _{𝐵(𝑧,𝑟)𝐵(𝑧,𝑟)}𝜌𝑑𝑥,𝑦𝜇𝐵𝑓𝑦𝑦(𝑥)=𝑥,4𝑑𝑥,𝑦𝑑𝜇𝑑𝜇_{𝐵(𝑧,𝑟)}𝑓𝑦×_{𝐵(𝑧,𝑟)}𝜌𝑑𝑥,𝑦𝜇𝐵(𝑥𝑦≤𝑥,4𝑑𝑥,𝑦𝑑𝜇)𝑑𝜇_{𝐵(𝑧,𝑟)}𝑓𝑦×^∞_{𝑗=0𝐵(𝑦,2−𝑗+1𝑟)⧵𝐵(𝑦,2−𝑗𝑟)}sup_{2−𝑗𝑟≤𝑠≤2−𝑗+1𝑟}𝜌(𝑠)𝜇𝐵𝑥,2^−𝑗+2𝑟(𝑥)𝑦≤×𝑑𝜇𝑑𝜇_{𝐵(𝑧,𝑟)}𝑓𝑦×^∞_{𝑗=0𝐵(𝑦,2−𝑗+1𝑟))}sup_{2−𝑗𝑟≤𝑠≤2−𝑗+1𝑟}𝜌(𝑠)𝜇𝐵𝑦,2^−𝑗+1𝑟×𝑑𝜇(𝑥)𝑑𝜇𝑦. (68) Since 𝜌 satisfies (66), we have _{𝐵(𝑧,𝑟)𝐵(𝑧,𝑟)}𝜌𝑑𝑥,𝑦𝜇𝐵𝑓𝑦𝑦(𝑥)𝑥,4𝑑𝑥,𝑦𝑑𝜇𝑑𝜇≤𝐶_{𝜌𝐵(𝑧,𝑟)}𝑓𝑦^∞_𝑗=0^{2−𝑗+1𝑘2𝑟}_{2−𝑗+1𝑘1𝑟}𝜌(𝑠)𝑠𝑑𝑡𝑑𝜇𝑦≤𝐶_𝜌^2𝑘2𝑟₀𝜌(𝑡)𝑡𝑑𝑡_{𝐵(𝑧,𝑟)}𝑓𝑦𝑑𝜇𝑦≤𝐶_𝜌𝜑(𝑟)𝜇(𝐵(𝑧,2𝑟))^2𝑘2𝑟₀𝜌(𝑡)𝑡𝑑𝑡𝑓_{𝐿(1,𝜑)(𝐺)}, (69) as required.

5. Proofs of the Theorems

We are now ready to prove our theorems.

Proof of Theorem 2. Let 𝑧∈𝐺 and 𝑟∈(0,𝑑_𝐺]. By the positivity of the kernel, we may assume that 𝑓≥0. We write 1𝜇(𝐵(𝑧,4𝑟))_{𝐵(𝑧,𝑟)}𝐼_𝜌𝑓(𝑥)(𝑥)≤1𝑑𝜇𝜇(𝐵(×𝑧,4𝑟))_{𝐵(𝑧,𝑟)𝐵(𝑥,𝑟)}𝜌𝑑𝑥,𝑦𝜇𝐵𝑦𝑦(𝑥)+1𝑥,4𝑑𝑥,𝑦×𝑓𝑑𝜇𝑑𝜇𝜇(𝐵(×𝑧,4𝑟))_{𝐵(𝑧,𝑟)𝐵(𝑥,𝑑𝐺)⧵𝐵(𝑥,𝑟)}𝜌𝑑𝑥,𝑦𝜇𝐵𝑦𝑦(𝑥)≤1𝑥,4𝑑𝑥,𝑦×𝑓𝑑𝜇𝑑𝜇𝜇(𝐵(×𝑧,4𝑟))_{B(𝑧,2𝑟)𝐵(𝑧,2𝑟)}𝜌𝑑𝑥,𝑦𝜇𝐵𝑦𝑦(𝑥)+1𝑥,4𝑑𝑥,𝑦×𝑓𝑑𝜇𝑑𝜇𝜇(𝐵(𝑧,4𝑟))×_{𝐵(𝑧,𝑟)𝐵(𝑥,𝑑𝐺)⧵𝐵(𝑥,𝑟)}𝜌𝑑𝑥,𝑦𝜇𝐵𝑦𝑦(𝑥)𝑥,4𝑑𝑥,𝑦×𝑓𝑑𝜇𝑑𝜇=𝐼₁+𝐼₂ (70) for 𝑧∈𝐺 and 0<𝑟≤𝑑_𝐺. By Lemma 9, we have 𝐼₁≤𝐶₁𝜑(𝑟)^4𝑘2𝑟₀𝜌(𝑡)𝑡𝑑𝑡𝑓_{𝐿(1,𝜑)(𝐺)}≤𝐶₁𝜓(𝑟)𝑓_{𝐿(1,𝜑)(𝐺)}. (71) Meanwhile, by Lemma 8 we have 𝐼₂≤𝐶₂^{4𝑘2𝑑𝐺}_2𝑘1𝑟𝜌(𝑡)𝜑(𝑡)𝑡𝑑𝑡𝑓_{𝐿(1,𝜑)(𝐺)}≤𝐶₂𝜓(𝑟)𝑓_{𝐿(1,𝜑)(𝐺)}. (72) Hence, it follows from (71) and (72) that 1𝜇(𝐵(𝑧,4𝑟))_{𝐵(𝑧,𝑟)}𝐼_𝜌𝑓(𝑥)𝑑𝜇(𝑥)≤𝐶𝜓(𝑟)𝑓_{𝐿(1,𝜑)(𝐺)}, (73) where 𝐶>0 depends only on 𝐶_𝜌, 𝑐_𝜑, 𝑘₁, and 𝑘₂.

Proof of Theorem 5. By Theorem 2, we have 1𝜇(𝐵(𝑧,4𝑟))_{𝐵(𝑧,𝑟)}𝐼_𝜌𝑓(𝑥)𝑑𝜇(𝑥)≤𝐶₁𝜓(𝑟)𝑓_{𝐿(1,𝜑)(𝐺)} (74) for 𝑧∈𝐺 and 0<𝑟≤𝑑_𝐺.
Let 𝑔∶=|𝑓|/‖𝑓‖_{𝐿(1,𝜑)(𝐺)}. For 𝑥∈𝐺 and 0<𝛿≤𝑑_𝐺, since 𝜌/𝜌 is decreasing, we have by Lemma 8𝐼_𝜌𝑔(𝑥)=_{𝐵(𝑥,𝛿)}𝜌𝑑𝑥,𝑦𝜇𝐵𝑔𝑦𝑦+𝑥,4𝑑𝑥,𝑦𝑑𝜇_{𝐵(𝑥,𝑑𝐺)⧵𝐵(𝑥,𝛿)}𝜌𝑑𝑥,𝑦𝜇𝐵𝑔𝑦𝑦≤𝜌(𝛿)𝑥,4𝑑𝑥,𝑦𝑑𝜇𝜌(𝛿)_{𝐵(𝑥,𝛿)}𝜌𝑑𝑥,𝑦𝜇𝐵𝑔𝑦𝑦𝑥,4𝑑𝑥,𝑦𝑑𝜇+𝐶₂^{4𝑘2𝑑𝐺}_2𝑘1𝛿𝜌(𝑡)𝜑(𝑡)𝑡≤𝜌(𝛿)𝑑𝑡𝜌(𝛿)𝐼_𝜌𝑔(𝑥)+𝐶₂𝜓₁(𝛿)≤𝜌4𝑘₂𝛿𝜌4𝑘₂𝛿𝐼_𝜌𝑔(𝑥)+𝐶₂𝜓₁(𝛿). (75) Hence, in view of the definition of 𝜅, we have 𝐼_𝜌𝑔(𝑥)≤𝜓₁(𝛿)𝜅(𝛿)𝐼_𝜌𝑔(𝑥)+𝐶₂𝜓₁(𝛿). (76) Now let 𝜅𝛿∶=⁻¹𝐼_𝜌𝑔(𝑥)when𝐼_𝜌𝑔(𝑥)≥𝜅𝑑_𝐺,𝑑_𝐺when𝐼_𝜌𝑔(𝑥)<𝜅𝑑_𝐺. (77) Observe that 𝜓₁(𝛿)=𝜓₁𝜅⁻¹𝐼_𝜌𝑔(𝑥)when𝐼_𝜌𝑔(𝑥)≥𝜅𝑑_𝐺,𝜓₁𝑑_𝐺when𝐼_𝜌𝑔(𝑥)<𝜅𝑑_𝐺, (78) by definition.
We claim that 𝜓₁(𝛿)𝜅(𝛿)𝐼_𝜌𝑔(𝑥)≤𝜓₁𝜅⁻¹𝐼_𝜌𝑔(𝑥)when𝐼_𝜌𝑔(𝑥)≥𝜅𝑑_𝐺,𝜓₁𝑑_𝐺when𝐼_𝜌𝑔(𝑥)<𝜅𝑑_𝐺. (79) Indeed, when 𝐼_𝜌𝑔(𝑥)<𝜅(𝑑_𝐺), we have 𝛿=𝑑_𝐺. Hence, 𝜓₁(𝛿)𝜅(𝛿)𝐼_𝜌𝑔(𝑥)=𝜓₁𝑑_𝐺×1𝜅𝑑_𝐺𝐼_𝜌𝑔(𝑥)≤𝜓₁𝑑_𝐺. (80) When 𝐼_𝜌𝑔(𝑥)≥𝜅(𝑑_𝐺), we have 𝛿=𝜅⁻¹(𝐼_𝜌𝑔(𝑥)). Hence, 𝜓₁(𝛿)𝜅(𝛿)𝐼_𝜌𝑔(𝑥)=𝜓₁𝜅⁻¹𝐼_𝜌𝑔(𝑥)𝐼_𝜌𝑔(𝑥)𝐼_𝜌𝑔(𝑥)=𝜓₁𝜅⁻¹𝐼_𝜌𝑔(𝑥). (81) Consequently our claim (79) is justified.
It follows from (76) and (79) that 𝐼_𝜌𝑔(𝑥)≤1+𝐶₂max𝜓₁𝜅⁻¹𝐼_𝜌𝑔(𝑥),𝜓₁𝑑_𝐺. (82) By (49), we obtain 𝜓₁∘𝜅⁻¹(𝑠)≤𝐶_𝐺Φ⁻¹(𝑠)for𝜅𝑑_𝐺≤𝑠<∞. (83) Hence, taking 𝐴∶=𝐶_𝐺(𝐶₁+1)(1+𝐶₂), we establish 𝐼_𝜌𝑓(𝑥)𝐴𝑓_{𝐿(1,𝜑)(𝐺)}≤𝐼_𝜌𝑔(𝑥)𝐴≤max𝜓₁𝜅⁻¹𝐼_𝜌𝑔(𝑥),𝜓₁𝑑_𝐺𝐶_𝐺𝐶₁=𝜓+1max₁𝜅⁻¹𝐼_𝜌𝑔(𝑥),𝜓₁𝜅⁻¹𝜅𝑑_𝐺𝐶_𝐺𝐶₁≤Φ+1max⁻¹𝐼_𝜌𝑔(𝑥),Φ⁻¹𝜅𝑑_𝐺𝐶₁.+1 (84) Since 𝜌/𝜌 is decreasing and 𝜌4𝑘₂𝑑_𝐺/𝜌4𝑘₂𝑑_𝐺𝜓₁𝑑_𝐺=𝜅𝑑_𝐺, (85) we see that 𝜓(𝑟)≥^{4𝑘2𝑑𝐺}_{2𝑘1𝑑𝐺}𝜌(𝑡)𝜑(𝑡)𝑡≥𝜌𝑑𝑡4𝑘₂𝑑_𝐺𝜌4𝑘₂𝑑_𝐺^{4𝑘2𝑑𝐺}_{2𝑘1𝑑𝐺}𝜌(𝑡)𝜑(𝑡)𝑡𝑑𝑑𝑡=𝜅_𝐺 (86) for all 0<𝑟≤𝑑_𝐺. Hence, with the aid of (74), we have 1𝜇(𝐵(𝑧,4𝑟))_{𝐵(𝑧,𝑟)}Φ𝐼_𝜌𝑓(𝑥)𝐴𝑓_{𝐿(1,𝜑)(𝐺)}(𝑥)≤1𝑑𝜇𝐶₁×1+1𝜇(𝐵(𝑧,4𝑟))_{𝐵(𝑧,𝑟)}max𝐼_𝜌𝑔(𝑥),𝜅𝑑_𝐺(𝑥)≤1𝑑𝜇𝐶₁×1+1𝜇(𝐵(𝑧,4𝑟))_{𝐵(𝑧,𝑟)}𝐼_𝜌𝑔(𝑥)(𝑥)+1𝑑𝜇𝜇(𝐵(𝑧,4𝑟))_{𝐵(𝑧,𝑟)}𝜅𝑑_𝐺(𝑥)≤1𝑑𝜇𝐶₁𝐶+1¹𝜓(𝑟)+𝜓(𝑟)=𝜓(𝑟), (87) which proves (50).

Proof of Theorem 7. Write 𝐼_𝜌𝑓(𝑥)−I_𝜌𝑓(𝑧)=_{𝐵(𝑥,2𝑑(𝑥,𝑧))}𝜌𝑑𝑥,𝑦𝜇𝐵𝑓𝑦𝑦−𝑥,4𝑑𝑥,𝑦𝑑𝜇_{𝐵(𝑥,2𝑑(𝑥,𝑧))}𝜌𝑑𝑧,𝑦𝜇𝐵𝑓𝑦𝑦+𝑧,4𝑑𝑧,𝑦𝑑𝜇_{𝐺⧵𝐵(𝑥,2𝑑(𝑥,𝑧))}𝜌𝑑𝑥,𝑦𝜇𝐵−𝜌𝑑𝑥,4𝑑𝑥,𝑦𝑧,𝑦𝜇𝐵𝑦𝑦.𝑧,4𝑑𝑧,𝑦×𝑓𝑑𝜇 (88) By (58), we have _{𝐵(𝑥,2𝑑(𝑥,𝑧))}𝜌𝑑𝑥,𝑦𝜇𝐵𝑓𝑦𝑦𝑥,4𝑑𝑥,𝑦𝑑𝜇≤𝐶₁𝜓(2𝑑(𝑥,𝑧))𝑓_{𝐿(1,𝜑)(𝐺)},_{𝐵(𝑥,2𝑑(𝑥,𝑧))}𝜌𝑑𝑧,𝑦𝜇𝐵𝑓𝑦𝑦≤𝑧,4𝑑𝑧,𝑦𝑑𝜇_{𝐵(𝑧,3𝑑(𝑥,𝑧))}𝜌𝑑𝑧,𝑦𝜇𝐵𝑓𝑦𝑦𝑧,4𝑑𝑧,𝑦𝑑𝜇≤𝐶^′₁𝜓(2𝑑(𝑥,𝑧))𝑓_{𝐿(1,𝜑)(𝐺)} (89)(90) for 𝑥,𝑧∈𝐺. On the other hand, we have by (52) and (59) _{𝐺⧵𝐵(𝑥,2𝑑(𝑥,𝑧))}𝜌𝑑𝑥,𝑦𝜇𝐵−𝜌𝑑𝑥,4𝑑𝑥,𝑦𝑧,𝑦𝜇𝐵×𝑓𝑦𝑦𝑧,4𝑑𝑧,𝑦𝑑𝜇≤𝐶^′_𝜌𝑑(𝑥,𝑧)^𝜃_{𝐺⧵𝐵(𝑥,2𝑑(𝑥,𝑧))}𝜌𝑑𝑥,𝑦𝜇𝐵𝑥,4𝑑𝑥,𝑦𝑑𝑥,𝑦^𝜃×𝑓𝑦𝑑𝜇𝑦≤𝐶₂𝑑(𝑥,𝑧)^{𝜃4𝑘2𝑑𝐺}_{4𝑘1𝑑(𝑥,𝑧)}𝜌(𝑡)𝜑(𝑡)𝑡^1+𝜃𝑑𝑡𝑓_{𝐿(1,𝜑)(𝐺)}≤𝐶₂𝜓(2𝑑(𝑥,𝑧))𝑓_{𝐿(1,𝜑)(𝐺)}. (91) Now from (89), (90), and (91), we establish 𝐼_𝜌𝑓(𝑥)−𝐼_𝜌𝑓(𝑧)≤𝐶𝜓(2𝑑(𝑥,𝑧))𝑓_{𝐿(1,𝜑)(𝐺)} (92) for 𝑥,𝑧∈𝐺, as required.

Appendix

A. Disproof of (15)

Inequality (15) can be disproved in terms of Besov spaces and Triebel-Lizorkin spaces. Let 𝜓∈𝐶^∞(ℝ^𝑛) satisfy 𝜒_𝐵(0,4)≤𝜓≤𝜒_𝐵(0,8). (A.1) Define 𝑇₀𝑓∶=ℱ⁻¹[𝜓⋅ℱ𝑓] for 𝑓∈𝒮^′(ℝ^𝑛). For parameters 𝑝∈(0,∞) and 𝑞∈(0,∞) and for 𝑓∈𝒮^′(ℝ^𝑛), the Besov norm ‖⋅‖_𝐵^𝑠_𝑝,𝑞 and the Triebel-Lizorkin norm ‖⋅‖_𝐹^𝑠_𝑝,𝑞 are defined by 𝑓_{𝐵𝑠𝑝,𝑞}∶=𝑇₀𝑓_𝐿𝑝+^∞_𝑗=1(2^𝑗𝑠𝑆_𝑗𝑓_𝐿𝑝)^𝑞1/𝑞,𝑓_{𝐹𝑠𝑝,𝑞}∶=𝑇₀𝑓_𝐿𝑝+^∞_𝑗=12^𝑗𝑠𝑆_𝑗𝑓^𝑞1/𝑞_𝐿𝑝, (A.2)(A.3) respectively, and for 𝑝∈(0,∞) and for 𝑓∈𝒮^′(ℝ^𝑛), the Besov norm ‖⋅‖_𝐵^𝑠_𝑝,∞ and the Triebel-Lizorkin norm ‖⋅‖_𝐹^𝑠_𝑝,∞ are defined by 𝑓_{𝐵𝑠𝑝,∞}∶=𝑇₀𝑓_𝐿𝑝+sup_𝑗∈ℕ2^𝑗𝑠𝑆_𝑗𝑓_𝐿𝑝,𝑓_{𝐹𝑠𝑝,∞}∶=𝑇₀𝑓_𝐿𝑝+sup_𝑗∈ℕ2^𝑗𝑠𝑆_𝑗𝑓_𝐿𝑝. (A.4) Meanwhile, by denoting 𝒫(ℝ^𝑛) the set of all polynomials, for parameters 𝑝∈(1,∞) and 𝑞∈(1,∞) and for a distribution 𝑓∈𝒮^′(ℝ^𝑛), the homogeneous Besov norm ‖⋅‖_̇𝐵^𝑠_𝑝,𝑞 and the homogeneous Triebel-Lizorkin norm ‖⋅‖_̇𝐹^𝑠_𝑝,𝑞 are defined by 𝑓_{̇𝐵𝑠𝑝,𝑞}∶=^∞_𝑗=−∞2^𝑗𝑠𝑆_𝑗𝑓_𝐿𝑝^𝑞1/𝑞,𝑓_{̇𝐹𝑠𝑝,𝑞}∶=^∞_𝑗=−∞2^𝑗𝑠𝑆_𝑗𝑓^𝑞1/𝑞_𝐿𝑝, (A.5)(A.6) respectively. Also, for 𝑝∈(1,∞) and 𝑓∈𝒮^′(ℝ^𝑛), one defines 𝑓_{̇𝐵𝑠𝑝,∞}∶=sup_𝑗∈ℕ2^𝑗𝑠𝑆_𝑗𝑓_𝐿𝑝,𝑓_{̇𝐹𝑠𝑝,∞}∶=sup_𝑗∈ℕ2^𝑗𝑠𝑆_𝑗𝑓_𝐿𝑝, (A.7)(A.8) respectively.

It follows from (A.7) and (A.8) that 𝑓_{̇𝐹𝑠𝑝,∞}≥𝑓_{̇𝐵𝑠𝑝,∞}𝑓∈𝒮^′ℝ^𝑛. (A.9)

Let 0<𝑝, 𝑞≤∞, and 𝑠∈ℝ. The inhomogeneous Besov space 𝐵^𝑠_𝑝,𝑞(ℝ^𝑛) (resp. the homogeneous Besov space 𝐵̇^𝑠_𝑝,𝑞(ℝ^𝑛)) is defined to be the set of all 𝑓∈𝒮^′(ℝ^𝑛) (resp. 𝑓∈𝒮^′(ℝ^𝑛)/𝒫(ℝ^𝑛)) for which the norm ‖𝑓‖_𝐵^𝑠_𝑝,𝑞 (resp. ‖𝑓‖_̇𝐵^𝑠_𝑝,𝑞) is finite, when 0<𝑝, 𝑞≤∞. Likewise, for 0<𝑝<∞, 0<𝑞≤∞ and 𝑠∈ℝ, the inhomogeneous Triebel-Lizorkin space 𝐹^𝑠_𝑝,𝑞(ℝ^𝑛) (resp. the homogeneous Triebel-Lizorkin space 𝐹̇^𝑠_𝑝,𝑞(ℝ^𝑛)) is defined to be the set of all 𝑓∈𝒮^′(ℝ^𝑛) (resp. 𝑓∈𝒮^′(ℝ^𝑛)/𝒫(ℝ^𝑛)) for which the norm ‖𝑓‖_𝐹^𝑠_𝑝,𝑞 (resp. ‖𝑓‖_̇𝐹^𝑠_𝑝,𝑞) is finite. To simplify the matters, even when we consider representatives in the function spaces 𝐵̇^𝑠_𝑝,𝑞(ℝ^𝑛) and 𝐹̇^𝑠_𝑝,𝑞(ℝ^𝑛), we forget that they are in equivalence classes, and we regard the function spaces 𝐵̇^𝑠_𝑝,𝑞(ℝ^𝑛) and 𝐹̇^𝑠_𝑝,𝑞(ℝ^𝑛) as subspaces of 𝒮^′(ℝ^𝑛).

Keeping this in mind, let us disprove (15). We have sup_𝑗∈ℤ𝑆_𝑗𝑓_𝐿1,𝜆≤𝑐sup_𝑗∈ℤ𝑆_𝑗𝑓_{𝐿𝑛/𝜆}=𝑐𝑓_{̇𝐵0𝑛/𝜆,∞}≤𝑐𝑓_{̇𝐹0𝑛/𝜆,∞} (A.10) from (5), (A.7), and (A.9).

However, according to [55, Theorem 11.2, (i), (11.2)], there exists 𝑓∈𝐹⁰_{𝑛/𝜆,∞}(ℝ^𝑛) such that it is not represented by 𝐿¹_loc(ℝ^𝑛)-functions: 𝑓∈𝐹⁰_{𝑛/𝜆,∞}ℝ^𝑛⧵𝐿¹_locℝ^𝑛. (A.11) If we consider ℱ⁻¹[(1−𝜓)⋅ℱ𝑓], where 𝜓∈𝐶^∞_c(ℝ^𝑛) is from (A.1), we can arrange that 𝑓∈𝐹⁰_{𝑛/𝜆,∞}(ℝ^𝑛) can be chosen so that supp(ℱ𝑓)∩𝐵(0,4)=∅. Indeed, ℱ⁻¹𝜓⋅ℱ𝑓∈𝐶^∞ℝ^𝑛. (A.12)

We suppose that the Fourier support of 𝑓 is away from 𝐵(0,4). Let us admit that 𝑓_{̇𝐹0𝑛/𝜆,∞}≤𝐶𝑓_{𝐹0𝑛/𝜆,∞}𝑓∈𝐹⁰_{𝑛/𝜆,∞}ℝ^𝑛 (A.13) under the understanding 𝐹̇⁰_{𝑛/𝜆,∞}(ℝ^𝑛)⊂𝒮^′(ℝ^𝑛). Note also that 𝐿_1,𝜆(ℝ^𝑛) is a subset of 𝐿¹_loc(ℝ^𝑛), hence our observation can be summarized as follows: 𝐿_1,𝜆ℝ^𝑛⊂𝐿¹_locℝ^𝑛,𝑓∈𝐹̇⁰_{𝑛/𝜆,∞}ℝ^𝑛⧵𝐿¹_locℝ^𝑛. (A.14) It then follows immediately that (15) fails since (15) implies 𝐿_1,𝜆ℝ^𝑛⊃𝐹̇⁰_{𝑛/𝜆,∞}ℝ^𝑛. (A.15) Inclusions (A.14) and (A.15) contradict obviously.

It remains to prove (A.13). Note that the frequency support of 𝑓 does not intersect with 𝐵(0,4). Observe also that 𝜑(2^−𝑗⋅) has the frequency support in 𝐵(0,4). Thus, we have 𝑆_𝑗𝑓=ℱ⁻¹[𝜑(2^−𝑗⋅)ℱ𝑓]=0(𝑗≤−1), and hence 𝑓_{̇𝐹0𝑛/𝜆,∞}=sup_𝑗∈ℤ𝑆_𝑗𝑓_{𝐿𝑛/𝜆}=sup_{𝑗∈ℕ∪{0}}𝑆_𝑗𝑓_{𝐿𝑛/𝜆}≤𝑓_{𝐹0𝑛/𝜆,∞}+𝑆₀𝑓_{𝐿𝑛/𝜆}. (A.16) Define 𝑊𝑓∶=ℱ⁻¹𝜑Ψ⋅ℱ𝑓,Ψ∶=𝜓+𝜑2⁻¹⋅. (A.17) In view of the size of frequency support, we conclude 𝑆₀𝑓=𝑉𝑇₀𝑓+𝑉𝑆₁𝑓. Now we invoke the following Planchrel-Polya-Nikolskii lemma.

Lemma A.1 (Planchrel-Polya-Nikols'kij [56, page 16]). Let 0<𝜂≤1. Assume that 𝑓∈𝒮^′(ℝ^𝑛) has frequency support in 𝑄(0,𝑅). Then, denoting by 𝑀 the Hardy-Littlewood maximal operator, we have sup_{𝑦∈ℝ𝑛}𝑓𝑥−𝑦1+𝑅𝑦^𝑛/𝜂≤𝐶𝑀𝑓^𝜂(𝑥)^1/𝜂, (A.18) where 𝐶 is independent of 𝑅>0.

According to Lemma A.1 with 𝜂=1/2 and 𝑅=16, we conclude that 𝑆₀𝑓(𝑥)≤𝐶_ℝ𝑛ℱ⁻¹Ψ𝑦⋅𝑇₀𝑓+𝑥−𝑦𝑑𝑦_ℝ𝑛ℱ⁻¹Ψ𝑦⋅𝑆₁𝑓𝑀𝑇𝑥−𝑦𝑑𝑦≤𝐶₀𝑓^1/2(𝑥)²+𝑀𝑆₁𝑓^1/2(𝑥)²×_ℝ𝑛ℱ⁻¹Ψ𝑦1+𝑦^2𝑛𝑀𝑇𝑑𝑦=𝐶₀𝑓^1/2(𝑥)²+𝑀𝑆₁𝑓^1/2(𝑥)², (A.19) where for the last inequality, we invoked ℱ⁻¹Ψ𝑦≤𝐶1+𝑦^3𝑛+1𝑦∈ℝ^𝑛. (A.20) By the fact that ‖𝐹‖_𝐿^𝑛/𝜆=‖|𝐹|‖_𝐿^2𝑛/𝜆2 and the 𝐿^2𝑛/𝜆(ℝ^𝑛)-boundedness of the Hardy-Littlewood maximal operator, we conclude 𝑆₀𝑓_{𝐿𝑛/𝜆}≤𝐶𝑀𝑇₀𝑓^1/2_{𝐿2𝑛/𝜆}²+𝑀𝑆₁𝑓^1/2_{𝐿2𝑛/𝜆}²≤𝐶𝑇₀𝑓^1/2_{𝐿2𝑛/𝜆}²+𝑆₁𝑓^1/2_{𝐿2𝑛/𝜆}²=𝐶𝑇₀𝑓_{𝐿𝑛/𝜆}+𝑆₁𝑓_{𝐿𝑛/𝜆}. (A.21) Combining (A.16) and (A.21), we obtain the desired result.

Acknowledgment

The authors are grateful to Professor Victor Burenkov for his kind suggestion about the relations of various definitions of the Morrey norms.