New Herz Type Besov and Triebel-Lizorkin Spaces with Variable Exponents

Abstract

The authors establish the boundedness of vector-valued Hardy-Littlewood maximal operator in Herz spaces with variable exponents. Then new Herz type Besov and Triebel-Lizorkin spaces with variable exponents are introduced. Finally, characterizations of these new spaces by maximal functions are given.

1. Introduction

Recent decades, variable exponent function spaces have attracted many attention. In fact, since the variable Lebesgue and Sobolev spaces were systemically studied by Kovác̆ik and Rákosník in [1], there are many spaces introduced, such as Bessel potential spaces with variable exponent, Besov and Trieble-Lizorkin spaces with variable exponents, and Morrey spaces with variable exponents; see [2–10] and reference therein. When the Hardy-Littlewood maximal operator is bounded on the variable Lebesgue spaces, many results in classic harmonic analysis and function theory are also hold for the variable exponent case; see [11–13]. All mentioned spaces have many applications in differential equations, here we refer the readers to the survey [14].

Meanwhile, there are another type function spaces, Herz type spaces, have attracted many authors' interests for the last three decades. We recommend the readers to the monograph [15] and references therein. Indeed, many results in classic Lebesgue spaces have been generalized to Herz type spaces; see [16–20]. Herz type Besov and Triebel-Lizorkin spaces have been developed; see [21–24]. In 2010, Izuki in [25] introduced Herz space with variable exponent 𝐾̇^𝛼,𝑞_𝑝(⋅)(ℝ^𝑛) and 𝐾^𝛼,𝑞_𝑝(⋅)(ℝ^𝑛) and proved the boundedness of sublinear operators on them. Then, in [26] Izuki obtained the boundedness of commutators on Herz spaces with variable exponent. In [27] Izuki established the boundedness of vector-valued sublinear operators on Herz spaces with variable exponent 𝐾̇^𝛼,𝑞_𝑝(⋅)(ℝ^𝑛) and 𝐾^𝛼,𝑞_𝑝(⋅)(ℝ^𝑛) and obtained equivalent norms and wavelet characterization of Herz-Sobolev spaces with variable exponent. In [28], Shi and the second author of this paper introduced Herz type Besov and Triebel-Lizorkin spaces with variable exponent, 𝐾̇^𝛼,𝑞_𝑝(⋅)(ℝ^𝑛)𝐵^𝑠_𝛽 and 𝐾^𝛼,𝑞_𝑝(⋅)(ℝ^𝑛)𝐵^𝑠_𝛽 and 𝐾̇^𝛼,𝑞_𝑝(⋅)(ℝ^𝑛)𝐹^𝑠_𝛽 and 𝐾^𝛼,𝑞_𝑝(⋅)(ℝ^𝑛)𝐹^𝑠_𝛽, and obtained their equivalent quasi-norms. Then, in [29], they gave a direct characterization of those spaces and its application. In 2012, Almeida and Drihem considered Herz spaces 𝐾̇^{𝛼(⋅),𝑞}_𝑝(⋅)(ℝ^𝑛) and 𝐾^{𝛼(⋅),𝑞}_𝑝(⋅)(ℝ^𝑛) in [30], where the exponent 𝛼 is variable as well, and gave boundedness results for a wide class of classical operators acting on such spaces.

Motivated by the previous papers, the goal of this paper is to introduce new Herz type Besov and Triebel-Lizorkin spaces with variable exponents. The structure of the paper is as follows. In Section 2 we will establish the boundedness of vector-valued Hardy-Littlewood maximal operator in spaces 𝐾̇^{𝛼(⋅),𝑞}_𝑝(⋅)(ℝ^𝑛) and 𝐾^{𝛼(⋅),𝑞}_𝑝(⋅)(ℝ^𝑛). In Section 3 we give characterizations of these Herz type Besov and Triebel-Lizorkin spaces with variable exponents by maximal functions.

2. Vector-Valued Estimates

In this section we will establish vector-valued estimates of Hardy-Littlewood maximal operator in Herz spaces with variable exponents. Before stating our result, we recall some definitions and notations. First, we give some convention. Throughout this paper |𝑆| denotes the Lebesgue measure and 𝜒_𝑆 the characteristics function for a measurable set 𝑆⊂ℝ^𝑛. We also use the notation 𝑎≲𝑏 if there exists a constant 𝑐>0 such that 𝑎⩽𝑐𝑏. If 𝑎≲𝑏 and 𝑏≲𝑎 we will write 𝑎≈𝑏. Finally we claim that 𝐶 is always a positive constant but it may change from line to line.

Definition 2.1. Let 𝐸 be a measurable set in ℝ^𝑛 with |𝐸|>0. Let 𝑝(⋅)∶𝐸→[1,∞) be a measurable function. Denote 𝐿^𝑝(⋅)(𝐸)∶=𝑓ismeasurableon𝐸∶𝜌_𝑝(⋅)𝑓𝜆<∞forsomeconstant𝜆>0, (2.1) where 𝜌_𝑝(⋅)(𝑓)∶=∫_𝐸|𝑓(𝑥)|^𝑝(𝑥)𝑑𝑥, and 𝑓_{𝐿𝑝(⋅)(𝐸)}∶=inf𝜆>0∶𝜌_𝑝(⋅)𝑓𝜆⩽1. (2.2) Then 𝐿^𝑝(⋅)(𝐸) is a Banach space with the norm ‖⋅‖_𝐿^𝑝(⋅)_(𝐸).

Let 𝐿¹_loc(ℝ^𝑛) be the collection of all locally integrable functions on ℝ^𝑛. Given a function 𝑓∈𝐿¹_loc(ℝ^𝑛), the Hardy-Littlewood maximal operator ℳ is defined by ℳ𝑓(𝑥)∶=sup_𝑟>0𝑟^−𝑛_{𝐵(𝑥,𝑟)}𝑓𝑦𝑑𝑦,∀𝑥∈ℝ^𝑛, (2.3) where 𝐵(𝑥,𝑟)∶={𝑦∈ℝ^𝑛∶|𝑥−𝑦|<𝑟}. We also use the following notation: 𝑝₋∶=essinf{𝑝(𝑥)∶𝑥∈ℝ^𝑛} and 𝑝₊∶=esssup{𝑝(𝑥)∶𝑥∈ℝ^𝑛}. The set 𝒫(ℝ^𝑛) consists of all 𝑝(⋅) satisfying 𝑝₋>1 and 𝑝₊<∞.  ℬ(ℝ^𝑛) is the set of 𝑝(⋅)∈𝒫(ℝ^𝑛) satisfying the condition that ℳ is bounded on 𝐿^𝑝(⋅)(ℝ^𝑛). It is well known that if 𝑝(⋅)∈𝒫(ℝ^𝑛) then 𝑝(⋅)∈ℬ(ℝ^𝑛) is equivalent to 𝑝^′(⋅)∈ℬ(ℝ^𝑛), where 𝑝^′(⋅) is the conjugate exponent to 𝑝(⋅), that means 𝑝^′(⋅)=𝑝(⋅)/(𝑝(⋅)−1); see [12]. For more about the set ℬ(ℝ^𝑛), one can see [11, 12, 31, 32]. Moreover, we define 𝒫⁰(ℝ^𝑛) to be the set of measurable functions 𝑝 on ℝ^𝑛 with the range in (0,∞) such that 𝑝₋∶=essinf{𝑝(𝑥)∶𝑥∈ℝ^𝑛}>0 and 𝑝₊∶=esssup{𝑝(𝑥)∶𝑥∈ℝ^𝑛}<∞. Given 𝑝(⋅)∈𝒫⁰(ℝ^𝑛), one can define the space 𝐿^𝑝(⋅)(ℝ^𝑛) as in Definition 2.1. This is equivalent to defining it to be the set of all functions 𝑓 such that |𝑓|^𝑝0∈𝐿^{𝑝(⋅)/𝑝0}(ℝ^𝑛), where 0<𝑝₀<𝑝₋ and 𝑝(⋅)/𝑝₀∈𝒫(ℝ^𝑛). We also define a quasi-norm on this space by ‖𝑓‖_𝐿^𝑝(⋅)_(ℝ^𝑛₎=‖|𝑓|^𝑝0‖^1/𝑝0_{𝐿0𝑝(⋅)/𝑝(ℝ𝑛)}.

Lemma 2.2 (see [25]). Let 𝑝(⋅)∈ℬ(ℝ^𝑛). Then there exist 0<𝛿₁,𝛿₂<1 depending only on 𝑝(⋅) and 𝑛 such that for balls 𝐵 in ℝ^𝑛 and all measurable subsets 𝑆⊂𝐵, 𝜒_{𝑆𝐿𝑝(⋅)(ℝ𝑛)}𝜒_{𝐵𝐿𝑝(⋅)(ℝ𝑛)}|𝑆|⩽𝐶|𝐵|^𝛿1,𝜒_{𝑆𝐿𝑝′(⋅)(ℝ𝑛)}𝜒_{𝐵𝐿𝑝′(⋅)(ℝ𝑛)}|𝑆|⩽𝐶|𝐵|^𝛿2. (2.4)

Lemma 2.3 (see [25]). Let 𝑝(⋅)∈ℬ(ℝ^𝑛). Then there exists a positive constant 𝐶>0 such that for balls 𝐵 in ℝ^𝑛, 1|𝐵|𝜒_{𝐵𝐿𝑝(⋅)(ℝ𝑛)}𝜒_{𝐵𝐿𝑝′(⋅)(ℝ𝑛)}⩽𝐶. (2.5)

Lemma 2.4 (see [11]). Let 𝑝(⋅)∈ℬ(ℝ^𝑛) and 1<𝑟<∞, then there exists a positive constant 𝐶 such that for all sequences {𝑓_𝑗}^∞_𝑗=1 of locally integrable functions on ℝ^𝑛, ^∞_𝑗=1ℳ𝑓_𝑗^𝑟1/𝑟_{𝐿𝑝(⋅)}⩽𝐶^∞_𝑗=1𝑓_𝑗^𝑟1/𝑟_{𝐿𝑝(⋅)}. (2.6)

For giving the definition of the Herz spaces with variable exponent, let us introduce the following notations. Let 𝑘∈ℤ.𝐵_𝑘∶={𝑥∈ℝ^𝑛∶|𝑥|⩽2^𝑘}.  𝑅_𝑘∶=𝐵_𝑘⧵𝐵_𝑘−1.  𝜒_𝑘∶=𝜒_𝑅𝑘.

ℕ₀ denotes the set of all nonnegative integers. For 𝑚∈ℕ₀, we denote 𝜒_𝑚∶=𝜒_𝑅𝑚 if 𝑚≥1 and 𝜒₀∶=𝜒_𝐵0.

Definition 2.5. Let 0<𝑞⩽∞,𝑝(⋅)∈𝒫(ℝ^𝑛) and 𝛼(⋅)∶ℝ^𝑛→ℝ with 𝛼(⋅)∈𝐿^∞(ℝ^𝑛).(i) The homogeneous Herz space 𝐾̇^{𝛼(⋅),𝑞}_𝑝(⋅)(ℝ^𝑛) is defined by 𝐾̇^{𝛼(⋅),𝑞}_𝑝(⋅)ℝ^𝑛∶=𝑓∈𝐿^𝑝(⋅)_locℝ^𝑛⧵0∶𝑓_̇𝐾^{𝛼(⋅),𝑞}_{𝑝(⋅)(ℝ𝑛)}<∞, (2.7) where 𝑓_̇𝐾^{𝛼(⋅),𝑞}_{𝑝(⋅)(ℝ𝑛)}∶=^∞_𝑘=−∞2^{𝛼(⋅)𝑘}𝑓𝜒_𝑘^𝑞_{𝐿𝑝(⋅)(ℝ𝑛)}^1/𝑞. (2.8)(ii) The nonhomogeneous Herz space 𝐾^{𝛼(⋅),𝑞}_𝑝(⋅)(ℝ^𝑛) is defined by 𝐾^{𝛼(⋅),𝑞}_𝑝(⋅)ℝ^𝑛∶=𝑓∈𝐿^𝑝(⋅)_locℝ^𝑛∶𝑓_𝐾^{𝛼(⋅),𝑞}_{𝑝(⋅)(ℝ𝑛)}<∞, (2.9) where 𝑓_𝐾^{𝛼(⋅),𝑞}_{𝑝(⋅)(ℝ𝑛)}∶=^∞_𝑚=02^{𝛼(⋅)𝑚}𝑓𝜒_𝑚^𝑞_{𝐿𝑝(⋅)(ℝ𝑛)}^1/𝑞. (2.10)

Definition 2.6. A function 𝛼(⋅)∶ℝ^𝑛→ℝ is called locally log-Hölder continuous on ℝ^𝑛 if there exists 𝑐_log>0 such that 𝛼(𝑥)𝑦⩽𝑐−𝛼^log,log𝑒+1/𝑥−𝑦 (2.11) for all 𝑥,𝑦∈ℝ^𝑛. If |𝛼(𝑥)−𝛼(0)|⩽𝑐_log/log(𝑒+1/|𝑥|) for all 𝑥∈ℝ^𝑛, then 𝛼(⋅) is called log- Hölder continuous at origin. 𝛼(⋅) is called satisfying the log-Hölder decay condition if there exist 𝛼_∞∈ℝ and a constant 𝑐_log>0 such that 𝛼(𝑥)−𝛼_∞⩽𝑐_log(|𝑥,log𝑒+|) (2.12) for all 𝑥∈ℝ^𝑛.

By ℓ^𝑞,𝑞∈(0,∞], we denote the discrete Lebesgue space equipped with the usual quasi-norm. Let us denote {𝑔_𝑘}_ℓ^𝑞_{>(𝐿𝑝(⋅))}∶=^∞_𝑘=0𝑔_𝑘^𝑞_𝑝(⋅)^1/𝑞,{𝑔_𝑘}_ℓ^𝑞_{<(𝐿𝑝(⋅))}∶=⁻¹_𝑘=−∞𝑔_𝑘^𝑞_𝑝(⋅)^1/𝑞, (2.13) for sequences {𝑔_𝑘}_𝑘∈ℤ of measurable functions (with the usual modification when 𝑞=∞.)

Proposition 2.7. Let 𝛼(⋅)∈𝐿^∞(ℝ^𝑛),𝑝(⋅)∈𝒫(ℝ^𝑛) and 𝑞∈(0,∞]. If 𝛼(⋅) satisfies log-Hölder decay condition at infinity, then 𝐾^{𝛼(⋅),𝑞}_𝑝(⋅)ℝ^𝑛=𝐾^𝛼∞,𝑞_𝑝(⋅)ℝ^𝑛. (2.14)
Additionally, if 𝛼(⋅) is log-Hölder continuous at the origin, then 𝑓_̇𝐾^{𝛼(⋅),𝑞}_{𝑝(⋅)(ℝ𝑛)}≈2^𝑘𝛼(0)𝑓𝜒_{𝑘ℓ𝑞<(𝐿𝑝(⋅))}+{2^𝑘𝛼∞𝑓𝜒_𝑘}_{ℓ𝑞>(𝐿𝑝(⋅))}. (2.15)

We have the boundedness of vector-valued Hardy-Littlewood maximal operator in Herz spaces with variable exponents, which generalizes the results in [27, 33].

Theorem 2.8. Let 𝑝(⋅)∈ℬ(ℝ^𝑛),0<𝑞<∞,𝛼(⋅)∶ℝ^𝑛→ℝ with 𝛼(⋅)∈𝐿^∞(ℝ^𝑛) and 𝛼(⋅) be log-Hölder continuous, both at the origin and at infinity, such that −𝑛𝛿₁<𝛼(0)⩽𝛼_∞<𝑛𝛿₂, where 0<𝛿₁,𝛿₂<1 are constants satisfying (2.4), 1<𝑟<∞, then there is a constant 𝐶 independent of sequences {𝑓_𝑗}^∞_𝑗=1 of locally integrable functions on ℝ^𝑛 such that ^∞_𝑘=1ℳ𝑓_𝑘^𝑟1/𝑟_{̇𝐾𝛼(⋅),𝑞𝑝(⋅)(ℝ𝑛)}⩽𝐶^∞_𝑘=1𝑓_𝑘^𝑟1/𝑟_{̇𝐾𝛼(⋅),𝑞𝑝(⋅)(ℝ𝑛)},_∞𝑘=1ℳ𝑓_𝑘^𝑟1/𝑟_{𝐾𝛼(⋅),𝑞𝑝(⋅)(ℝ𝑛)}⩽𝐶^∞_𝑘=1𝑓_𝑘^𝑟1/𝑟_{𝐾𝛼(⋅),𝑞𝑝(⋅)(ℝ𝑛)}. (2.16)

Proof. Here we only prove the result for homogeneous Herz space, for nonhomogeneous Herz space it can be proved by similar way. We only consider sequences {𝑓_𝑗}^∞_𝑗=1 of locally measurable functions on ℝ^𝑛 such that (∑^∞_𝑘=1|𝑓_𝑘|^𝑟)^1/𝑟∈𝐾̇^{𝛼(⋅),𝑞}_𝑝(⋅)(ℝ^𝑛).
Firstly, by Proposition 2.7 we have ^∞_𝑗=1ℳ𝑓_𝑗^𝑟1/𝑟_{̇𝐾𝛼(⋅),𝑞𝑝(⋅)(ℝ𝑛)}≈2^{𝑘𝛼(0)∞}_𝑗=1ℳ𝑓_𝑗^𝑟1/𝑟𝜒_{𝑘ℓ𝑞<(𝐿𝑝(⋅))}+2^{𝑘𝛼∞∞}_𝑗=1ℳ𝑓_𝑗^𝑟1/𝑟𝜒_{𝑘ℓ𝑞>(𝐿𝑝(⋅))}∶=𝐴_<+𝐴_>. (2.17) To continue, we let 𝑓^𝑖_𝑗=𝜒_𝑖𝑓_𝑗. Now we estimate 𝐴_< by Minkowski inequality as follows: 𝐴_<∶=⁻¹_𝑘=−∞2^{𝛼(0)𝑞𝑘}𝜒_𝑘^∞_𝑗=1ℳ𝑓_𝑗^{𝑟1/𝑟𝑞}_{𝐿𝑝(⋅)}^1/𝑞=⁻¹_𝑘=−∞2^{𝛼(0)𝑞𝑘}𝜒_𝑘^∞_𝑗=1ℳ^∞_𝑖=−∞𝑓^𝑖_𝑗^{𝑟1/𝑟𝑞}_{𝐿𝑝(⋅)}^1/𝑞⩽⁻¹_𝑘=−∞2^{𝛼(0)𝑞𝑘}𝜒_𝑘^∞_𝑖=−∞^∞_𝑗=1ℳ𝑓^𝑖_𝑗^{𝑟1/𝑟𝑞}_{𝐿𝑝(⋅)}^1/𝑞≲⁻¹_𝑘=−∞2^{𝛼(0)𝑞𝑘}𝜒_𝑘^𝑘−2_𝑖=−∞^∞_𝑗=1ℳ𝑓^𝑖_𝑗^{𝑟1/𝑟𝑞}_{𝐿𝑝(⋅)}^1/𝑞+⁻¹_𝑘=−∞2^{𝛼(0)𝑞𝑘}𝜒_𝑘^𝑘+1_{𝑖=𝑘−1}^∞_𝑗=1ℳ𝑓^𝑖_𝑗^{𝑟1/𝑟𝑞}_{𝐿𝑝(⋅)}^1/𝑞+⁻¹_𝑘=−∞2^{𝛼(0)𝑞𝑘}𝜒_𝑘^∞_𝑖=𝑘+2^∞_𝑗=1ℳ𝑓^𝑖_𝑗^{𝑟1/𝑟𝑞}_{𝐿𝑝(⋅)}^1/𝑞∶=𝐼_<+𝐼𝐼_<+𝐼𝐼𝐼_<. (2.18)
By the same way we consider 𝐴_>, 𝐴_>≲^∞_𝑘=02^{𝛼∞𝑞𝑘}𝜒_𝑘^𝑘−2_𝑖=−∞^∞_𝑗=1ℳ𝑓^𝑖_𝑗^{𝑟1/𝑟𝑞}_{𝐿𝑝(⋅)}^1/𝑞+^∞_𝑘=02^{𝛼∞𝑞𝑘}𝜒_𝑘^𝑘+1_{𝑖=𝑘−1}^∞_𝑗=1|ℳ𝑓^𝑖_𝑗|^{𝑟1/𝑟𝑞}_{𝐿𝑝(⋅)}^1/𝑞+^∞_𝑘=02^{𝛼∞𝑞𝑘}𝜒_𝑘^∞_𝑖=𝑘+2^∞_𝑗=1ℳ𝑓^𝑖_𝑗^{𝑟1/𝑟𝑞}_{𝐿𝑝(⋅)}^1/𝑞∶=𝐼_>+𝐼𝐼_>+𝐼𝐼𝐼_>. (2.19)
So we get ^∞_𝑗=1ℳ𝑓_𝑗^𝑟1/𝑟_{̇𝐾𝛼(⋅),𝑞𝑝(⋅)(ℝ𝑛)}≲𝐼+𝐼𝐼+𝐼𝐼𝐼, (2.20) with 𝐼∶=𝐼_<+𝐼_>,𝐼𝐼∶=𝐼𝐼_<+𝐼𝐼_>,𝐼𝐼𝐼∶=𝐼𝐼𝐼_<+𝐼𝐼𝐼_>.
Secondly, we divide the proof into three steps.
Step  1. We consider 𝐼𝐼. By Lemma 2.4 we have 𝐼𝐼_<≲⁻¹_𝑘=−∞2^{𝛼(0)𝑞𝑘𝑘+1}_{𝑖=𝑘−1}^∞_𝑗=1𝑓^𝑖_𝑗^{𝑟1/𝑟𝑞}_{𝐿𝑝(⋅)}^1/𝑞≲⁻¹_𝑘=−∞2^{𝛼(0)𝑞𝑘𝑘+1}_{𝑖=𝑘−1}𝜒_𝑖^∞_𝑗=1𝑓_𝑗^{𝑟1/𝑟𝑞}_{𝐿𝑝(⋅)}^1/𝑞≲⁻¹_𝑘=−∞2^{𝛼(0)𝑞𝑘}𝜒_𝑘−1^∞_𝑗=1𝑓_𝑗^{𝑟1/𝑟𝑞}_{𝐿𝑝(⋅)}^1/𝑞+⁻¹_𝑘=−∞2^{𝛼(0)𝑞𝑘}𝜒_𝑘^∞_𝑗=1𝑓_𝑗^{𝑟1/𝑟𝑞}_{𝐿𝑝(⋅)}^1/𝑞+⁻¹_𝑘=−∞2^{𝛼(0)𝑞𝑘}𝜒_𝑘+1^∞_𝑗=1𝑓_𝑗^{𝑟1/𝑟𝑞}_{𝐿𝑝(⋅)}^1/𝑞≲⁻²_𝑘=−∞2^{𝛼(0)𝑞𝑘}𝜒_𝑘^∞_𝑗=1𝑓_𝑗^{𝑟1/𝑟𝑞}_{𝐿𝑝(⋅)}^1/𝑞+2^{𝛼(0)𝑘∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_{𝑘ℓ𝑞<(𝐿𝑝(⋅))}+⁰_𝑘=−∞2^{𝛼(0)𝑞𝑘}𝜒_𝑘^∞_𝑗=1𝑓_𝑗^{𝑟1/𝑟𝑞}_{𝐿𝑝(⋅)}^1/𝑞≲2^{𝛼(0)𝑘∞}_𝑗=1|𝑓_𝑗|^𝑟1/𝑟𝜒_{𝑘ℓ𝑞<(𝐿𝑝(⋅))}. (2.21) Similarly, we have 𝐼𝐼_>≲^∞_𝑘=−12^{𝛼∞𝑞𝑘}𝜒_𝑘^∞_𝑗=1𝑓_𝑗^{𝑟1/𝑟𝑞}_{𝐿𝑝(⋅)}^1/𝑞+2^{𝛼∞𝑘∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_{𝑘ℓ𝑞>(𝐿𝑝(⋅))}+^∞_𝑘=12^{𝛼∞𝑞𝑘}𝜒_𝑘^∞_𝑗=1𝑓_𝑗^{𝑟1/𝑟𝑞}_{𝐿𝑝(⋅)}^1/𝑞≲2^{𝛼∞𝑘∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_{𝑘ℓ𝑞>(𝐿𝑝(⋅))}. (2.22) Thus 𝐼𝐼∶=𝐼𝐼_<+𝐼𝐼_>≲^∞_𝑗=1𝑓_𝑗^𝑟1/𝑟_{̇𝐾𝛼(⋅),𝑞𝑝(⋅)}. (2.23)Step  2. We consider 𝐼. We recall that the Hardy-Littlewood maximal operator satisfies the size condition ℳ𝑓(𝑥)⩽𝐶_ℝ𝑛𝑥−𝑦^−𝑛𝑓𝑦𝑑𝑦, (2.24) for all 𝑓∈𝐿¹_loc(ℝ^𝑛) with compact support and a.e 𝑥∉supp𝑓.
Therefore, for for all 𝑖⩽𝑘−2,𝑥∈𝑅_𝑘,1<𝑟<∞, by the size condition and the generalized Minkowski inequality, we obtain ^∞_𝑗=1ℳ^𝑟𝑓^𝑖_𝑗(𝑥)^1/𝑟≲^∞_𝑗=12^−𝑘𝑛_ℝ𝑛𝑓^𝑖_𝑗𝑦𝑑𝑦^𝑟1/𝑟≲2^−𝑘𝑛_ℝ𝑛^∞_𝑗=1𝑓^𝑖_𝑗^𝑟1/𝑟𝑑𝑦. (2.25)
By Hölder's inequality we get 𝐼_<≲⁻¹_𝑘=−∞2^{𝛼(0)𝑞𝑘}𝜒_𝑘^𝑘−2_𝑖=−∞2^−𝑘𝑛_ℝ𝑛^∞_𝑗=1𝑓^𝑖_𝑗^𝑟1/𝑟𝑑𝑦^𝑞_{𝐿𝑝(⋅)}^1/𝑞≈⁻¹_𝑘=−∞2^{𝛼(0)𝑞𝑘}𝜒_𝑘^𝑞_{𝐿𝑝(⋅)}^𝑘−2_𝑖=−∞2^−𝑘𝑛_ℝ𝑛^∞_𝑗=1𝑓^𝑖_𝑗^𝑟1/𝑟𝑑𝑦^𝑞1/𝑞≲⁻¹_𝑘=−∞2^{𝛼(0)𝑞𝑘}𝜒_𝑘^𝑞_{𝐿𝑝(⋅)}×^𝑘−2_𝑖=−∞2^{−𝑘𝑛∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_{𝑖𝐿𝑝(⋅)}𝜒_{𝑖𝐿𝑝′(⋅)}^𝑞1/𝑞. (2.26)
On the other hand, by using Lemmas 2.2 and 2.3, we have 2^−𝑘𝑛𝜒_{𝑘𝐿𝑝(⋅)}𝜒_{𝑖𝐿𝑝′(⋅)}≲2^−𝑘𝑛𝜒_{𝐵𝑖𝐿𝑝′(⋅)}𝐵_𝑘𝜒_𝐵k𝐿⁻¹_𝑝′(⋅)≲2^{𝑛𝛿2(𝑖−𝑘)}. (2.27)
We put (2.27) into (2.26) and get 𝐼_<≲⁻¹_𝑘=−∞2^{𝛼(0)𝑘𝑞𝑘−2}_𝑖=−∞^∞_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_{𝑖𝐿𝑝(⋅)}2^{𝑛𝛿2(𝑖−𝑘)𝑞1/𝑞}≈⁻¹_𝑘=−∞^𝑘−2_𝑖=−∞2^{𝛼(0)𝑘∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_{𝑖𝐿𝑝(⋅)}2^{𝑛𝛿2(𝑖−𝑘)𝑞1/𝑞}≈⁻¹_𝑘=−∞^𝑘−2_𝑖=−∞2^{𝛼(0)𝑖∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_{𝑖𝐿𝑝(⋅)}2^{𝑏(𝑖−𝑘)𝑞1/𝑞}, (2.28) here 𝑏=𝑛𝛿₂−𝛼(0)>0.
For 1<𝑞<∞, we can make (2.28) further calculation by Hölder's inequality 𝐼_<≲⁻¹_𝑘=−∞^𝑘−2_𝑖=−∞2^{𝛼(0)𝑖𝑞∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_𝑖^𝑞_{𝐿𝑝(⋅)}2^{𝑏𝑞(𝑖−𝑘)/2𝑘−2}_𝑖=−∞2^{𝑏𝑞′(𝑖−𝑘)/2𝑞/𝑞′1/𝑞}≲⁻¹_𝑘=−∞^𝑘−2_𝑖=−∞2^{𝛼(0)𝑖𝑞∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_𝑖^𝑞_{𝐿𝑝(⋅)}2^{𝑏𝑞(𝑖−𝑘)/21/𝑞}≈⁻³_𝑖=−∞2^{𝛼(0)𝑖𝑞∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_𝑖^𝑞_{𝐿𝑝(⋅)}⁻¹_𝑘=𝑖+22^{𝑏𝑞(𝑖−𝑘)/21/𝑞}≲⁻³_𝑖=−∞2^{𝛼(0)𝑖𝑞∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_𝑖^𝑞_{𝐿𝑝(⋅)}^1/𝑞≲2^{𝛼(0)𝑘∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_{𝑘ℓ𝑞<(𝐿𝑝(⋅))}. (2.29) For 0<𝑞⩽1, we have 𝐼_<≲⁻¹_𝑘=−∞^𝑘−2_𝑖=−∞2^{𝛼(0)𝑖𝑞∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_𝑖^𝑞_{𝐿𝑝(⋅)}2^{𝑏𝑞(𝑖−𝑘)1/𝑞}≈⁻³_𝑖=−∞2^{𝛼(0)𝑖𝑞∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_𝑖^𝑞_{𝐿𝑝(⋅)}⁻¹_𝑘=𝑖+22^{𝑏𝑞(𝑖−𝑘)1/𝑞}≲⁻³_𝑖=−∞2^{𝛼(0)𝑖𝑞∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_𝑖^𝑞_{𝐿𝑝(⋅)}^1/𝑞≲2^{𝛼(0)𝑘∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_{𝑘ℓ𝑞<(𝐿𝑝(⋅))}. (2.30)
Similarly, we have 𝐼_>≲^∞_𝑘=0^𝑘−2_𝑖=−∞2^{𝛼∞𝑖∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_{𝑖𝐿𝑝(⋅)}2^{𝑏(𝑖−𝑘)𝑞1/𝑞}, (2.31) here 𝑏=𝑛𝛿₂−𝛼_∞>0.
For 1<𝑞<∞, we have 𝐼_>≲^∞_𝑘=0^𝑘−2_𝑖=−∞2^{𝛼∞𝑖𝑞∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_𝑖^𝑞_{𝐿𝑝(⋅)}2^{𝑏𝑞(𝑖−𝑘)/21/𝑞}≈⁻²_𝑖=−∞2^{𝛼∞𝑖𝑞∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_𝑖^𝑞_{𝐿𝑝(⋅)}^∞_𝑘=02^{𝑏𝑞(𝑖−𝑘)/2}+^∞_𝑖=−12^{𝛼∞𝑖𝑞∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_𝑖^𝑞_{𝐿𝑝(⋅)}^∞_𝑘=𝑖+22^{𝑏𝑞(𝑖−𝑘)/21/𝑞}≲⁻²_𝑖=−∞2^{(𝛼∞+𝑛𝛿2)𝑖𝑞/2∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_𝑖^𝑞_{𝐿𝑝(⋅)}+^∞_𝑖=−12^{𝛼∞𝑖𝑞∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_𝑖^𝑞_{𝐿𝑝(⋅)}^1/𝑞≲⁻²_𝑖=−∞2^{𝛼(0)𝑖𝑞+((𝛼∞−2𝛼(0)+𝑛𝛿2)𝑖𝑞)/2∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_𝑖^𝑞_{𝐿𝑝(⋅)}+^∞_𝑖=−12^{𝛼∞𝑖𝑞∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_𝑖^𝑞_{𝐿𝑝(⋅)}^1/𝑞≲⁻²_𝑖=−∞2^{𝛼(0)𝑖𝑞∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_𝑖^𝑞_{𝐿𝑝(⋅)}+^∞_𝑖=−12^{𝛼∞𝑖𝑞∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_𝑖^𝑞_{𝐿𝑝(⋅)}^1/𝑞≲2^{𝛼(0)𝑘∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_{𝑘ℓ𝑞<(𝐿𝑝(⋅))}+2^{𝛼∞𝑘∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_{𝑘ℓ𝑞>(𝐿𝑝(⋅))}. (2.32) For 0<𝑞⩽1, we have 𝐼_>≲^∞_𝑘=0^𝑘−2_𝑖=−∞2^{𝛼∞𝑖𝑞∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_𝑖^𝑞_{𝐿𝑝(⋅)}2^{𝑏𝑞(𝑖−𝑘)1/𝑞}≈⁻²_𝑖=−∞2^{𝛼∞𝑖𝑞∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_𝑖^𝑞_{𝐿𝑝(⋅)}^∞_𝑘=02^{𝑏𝑞(𝑖−𝑘)}+^∞_𝑖=−12^{𝛼∞𝑖𝑞∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_𝑖^𝑞_{𝐿𝑝(⋅)}^∞_𝑘=𝑖+22^{𝑏𝑞(𝑖−𝑘)1/𝑞}≲⁻²_𝑖=−∞2^{𝑛𝛿2𝑖𝑞∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_𝑖^𝑞_{𝐿𝑝(⋅)}+^∞_𝑖=−12^{𝛼∞𝑖𝑞∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_𝑖^𝑞_{𝐿𝑝(⋅)}^1/𝑞≲⁻²_𝑖=−∞2^{𝛼(0)𝑖𝑞+(𝑛𝛿2−𝛼(0))𝑖𝑞∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_𝑖^𝑞_{𝐿𝑝(⋅)}+^∞_𝑖=−12^{𝛼∞𝑖𝑞∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_𝑖^𝑞_{𝐿𝑝(⋅)}^1/𝑞≲⁻²_𝑖=−∞2^{𝛼(0)𝑖𝑞∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_𝑖^𝑞_{𝐿𝑝(⋅)}+^∞_𝑖=−12^{𝛼∞𝑖𝑞∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_𝑖^𝑞_{𝐿𝑝(⋅)}^1/𝑞≲2^{𝛼(0)𝑘∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_{𝑘ℓ𝑞<(𝐿𝑝(⋅))}+2^{𝛼∞𝑘∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_{𝑘ℓ𝑞>(𝐿𝑝(⋅))}. (2.33) Thus 𝐼∶=𝐼_<+𝐼_>≲^∞_𝑗=1𝑓_𝑗^𝑟1/𝑟_{̇𝐾𝛼,𝑞𝑝(⋅)}. (2.34)Step  3. We consider 𝐼𝐼𝐼.
For for all 𝑖≥𝑘+2,𝑥∈𝑅_𝑘, by size condition and generalized Minkowski inequality, we have ^∞_𝑗=1𝑀^𝑟𝑓^𝑖_𝑗(𝑥)^1/𝑟≲^∞_𝑗=12^−𝑖𝑛_ℝ𝑛𝑓^𝑖_𝑗𝑦𝑑𝑦^𝑟1/𝑟≈2^{−𝑖𝑛∞}_{𝑗=1ℝ𝑛}𝑓^𝑖_𝑗𝑦𝑑𝑦^𝑟1/𝑟≲2^−𝑖𝑛_ℝ𝑛^∞_𝑗=1𝑓^𝑖_𝑗^𝑟1/𝑟𝑑𝑦. (2.35)
By Hölder's inequality we get 𝐼𝐼𝐼_<≲⁻¹_𝑘=−∞2^{𝛼(0)𝑞𝑘}𝜒_𝑘^∞_𝑖=𝑘+22^−𝑖𝑛_ℝ𝑛^∞_𝑗=1𝑓^𝑖_𝑗^𝑟1/𝑟𝑑𝑦^𝑞_{𝐿𝑝(⋅)}^1/𝑞≈⁻¹_𝑘=−∞2^{𝛼(0)𝑞𝑘}𝜒_𝑘^𝑞_{𝐿𝑝(⋅)}^∞_𝑖=𝑘+22^−𝑖𝑛_ℝ𝑛^∞_𝑗=1𝑓^𝑖_𝑗^𝑟1/𝑟𝑑𝑦^𝑞1/𝑞≲⁻¹_𝑘=−∞2^{𝛼(0)𝑞𝑘}𝜒_𝑘^𝑞_{𝐿𝑝(⋅)}×^∞_𝑖=𝑘+22^{−𝑖𝑛∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_{𝑖𝐿𝑝(⋅)}𝜒_{𝑖𝐿𝑝′(⋅)}^𝑞1/𝑞. (2.36)
On the other hand, by using Lemmas 2.2 and 2.3, we have 2^−𝑖𝑛𝜒_{𝑘𝐿𝑝(⋅)}𝜒_{𝑖𝐿𝑝′(⋅)}⩽2^−𝑖𝑛𝜒_{𝐵𝑘𝐿𝑝(⋅)}𝜒_{𝐵𝑖𝐿𝑝′(⋅)}≲2^−𝑖𝑛𝜒_{𝐵𝑘𝐿𝑝(⋅)}𝐵_𝑖𝜒_𝐵𝑖𝐿⁻¹_𝑝(⋅)≲2^{𝑛𝛿1(𝑘−𝑖)}. (2.37)
We put (2.37) into (2.36) and get 𝐼𝐼𝐼_<≲⁻¹_𝑘=−∞2^{𝛼(0)𝑘𝑞∞}_𝑖=𝑘+2^∞_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_{𝑖𝐿𝑝(⋅)}2^{𝑛𝛿1(𝑘−𝑖)𝑞1/𝑞}≈⁻¹_𝑘=−∞^∞_𝑖=𝑘+22^{𝛼(0)𝑘∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_{𝑖L𝑝(⋅)}2^{𝑛𝛿1(𝑘−𝑖)𝑞1/𝑞}≈⁻¹_𝑘=−∞^∞_𝑖=𝑘+22^{𝛼(0)𝑖∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_{𝑖𝐿𝑝(⋅)}2^{𝑏(𝑘−𝑖)𝑞1/𝑞}, (2.38) here 𝑏=𝑛𝛿₁+𝛼(0)>0.
Similar to Step  2, we can make (2.38) further calculation and obtain 𝐼𝐼𝐼_<≲2^{𝛼(0)𝑘∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_{𝑘ℓ𝑞<(𝐿𝑝(⋅))}+2^{𝛼∞𝑘∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_{𝑘ℓ𝑞>(𝐿𝑝(⋅))}. (2.39)
Similarly, we have 𝐼𝐼𝐼_>≲^∞_𝑘=0^∞_𝑖=𝑘+22^{𝛼∞𝑖∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_{𝑖𝐿𝑝(⋅)}2^{𝑏(𝑘−𝑖)𝑞1/𝑞}, (2.40) here 𝑏=𝑛𝛿₁+𝛼_∞>0. By the same argument followed (2.28), we have 𝐼𝐼𝐼_>≲2^{𝛼∞𝑘∞}_𝑗=1𝑓_𝑗^𝑟1/𝑟𝜒_{𝑘ℓ𝑞>(𝐿𝑝(⋅))}. (2.41)
Thus 𝐼𝐼𝐼∶=𝐼𝐼𝐼_<+𝐼𝐼𝐼_>≲^∞_𝑗=1𝑓_𝑗^𝑟1/𝑟_{̇𝐾𝛼(⋅),𝑞𝑝(⋅)}. (2.42) We get what we want by putting 𝐼, 𝐼𝐼, and 𝐼𝐼𝐼 into (2.20).

We remark that there is an analogue of Theorem 2.8 for sublinear operators.

Theorem 2.9. Let 𝑝(⋅)∈ℬ(ℝ^𝑛),0<𝑞<∞ and 𝛼∈𝐿^∞(ℝ^𝑛) be log-Hölder continuous, both at the origin and at infinity, such that −𝑛𝛿₁<𝛼(0)⩽𝛼_∞<𝑛𝛿₂, where 0<𝛿₁,𝛿₂<1 are constants satisfying (2.4), Suppose that 𝑇 is a sublinear and bounded operator on 𝐿^𝑝(⋅)(ℝ^𝑛) satisfying size condition 𝑇𝑓(𝑥)⩽𝐶_ℝ𝑛𝑥−𝑦^−𝑛𝑓𝑦𝑑𝑦, (2.43) for all 𝑓∈𝐿¹_loc(ℝ^𝑛) with compact support and a.e. 𝑥∉supp𝑓, and for any 1<𝑟<∞, there exists a positive constant 𝐶 such that for all sequences {𝑓_𝑗}^∞_𝑗=1 of locally integrable functions on ℝ^𝑛, ^∞_𝑗=1𝑇𝑓_𝑗^𝑟1/𝑟_{𝐿𝑝(⋅)}⩽𝐶^∞_𝑗=1𝑓_𝑗^𝑟1/𝑟_{𝐿𝑝(⋅)}. (2.44) Then ^∞_𝑘=1𝑇𝑓_𝑘^𝑟1/𝑟_{̇𝐾𝛼(⋅),𝑞𝑝(⋅)(ℝ𝑛)}⩽𝐶^∞_𝑘=1𝑓_𝑘^𝑟1/𝑟_{̇𝐾𝛼(⋅),𝑞𝑝(⋅)(ℝ𝑛)},_∞𝑘=1𝑇𝑓_𝑘^𝑟1/𝑟_{𝐾𝛼(⋅),𝑞𝑝(⋅)(ℝ𝑛)}⩽𝐶^∞_𝑘=1𝑓_𝑘^𝑟1/𝑟_{𝐾𝛼(⋅),𝑞𝑝(⋅)(ℝ𝑛)}, (2.45) for all sequences {𝑓_𝑗}^∞_𝑗=1 of locally integrable functions on ℝ^𝑛, where 𝐶 is independent of {𝑓_𝑗}^∞_𝑗=1.

3. The Quasi-Norm Characterizations

Let 𝒮(ℝ^𝑛) be the Schwartz space on ℝ^𝑛,𝒮^′(ℝ^𝑛) be its dual space on ℝ^𝑛. For 𝜑∈𝒮(ℝ^𝑛),  𝜑 denotes its Fourier transform, and 𝜑^∨ denotes its inverse Fourier transform. Take 𝜑₀∈𝒮(ℝ^𝑛) with 𝜑₀(𝑥)≥0 and 𝜑₀(𝑥)=|𝑥||𝑥|1,⩽1,0,≥2. (3.1)

Now define 𝜑(𝑥)=𝜑₀(𝑥)−𝜑₀(2𝑥) and set 𝜑_𝑗(𝑥)=𝜑(2^−𝑗𝑥) for all 𝑗∈ℕ. Let ℕ₀=ℕ∪{0}. Then {𝜑_𝑗}_𝑗∈ℕ0 is a resolution of unity, that means ∑^∞_𝑗=0𝜑_𝑗(𝑥)=1 for all 𝑥∈ℝ^𝑛.

Now, we introduce the Herz-type Besov spaces and Triebel-Lizorkin spaces with variable exponents as following.

Definition 3.1. Let {𝜑_𝑗}_𝑗∈ℕ0 be a resolution of unity as above, 𝑠∈ℝ,0<𝛽,𝑞≤∞,𝑝(⋅)∈𝒫⁰(ℝ^𝑛) and 𝛼(⋅)∶ℝ^𝑛→ℝ with 𝛼(⋅)∈𝐿^∞(ℝ^𝑛).
(i) The set 𝑓∈𝒮^′ℝ^𝑛∶2^𝑠𝑗𝜑^∨_𝑗∗𝑓^∞_{𝑗=0ℓ𝛽(𝐾𝛼(⋅),𝑞𝑝(⋅))}<∞ (3.2) is named to the Herz-type Besov space with variable exponents and denoted by 𝐾^{𝛼(⋅),𝑞}_𝑝(⋅)𝐵^𝑠_𝛽(ℝ^𝑛). The quasi-norm of 𝑓 in this space is denoted by 𝑓_𝐾^{𝛼(⋅),𝑞}_{𝑝(⋅)𝐵𝑠𝛽(ℝ𝑛)}∶=2^𝑠𝑗𝜑^∨_𝑗∗𝑓^∞_{𝑗=0ℓ𝛽(𝐾𝛼(⋅),𝑞𝑝(⋅))} (3.3)
(ii) For 𝑝⁺<∞, the set 𝑓∈𝒮^′ℝ^𝑛∶2^𝑠𝑗𝜑^∨_𝑗∗𝑓^∞_{𝑗=0𝐾𝛼(⋅),𝑞𝑝(⋅)(ℓ𝛽)}<∞ (3.4) is named to the Herz-type Triebel-Lizorkin space with variable exponents and denoted by 𝐾^{𝛼(⋅),𝑞}_𝑝(⋅)𝐹^𝑠_𝛽(ℝ^𝑛). The quasi-norm of 𝑓 in this space is denoted by 𝑓_𝐾^{𝛼(⋅),𝑞}_{𝑝(⋅)𝐹𝑠𝛽(ℝ𝑛)}∶=2^𝑠𝑗𝜑^∨_𝑗∗𝑓^∞_{𝑗=0𝐾𝛼(⋅),𝑞𝑝(⋅)(ℓ𝛽)}. (3.5)

Here ℓ_𝛽(𝐾^{𝛼(⋅),𝑞}_𝑝(⋅)) and 𝐾^{𝛼(⋅),𝑞}_𝑝(⋅)(ℓ_𝛽) are the spaces of all sequences {𝑔_𝑗} of measurable functions on ℝ^𝑛 with finite quasi-norms {𝑔_𝑗}_{ℓ𝛽(𝐾𝛼(⋅),𝑞𝑝(⋅))}∶=^∞_𝑗=0𝑔_𝑗^𝛽_{𝐾𝛼(⋅),𝑞𝑝(⋅)}^1/𝛽,𝑔_𝑗^∞_{𝑗=0𝐾𝛼(⋅),𝑞𝑝(⋅)(ℓ𝛽)}∶=^∞_𝑗=0𝑔_𝑗^𝛽1/𝛽_{𝐾𝛼(⋅),𝑞𝑝(⋅)}. (3.6)

The spaces 𝐾̇^{𝛼(⋅),𝑞}_𝑝(⋅)𝐵^𝑠_𝛽(ℝ^𝑛) and 𝐾̇^{𝛼(⋅),𝑞}_𝑝(⋅)𝐹^𝑠_𝛽(ℝ^𝑛) are defined similarly by replacing the norm 𝐾^{𝛼(⋅),𝑞}_𝑝(⋅) with 𝐾̇^{𝛼(⋅),𝑞}_𝑝(⋅). To make these spaces definite, we need to show them independent of the choice of the resolution of unity {𝜑_𝑗}_𝑗∈ℕ0. To this aim we need more notation.

Let Ψ₀,Ψ∈𝒮(ℝ^𝑛),𝜖>0, integer 𝑆≥−1 be such that Ψ₀(𝜉𝜉Ψ(𝜉𝜀)>0on<2𝜀,)>0on2<𝜉𝐷<2𝜀,_𝜏Ψ(0)=0∀|𝜏|⩽𝑆. (3.7)(3.8)(3.9) Here (3.7) and (3.8) are Tauberian conditions, while (3.9) expresses moment conditions on Ψ.

Let us recall the classical Peetre maximal operator introduced in [34]. Given a sequence of function {Ψ_𝑘}_𝑘∈ℤ⊂𝒮(ℝ^𝑛), a tempered distribution 𝑓∈𝒮^′(ℝ^𝑛) and a positive number 𝑎>0, we define the system of maximal functions Ψ^∗_𝑘𝑎𝑓(𝑥)∶=sup_{𝑦∈ℝ𝑛}Ψ_𝑘∗𝑓𝑥+𝑦1+2^𝑘𝑦^𝑎,𝑥∈ℝ^𝑛,𝑘∈ℤ. (3.10) Since Ψ_𝑘∗𝑓(𝑦) makes sense pointwise everything is well defined. We will often use dilates Ψ_𝑘(𝑥)=2^𝑘𝑛Ψ(2^𝑘𝑥) of a fixed function Ψ∈𝒮(ℝ^𝑛), where Ψ₀(𝑥) might be given by a separate function. Also continuous dilates are needed. Let Ψ_𝑡(⋅)∶=𝑡^−𝑛Ψ(𝑡⁻¹⋅). We define (Ψ^∗_𝑡)_𝑎𝑓(𝑥) by Ψ^∗_𝑡𝑎𝑓(𝑥)∶=sup_{𝑦∈ℝ𝑛}Ψ_𝑡∗𝑓𝑥+𝑦1+𝑦/𝑡^𝑎𝑥∈ℝ^𝑛,𝑡>0. (3.11)

Now we have equivalent quasi-norms for these new spaces.

Theorem 3.2. Let 𝑠<𝑆+1,0<𝛽,𝑞⩽∞,𝑝(⋅)∈𝒫⁰(ℝ^𝑛) for 𝑝(⋅)/𝑝₀∈ℬ(ℝ^𝑛) with 𝑝₀<min(𝑝₋,1), 𝛼(⋅)∶ℝ^𝑛→ℝ with 𝛼(⋅)∈𝐿^∞(ℝ^𝑛) and 𝛼(⋅) be log-Hölder continuous, both at the origin and at infinity, such that −𝑛𝛿₁<𝛼(0)𝑝₀⩽𝛼_∞𝑝₀<𝑛𝛿₂, where 0<𝛿₁,𝛿₂<1 are constants satisfying (2.4) for 𝑝(⋅)/𝑝₀. And let 𝑎∈ℝ. Further, let Φ₀,Φ belong to 𝒮(ℝ^𝑛) be given by (3.7), (3.8), and (3.9).
(i) For 𝑎>𝑛/𝑝₀, then the space 𝐾^{𝛼(⋅),𝑞}_𝑝(⋅)𝐵^𝑠_𝛽 can be characterized by 𝐾_𝑝(⋅)^{𝛼(⋅),𝑞}𝐵^𝑠_𝛽ℝ^𝑛=𝑓∈𝒮^′ℝ^𝑛∶𝑓_𝐾^(𝑖)_{𝑝(⋅)𝛼(⋅),𝑞𝐵𝑠𝛽(ℝ𝑛)}<∞,𝑖=1,…,4, (3.12) where 𝑓_𝐾⁽¹⁾_{𝛼(⋅),𝑞𝑝(⋅)𝐵𝑠𝛽(ℝ𝑛)}∶=Φ₀∗𝑓_𝐾^{𝛼(⋅),𝑞}_𝑝(⋅)+¹₀𝑡^−𝑠𝛽Φ_𝑡∗𝑓(⋅)^𝛽_{𝐾𝑝(⋅)𝛼(⋅),𝑞(ℝ𝑛)}𝑑𝑡𝑡^1/𝛽,𝑓_𝐾⁽²⁾_{𝛼(⋅),𝑞𝑝(⋅)𝐵𝑠𝛽(ℝ𝑛)}∶=Φ^∗_0𝑎𝑓_𝐾^{𝛼(⋅),𝑞}_𝑝(⋅)+¹₀𝑡^−𝑠𝛽Φ^∗_𝑡𝑎𝑓^𝛽_{𝐾𝛼(⋅),𝑞𝑝(⋅)(ℝ𝑛)}𝑑𝑡𝑡^1/𝛽,𝑓_𝐾⁽³⁾_{𝛼(⋅),𝑞𝑝(⋅)𝐵𝑠𝛽(ℝ𝑛)}∶=^∞_𝑘=02^𝑠𝑘𝛽Φ^∗_𝑘𝑎𝑓^𝛽_{𝐾𝛼(⋅),𝑞𝑝(⋅)(ℝ𝑛)}^1/𝛽,𝑓_𝐾⁽⁴⁾_{𝛼(⋅),𝑞𝑝(⋅)𝐵𝑠𝛽(ℝ𝑛)}∶=^∞_𝑘=02^𝑠𝑘𝛽Φ_𝑘∗𝑓^𝛽_{𝐾𝛼(⋅),𝑞𝑝(⋅)(ℝ𝑛)}^1/𝛽. (3.13) Furthermore, {‖⋅‖_𝐾^(𝑖)_{𝛼(⋅),𝑞𝑝(⋅)𝐵𝑠𝛽(ℝ𝑛)}}⁴_𝑖=1 are equivalent.
(ii) If 𝑝₀<𝛽, then for 𝑎>𝑛/𝑝₀ the space 𝐾^{𝛼(⋅),𝑞}_𝑝(⋅)𝐹^𝑠_𝛽 can be characterized by 𝐾_𝑝(⋅)^{𝛼(⋅),𝑞}𝐹^𝑠_𝛽ℝ^𝑛=𝑓∈𝒮^′ℝ^𝑛∶𝑓_𝐾^(𝑖)_{𝑝(⋅)𝛼(⋅),𝑞𝐹𝑠𝛽(ℝ𝑛)}<∞,𝑖=1,…,5, (3.14) where 𝑓_𝐾⁽¹⁾_{𝑝(⋅)𝛼(⋅),𝑞𝐹𝑠𝛽(ℝ𝑛)}∶=Φ₀∗𝑓_𝐾^{𝛼(⋅),𝑞}_𝑝(⋅)+¹₀𝑡^−𝑠𝛽Φ_𝑡∗𝑓(⋅)^𝛽𝑑𝑡𝑡^1/𝛽_{𝐾𝛼(⋅),𝑞𝑝(⋅)(ℝ𝑛)},𝑓_𝐾⁽²⁾_{𝑝(⋅)𝛼(⋅),𝑞𝐹𝑠𝛽(ℝ𝑛)}∶=Φ^∗_0𝑎𝑓_𝐾^{𝛼(⋅),𝑞}_𝑝(⋅)+¹₀𝑡^−𝑠Φ^∗_𝑡𝑎𝑓^𝛽𝑑𝑡𝑡^1/𝛽_{𝐾𝛼(⋅),𝑞𝑝(⋅)(ℝ𝑛)},𝑓_𝐾⁽³⁾_{𝑝(⋅)𝛼(⋅),𝑞𝐹𝑠𝛽(ℝ𝑛)}∶=Φ₀∗𝑓_𝐾^{𝛼(⋅),𝑞}_{𝑝(⋅)(ℝ𝑛)}+¹₀𝑡^−𝑠𝛽_|𝑧|<𝑡Φ_𝑡∗𝑓(⋅+𝑧)^𝛽𝑑𝑧𝑑𝑡𝑡^𝑛+11/𝛽_{𝐾𝛼(⋅),𝑞𝑝(⋅)(ℝ𝑛)},𝑓_𝐾⁽⁴⁾_{𝑝(⋅)𝛼(⋅),𝑞𝐹𝑠𝛽(ℝ𝑛)}∶=^∞_𝑘=02^𝑘𝑠𝛽Φ^∗_𝑘𝑎𝑓^𝛽1/𝛽_{𝐾𝛼(⋅),𝑞𝑝(⋅)(ℝ𝑛)},𝑓_𝐾⁽⁵⁾_{𝑝(⋅)𝛼(⋅),𝑞𝐹𝑠𝛽(ℝ𝑛)}∶=^∞_𝑘=02^𝑘𝑠𝛽Φ_𝑘∗𝑓^𝛽1/𝛽_{𝐾𝛼(⋅),𝑞𝑝(⋅)}. (3.15)(3.16)(3.17)(3.18)(3.19) Furthermore, {‖⋅‖_𝐾^(𝑖)_{𝛼(⋅),𝑞𝑝(⋅)𝐹𝑠𝛽(ℝ𝑛)}}⁵_𝑖=1 are equivalent.

Theorem 3.2 is the generalization of Theorem 1 in [28] for the Herz type space 𝐾^𝛼,𝑞_𝑝(⋅) only one variable exponent 𝑝(⋅). For readers' convenience, we give the proof for 𝐹-parts in outline. The proof of 𝐵-parts is similar and a bit simpler. The idea of the proof goes back to [35]. In fact, we will use the argument in [35]. To go on, we need some lemmas.

Lemma 3.3 (see [36]). Let 𝜇,𝜈∈𝒮(ℝ^𝑛),−1⩽𝑀∈ℤ, 𝐷^𝜏𝜇(0)=0∀|𝜏|⩽𝑀. (3.20) Then for any 𝑁>0 there is a constant 𝐶_𝑁 so that sup_{𝑧∈ℝ𝑛}𝜇_𝑡∗𝜈(𝑧)(1+|𝑧|)^𝑁⩽𝐶_𝑁𝑡^𝑀+1, (3.21) where 𝜇_𝑡(𝑥)=𝑡^−𝑛𝜇(𝑥/𝑡) for all 0<𝑡⩽2.

Lemma 3.4 (see [36]). Let 0<𝑞⩽∞,𝛿>0. For any sequence {𝑔_𝑗}^∞₀ of nonnegative numbers denote 𝐺_𝑗=^∞_𝑘=02^{−|𝑘−𝑗|𝛿}𝑔_𝑘. (3.22) Then {𝐺_𝑗}^∞_0ℓ𝑞⩽𝐶{𝑔_𝑗}^∞_0ℓ𝑞, (3.23) holds, where 𝐶 is constant only depend on 𝑞,𝛿.

Lemma 3.5. Let 0<𝛽⩽∞,𝛿>0,0<𝑞⩽∞, 𝑝(⋅)∈𝒫⁰(ℝ^𝑛) and 𝛼(⋅)∶ℝ^𝑛→ℝ with 𝛼(⋅)∈𝐿^∞(ℝ^𝑛). For any sequence {𝑔_𝑗}^∞₀ of nonnegative measurable functions on ℝ^𝑛 denote 𝐺_𝑗(𝑥)=^∞_𝑘=02^{−|𝑘−𝑗|𝛿}𝑔_𝑘(𝑥),𝑥∈ℝ^𝑛. (3.24) Then {𝐺_𝑗}^∞_{0𝐾𝛼(⋅),𝑞𝑝(⋅)(ℓ𝛽)}⩽𝐶₁{𝑔_𝑗}^∞_{0𝐾𝛼(⋅),𝑞𝑝(⋅)(ℓ𝛽)},{𝐺_𝑗}^∞_{0ℓ𝛽(𝐾𝛼(⋅),𝑞𝑝(⋅))}⩽𝐶₂{𝑔_𝑗}^∞_{0ℓ𝛽(𝐾𝛼(⋅),𝑞𝑝(⋅))}, (3.25)(3.26) hold with some constants 𝐶₁=𝐶₁(𝑞,𝛿) and 𝐶₂=𝐶₂(𝑝(⋅),𝑞,𝛿).

Proof. By Lemma 3.4, (3.25) follows immediately from (3.23). Now we prove (3.26).
Let 𝑝(⋅)∈𝒫⁰(ℝ^𝑛). Here we can divide it into two cases.
Case  1 ( 𝑝₋≥1,𝑞≥1). Since ‖⋅‖_𝐾^{𝛼(⋅),𝑞}_𝑝(⋅) is a norm, we have 𝐺_{𝑗𝐾𝛼(⋅),𝑞𝑝(⋅)}⩽^∞_𝑘=02^{−|𝑘−𝑗|𝛿}𝑔_𝑘𝐾^{𝛼(⋅),𝑞}_𝑝(⋅). (3.27) By Lemma 3.4, we obtain (3.26).
Case  2. If 𝑞<1, take 𝑝₀<min(𝑝₋,𝑞). Then 𝐺_𝑗^𝑝0_{𝐾𝛼(⋅),𝑞𝑝(⋅)}=𝐺_𝑗^𝑝0_{𝐾000𝛼(⋅)𝑝,𝑞/𝑝𝑝(⋅)/𝑝}⩽^∞_𝑘=02^{−|𝑘−𝑗|𝑝0𝛿}𝑔_𝑘^𝑝0_{𝐾000𝛼(⋅)𝑝,𝑞/𝑝𝑝(⋅)/𝑝}⩽^∞_𝑘=02^{−|𝑘−𝑗|𝑝0𝛿}𝑔_𝑘^𝑝0_{𝐾000𝛼(⋅)𝑝,𝑞/𝑝𝑝(⋅)/𝑝}. (3.28) By Lemma 3.4, we obtain {𝐺_𝑗}^𝑝0_{ℓ𝛽(𝐾𝛼(⋅),𝑞𝑝(⋅))}=𝐺_𝑗^𝑝0_{ℓ0𝛽/𝑝(𝐾000𝛼(⋅)𝑝,𝑞/𝑝𝑝(⋅)/𝑝)}≲𝑔_𝑘^𝑝0_{ℓ0𝛽/𝑝(𝐾000𝛼(⋅)𝑝,𝑞/𝑝𝑝(⋅)/𝑝)}=𝐶{𝑔_𝑘}^𝑝0_{ℓ𝛽𝐾𝑝(⋅)𝛼(⋅),𝑞}. (3.29) In the last inequality, we used (3.26) has been proved for space 𝐾^{𝛼(⋅)𝑝0,𝑞/𝑝0}_{𝑝(⋅)/𝑝0}. Raising to power 1/𝑝₀, we obtain (3.26).

Lemma 3.6 (Theorem 6 in [37]). Let {𝜑_𝑗}_𝑗∈ℕ0 is a resolution of unity and let 𝑅∈ℕ. Then there exist functions 𝜃₀,𝜃∈𝒮(ℝ^𝑛) satisfy supp𝜃₀,supp𝜃⊆𝑥∈ℝ^𝑛∶|𝑥|,𝜃⩽1₀(𝜉𝜉,𝜃(𝜉𝜖)>0𝑜𝑛<2𝜖)>0𝑜𝑛2<𝜉<2𝜖_ℝ𝑛,𝑥^𝛾𝜃(𝑥)𝑑𝑥=0for0<𝛾⩽𝑅 (3.30) such that 𝜃₀(𝜉)𝜓₀(𝜉)+^∞_𝑗=1𝜃2^−𝑗𝜉𝜓2^−𝑗𝜉=1,∀𝜉∈ℝ^𝑛, (3.31) where the functions 𝜓₀,𝜓∈𝒮(ℝ^𝑛) are defined via 𝜓₀(𝜉)=𝜑₀(𝜉)/𝜃₀(𝜉) and 𝜓(𝜉)=𝜑₁(2𝜉)/𝜃(𝜉).

Proof of Theorem 3.2. We divide the total proof into four steps. First, we prove the equivalence of (3.15) and (3.16). The next step is to build the bridge between (3.16) and (3.18) and to change from the system (Φ₀,Φ) to a system (Ψ₀,Ψ). The equivalence of (3.18) and (3.19) goes parallel to (3.15) and (3.16). Indeed, Definition 3.1 can be seen as a special case of (3.19). Finally, we prove (3.19) is equivalent to the rest.
Step  1. We are going to prove that for every 𝑓∈𝒮^′(ℝ^𝑛)𝑓_𝐾⁽²⁾_{𝛼(⋅),𝑞𝑝(⋅)𝐹𝑠𝛽}≲𝑓_𝐾⁽¹⁾_{𝛼(⋅),𝑞𝑝(⋅)𝐹𝑠𝛽}≲𝑓_𝐾⁽²⁾_{𝛼(⋅),𝑞𝑝(⋅)𝐹𝑠𝛽}. (3.32)
From Lemmas 3.3 and 3.6, we have that, see [35], for 𝑟<min{𝑝₋,𝛽},𝑁∈ℕ, there exists positive constant 𝐶 for 𝑓∈𝒮(ℝ^𝑛), ²₁2^𝑙𝑠Φ^∗_2−𝑙𝑡𝑓_𝑎(𝑥)^𝛽𝑑𝑡𝑡^𝑟/𝛽⩽𝐶_{𝑘∈𝑙+ℕ0}2^{(𝑙−𝑘)(𝑁𝑟−𝑛+𝑟𝑠)}2^𝑘𝑟𝑠×ℳ²₁Φ_𝑘𝑡∗𝑓(⋅)^𝛽𝑑𝑡𝑡^𝑟/𝛽(𝑥). (3.33)
Take now 𝑛/𝑎<𝑟=𝑝₀<min{𝑝₋,𝛽},𝑁>max{0,−𝑠}+𝑎 and put 𝛿=𝑁+𝑠−𝑑/𝑟>0. We obtain for 𝑙∈ℕ²₁2^𝑙𝑠Φ^∗_2−𝑙𝑡𝑓_𝑎(𝑥)^𝛽𝑑𝑡𝑡^𝑟/𝛽≲_{𝑘∈𝑙+ℕ0}2^{−𝛿𝑟|𝑙−𝑘|}2^𝑘𝑟𝑠ℳ²₁Φ_𝑘𝑡∗𝑓(⋅)^𝛽𝑑𝑡𝑡^𝑟/𝛽(𝑥). (3.34)
Now we apply Lemma 3.5 in 𝐾^{𝑟𝛼(⋅),𝑞/𝑟}_{𝑝(⋅)/𝑟}(ℓ_𝛽/𝑟) which yields ²₁2^𝑙𝑠Φ^∗_2−𝑙𝑡𝑓_𝑎(𝑥)^𝛽𝑑𝑡𝑡^𝑟/𝛽_{𝑙∈ℕ𝐾𝑟𝛼(⋅),𝑞/𝑟𝑝(⋅)/𝑟(ℓ𝛽/𝑟)}≲ℳ²₁2^𝑘𝑠Φ_𝑘𝑡∗𝑓(⋅)^𝛽𝑑𝑡𝑡^𝑟/𝛽_{𝑙∈ℕ𝐾𝑟𝛼(⋅),𝑞/𝑟𝑝(⋅)/𝑟(ℓ𝛽/𝑟)}. (3.35)
Next, using Theorem 2.8, we can obtain ²₁2^𝑙𝑠Φ^∗_2−𝑙𝑡𝑓_𝑎(𝑥)^𝛽𝑑𝑡𝑡^𝑟/𝛽_{𝑙∈ℕ𝐾𝑟𝛼(⋅),𝑞/𝑟𝑝(⋅)/𝑟(ℓ𝛽/𝑟)}≲ℳ²₁2^𝑘𝑠Φ_𝑘𝑡∗𝑓(⋅)^𝛽𝑑𝑡𝑡^𝑟/𝛽_{𝑙∈ℕ𝐾𝑟𝛼(⋅),𝑞/𝑟𝑝(⋅)/𝑟(ℓ𝛽/𝑟)}≲²₁2^𝑘𝑠Φ_𝑘𝑡∗𝑓(⋅)^𝛽𝑑𝑡𝑡^𝑟/𝛽_{𝑘∈ℕ𝐾𝑟𝛼(⋅),𝑞/𝑟𝑝(⋅)/𝑟(ℓ𝛽/𝑟)}=²₁2^𝑘𝑠Φ_𝑘𝑡∗𝑓(⋅)^𝛽𝑑𝑡𝑡^1/𝛽_𝑘∈ℕ^𝑟_{𝐾𝛼(⋅),𝑞𝑝(⋅)(ℓ𝛽)}. (3.36)
Hence, we obtain ¹₀𝜆^−𝑠Φ^∗_𝜆𝑓_𝑎(⋅)^𝛽𝑑𝜆𝜆^1/𝛽_{𝐾𝛼(⋅),𝑞𝑝(⋅)}≈^∞_𝑙=1²₁2^𝑙𝑠Φ^∗_2−𝑙𝑡𝑓_𝑎(⋅)^𝛽𝑑𝑡𝑡^1/𝛽_{𝐾𝛼(⋅),𝑞𝑝(⋅)}≲²₁2^𝑙𝑠Φ₂^−𝑙_𝑡∗𝑓(⋅)^𝛽𝑑𝑡𝑡^1/𝛽_{𝑙∈ℕ𝐾𝛼(⋅),𝑞𝑝(⋅)(ℓ𝛽)}∼¹₀𝜆^−𝑠Φ_𝜆∗𝑓(⋅)^𝛽𝑑𝜆𝜆^1/𝛽_{𝐾𝛼(⋅),𝑞𝑝(⋅)}. (3.37) This proves ‖𝑓‖_𝐾⁽²⁾_{𝛼(⋅),𝑞𝑝(⋅)𝐹𝑠𝛽}≲‖𝑓‖_𝐾⁽¹⁾_{𝛼(⋅),𝑞𝑝(⋅)𝐹𝑠𝛽}. Since the reverse inequality is trivial, this finishes Step  1.
Step  2. Let Ψ₀,Ψ∈𝒮^′(ℝ^𝑛) be functions satisfying (3.25).
First, we are going to prove for all 𝑓∈𝒮^′(ℝ^𝑛)𝑓_𝐾⁽⁴⁾_{𝛼(⋅),𝑞𝑝(⋅)𝐹𝑠𝛽(ℝ𝑛,Ψ)}≲𝑓_𝐾⁽²⁾_{𝑝(⋅)𝛼(⋅),𝑞𝐹𝑠𝛽(ℝ𝑛),Φ}. (3.38)
Again from Lemmas 3.6 and 3.3, we have that, see [35], let 𝛿=min{1,𝑆+1−𝑠}, there exists a positive constant 𝐶 for any 𝑓∈𝒮(ℝ^𝑛), 2^𝑙𝑠Ψ^∗_𝑙𝑓_𝑎(𝑥)⩽𝐶_𝑘∈ℕ02^{−|𝑘−𝑙|𝛿}2^𝑘𝑠Φ^∗_2−𝑘𝑡𝑓_𝑎(𝑥), (3.39) for all 𝑥∈ℝ^𝑛 and all 𝑡∈[1,2].
Suppose first that 𝛽≥1. Then we take on both sides (∫²₁|⋅|^𝛽𝑑𝑡/𝑡)^1/𝛽, which gives 2^𝑙𝑠Ψ^∗_𝑙𝑓_𝑎(𝑥)≲_𝑘∈ℕ02^{−|𝑘−𝑙|𝛿}2^𝑘𝑠2₁Φ^∗_2−𝑘𝑡𝑓_𝑎(𝑥)^𝛽𝑑𝑡𝑡^1/𝛽. (3.40) Applying Lemma 3.5 we obtain that 2^𝑙𝑠Ψ^∗_𝑙𝑓_{𝑎𝑙∈ℕ𝐾𝛼(⋅),𝑞𝑝(⋅)(ℓ𝛽)}≲^∞_𝑘=12^𝑘𝑠𝛽Φ^∗_2−𝑘𝑡𝑓_𝑎(𝑥)^𝛽𝑑𝑡𝑡^1/𝛽_{𝐾𝛼(⋅),𝑞𝑝(⋅)}, (3.41) which gives the desired result.
In case 𝛽<1 we argue as follows. The quantity (∫²₁|⋅|^𝛽𝑑𝑡/𝑡)^1/𝛽 is no longer a norm. This gives 2^𝑙𝑠Ψ^∗_𝑙𝑓_𝑎(𝑥)^𝛽≲_𝑘∈ℕ02^{−𝛽|𝑘−𝑙|𝛿}2^{𝑘𝑠𝛽2}₁Φ^∗_2−𝑘𝑡𝑓_𝑎(𝑥)^𝛽𝑑𝑡𝑡. (3.42) Notice that the right-hand side is nothing more than a convolution (𝛾∗𝛼(⋅))_ℓ of the sequences 𝛾_𝑘=2^{−|𝑘|𝛿𝛽},𝛼(⋅)_𝑘=2^{𝑘𝑠𝛽2}₁Φ^∗_2−𝑘𝑡𝑓_𝑎(𝑥)^𝛽𝑑𝑡𝑡. (3.43) Now we apply the ℓ₁-norm to both sides and get for all 𝑥∈ℝ^𝑛2^𝑙𝑠Ψ^∗_𝑙𝑓_𝑎(𝑥)^𝛽_ℓ𝛽⩽𝛾_ℓ1⋅‖𝛼(⋅)‖_ℓ1≲^∞_𝑘=12^{𝑘𝑠𝛽2}₁Φ^∗_2−𝑘𝑡𝑓_𝑎(𝑥)^𝛽𝑑𝑡𝑡. (3.44)
We take both sides to power (⋯)^1/𝛽 and apply the 𝐾^{𝛼(⋅),𝑞}_𝑝(⋅)(ℝ^𝑛)-norm. This gives (3.38).
Similarly, we obtain for all 𝑓∈𝒮^′(ℝ^𝑛)𝑓_𝐾⁽²⁾_{𝛼(⋅),𝑞𝑝(⋅)𝐹𝑠𝛽(ℝ𝑛,Φ)}≲𝑓_𝐾⁽⁴⁾_{𝛼(⋅),𝑞𝑝(⋅)𝐹𝑠𝛽(ℝ𝑛,Ψ)}. (3.45)Step  3. Choosing 𝑡=1 in Step  1 and omitting the integration over 𝑡 we see immediately 𝑓_𝐾⁽⁵⁾_{𝛼(⋅),𝑞𝑝(⋅)𝐹𝑠𝛽(ℝ𝑛)}≲𝑓_𝐾⁽⁴⁾_{𝛼(⋅),𝑞𝑝(⋅)𝐹𝑠𝛽(ℝ𝑛)}≲𝑓_𝐾⁽⁵⁾_{𝛼(⋅),𝑞𝑝(⋅)𝐹𝑠𝛽(ℝ𝑛)}. (3.46)Step  4. What remains is to show that (3.19) is equivalent to the rest.
First, let us prove that for any 𝑓∈𝒮(ℝ^𝑛)𝑓_𝐾⁽²⁾_{𝛼(⋅),𝑞𝑝(⋅)𝐹𝑠𝛽(ℝ𝑛)}≲𝑓_𝐾⁽³⁾_{𝛼(⋅),𝑞𝑝(⋅)𝐹𝑠𝛽(ℝ𝑛)}. (3.47)
From [35], for 0<𝑟<min{𝑝₋,𝛽}, there exists a positive constant 𝐶 for any 𝑓∈𝒮(ℝ^𝑛)²₁Ψ^∗_2−𝑙𝑡𝑓_𝑎(𝑥)^𝛽𝑑𝑡𝑡^𝑟/𝛽⩽𝐶_𝑘∈ℕ02^{−𝑘𝑁𝑠}2^{(𝑘+𝑙)𝑛}_ℝ𝑛∫²₁∫_{|𝑧|<2−(𝑘+𝑙)𝑡}Φ_{𝑘+𝑙𝑡}∗𝑓𝑧+𝑦^𝛽𝑑𝑧(𝑑𝑡/𝑡^𝑛+1)^𝑟/𝛽1+2^𝑙𝑥−𝑦^𝑎𝑟𝑑𝑦. (3.48)
If 𝑎𝑟>𝑛 then we have 𝑔_𝑙𝑦=2^𝑛𝑙1+2^𝑙𝑦^𝑎𝑟∈𝐿₁ℝ^𝑛. (3.49) Thus we have ²₁2^𝑙𝑠Φ^∗_2−𝑙𝑡𝑓_𝑎(𝑥)^𝛽𝑑𝑡𝑡^𝑟/𝛽≲_𝑘∈ℕ02^{−𝑘𝑁𝑟}2^𝑘𝑛2^𝑙𝑠𝑟𝑔_𝑙∗²_{1|𝑧|<2−(𝑘+𝑙)𝑡}Φ_{𝑘+𝑙𝑡}∗𝑓(𝑧+⋅)^𝛽𝑑𝑧𝑑𝑡𝑡^{𝑛+1𝑟/𝛽}(𝑥). (3.50)
Now we use the majorant property of the Hardy-Littlewood maximal operator in [38] and continue estimating ²₁2^𝑙𝑠Φ^∗_2−𝑙𝑡𝑓_𝑎(𝑥)^𝛽𝑑𝑡𝑡^𝑟/𝛽≲_𝑘∈ℕ02^𝑙𝑠𝑟2^{𝑘(−𝑁𝑟+𝑛)}ℳ²_{1|𝑧|<2−(𝑘+𝑙)𝑡}Φ_{𝑘+𝑙𝑡}∗𝑓(𝑧+⋅)^𝛽𝑑𝑧𝑑𝑡𝑡^{𝑛+1𝑟/𝛽}(𝑥). (3.51) An index shift on the right-hand side gives ²₁2^𝑙𝑠Φ^∗_2−𝑙𝑡𝑓_𝑎(𝑥)^𝛽𝑑𝑡𝑡^𝑟/𝛽≲_{𝑘∈𝑙+ℕ0}2^𝑙𝑠𝑟2^{(𝑘−𝑙)(−𝑁𝑟+𝑛)}ℳ²_{1|𝑧|<2−𝑘𝑡}Φ_𝑘𝑡∗𝑓(𝑧+⋅)^𝛽𝑑𝑧𝑑𝑡𝑡^{𝑛+1𝑟/𝛽}(𝑥)=𝐶_{𝑘∈𝑙+ℕ0}2^{(𝑙−𝑘)(𝑁𝑟−𝑛+𝑟𝑠)}2^𝑘𝑟𝑠ℳ²_{1|𝑧|<2−𝑘𝑡}Φ_𝑘𝑡∗𝑓(𝑧+⋅)^𝛽𝑑𝑧𝑑𝑡𝑡^{𝑛+1𝑟/𝛽}(𝑥). (3.52) Then with analogous arguments as after (3.39), we obtain (3.47).
Second, we prove ‖𝑓‖_𝐾⁽³⁾_{𝛼(⋅),𝑞𝑝(⋅)𝐹𝑠𝛽(ℝ𝑛)}≲‖𝑓‖_𝐾⁽²⁾_{𝛼(⋅),𝑞𝑝(⋅)𝐹𝑠𝛽(ℝ𝑛)}. Since for all 𝑡>01𝑡^𝑛_|𝑧|<𝑡Φ_𝑡∗𝑓(𝑥+𝑧)𝑑𝑧≲sup_|𝑧|<𝑡Φ_𝑡∗𝑓(𝑥+𝑧)(1+1/𝑡|𝑧|)^𝑎≲Φ^∗_𝑡𝑓_𝑎(𝑥). (3.53) Therefore we can obtain what we want. The proof is complete.

Remark 3.7. One can define homogenous spaces 𝐾^{𝛼(⋅),𝑞}_𝑝(⋅)𝐹̇^𝑠_𝛽(ℝ^𝑛),𝐾^{𝛼(⋅),𝑞}_𝑝(⋅)𝐵̇^𝑠_𝛽(ℝ^𝑛),𝐾̇^{𝛼(⋅),𝑞}_𝑝(⋅)𝐹̇^𝑠_𝛽(ℝ^𝑛) and 𝐾̇^{𝛼(⋅),𝑞}_𝑝(⋅)𝐵̇^𝑠_𝛽(ℝ^𝑛). Then there is an analogue of Theorem 3.2 for these spaces. We omit the detail here and leave it for interested readers.

Acknowledgments

The authors would like to thank the referee for his carefully reading and suggestions which made the presentation more readable. J. Xu is supported by the National Natural Science Foundation of China (Grant no. 11071064 and 11226167) and the Natural Science Foundation of Hainan Province (no. 111006) and the Scientific Research Foundation of Hainan Province Education Bureau (no. Hjkj2011-19).