The aim of this paper is to establish a characterization of the anisotropic variable Herz spaces K˙p(·)α(·),q(·)(A Rn) associated with non-isotropic dilations A on Rn in terms of block decompositions. All exponents in the considered spaces are variable. First, we define the set of variable exponents as follows:

P0Rn:=p measurable: p·:Rn→c,∞ for some c>0.

The subset of variable exponents with a range of [1,∞) is denoted by P(Rn). For p∈P0(Rn), we introduce the notation

p-=essinfx∈Rnp(x), p+=esssupx∈Rnp(x).

Now we give the definition of variable Lebesgue spaces.

Definition 1.1. Let p∈P0(Rn). The Lebesgue space Lp(·)(Rn), with a variable exponent is the class of all measurable functions f on Rn such that the modular

ϱp(·)(f):=∫Rn∣f(x)∣p(x)dx

is finite. This space is a quasi-Banach function space equipped with the norm

‖f‖p(·):=infμ>0:ϱp(·)(1μf)⩽1.

If p(x)≡p is constant, then Lp(·)(Rn)=Lp(Rn) is the classical Lebesgue space. We refer to the monographs [3] and [4] for further details and references on recent developments on variable Lebesgue spaces.

We present the most important condition on the exponent in the study of variable exponent spaces.

Definition 1.2. We say that a function g:Rn→R is locally log-Hölder continuous if there exists a constant clog>0 such that

∣g(x)-g(y)∣⩽cloglog(e+1/∣x-y∣)

for all x,y∈Rn. In particular, if

∣g(x)-g(0)∣⩽cloglog(e+1/∣x∣)

for all x∈Rn, then we say that g is log-Hölder continuous at the origin (or has a log decay at the origin). Additionally, if there exist g∞∈R and clog>0 such that

∣g(x)-g∞∣⩽cloglog(e+∣x∣)

for all x∈Rn, then we say that g is log-Hölder continuous at infinity (or has a log decay at infinity).

For some examples of a function locally log-Hölder continuous, see E. Nakai and Y. Sawano [7, Example 1.3].

The sets P0log(Rn) and P∞log(Rn) consist of all exponents p∈P(Rn) that have a log decay at the origin and at infinity, respectively. The set Plog(Rn) is used for all those exponents p∈P(Rn) that are locally log-Hölder continuous and have a log decay at infinity, with p∞:=lim∣x∣→∞p(x).

It is well known that if p∈Plog(Rn), then p′∈Plog(Rn), where p′ denotes the conjugate exponent of p given by 1/p(·)+1/p′(·)=1.

Definition 1.3. Let p,q∈P0(Rn). The mixed Lebesgue-sequence space ℓq(·)(Lp(·)) is defined on sequences of Lp(·)-functions by the modular

ϱℓq(·)(Lp·)((fv)v)=∑vinfλv>0:ϱp(·)(fvλv1/q(·))⩽1.

A (quasi)-norm is defined from this as usual:

(1.4) fvvlq(⋅)(Lp⋅)=infγ>0:ϱlq(⋅)(Lp(⋅))(1γ(fv)v)⩽1.

If q(·) satisfies q+<∞ , then we can replace (1.4) by the simpler expression

ϱlq(⋅)(Lp(⋅))((fv)v)=∑ v|fv |q(⋅)p(⋅)q(⋅).

If E⊂Rn is a measurable set, then ∣E∣ stands for the (Lebesgue) measure of E and χE denotes its characteristic function.

In the following, we introduce some basic notation and definitions of non-isotropic spaces associated with general expansive dilations.

Definition 1.5. A dilation is n×n real matrix A, such that all eigenvalues λ of A satisfy ∣λ∣>1. We suppose λ1,λ2,...,λn are eigenvalues of A so that 1<∣λ1∣⩽...⩽∣λn∣. Let λ-,λ+ be any numbers so that

1<λ-<∣λ1∣⩽...⩽∣λn∣<λ+.

A set △⊂Rn is said to be an ellipsoid if

△=x∈Rn:∣Px∣<1

for some nondegenerate n×n matrix P, where ∣·∣ denotes the Euclidean norm in Rn.

In [2, Lemma 2.2], it is demonstrated that for a dilation A, there exists an ellipsoid △ and r>1 satisfying

(1.6)△⊂r△⊂A△,  where ∣△∣=1.

For convenience, we set

Bk=Ak△ for k∈Z,

then, by (1.6) we obtain

Bk⊂rBk⊂Bk+1, ∣Bk∣=bk,

where A∣>1.

Definition 1.7. A homogeneous quasi-norm associated with a dilation A is a measurable mapping σA:ℝn→0,∞, so that

• σ​Ax>0 for x≠0,

• σ​AAx=bσAx for all x∈Rn,

• there is c>0 so that σAx+y⩽cσAx+σAy for all x,y∈Rn.

For a fixed dilation A, we define the “canonical” quasi-norm σ.

Definition 1.8. Define the step homogeneous quasi-norm σ on ℝn induced by the dilation A as

σx=bjifx∈Bj+1\Bj, j∈ℤ0ifx=0.

For any x,y∈Rn, we have

σx+y⩽bθσx+σy,

where θ is the smallest integer so that

2B0⊂AθB0=Bθ.

Also, we use the following notation

Rk:=Bk∖Bk-1 and χk=χRk,k∈Z.

Now, we define the anisotropic Herz spaces with variable exponent.

Definition 1.9. Let p,q∈P0(Rn) and α:Rn→R with α∈L∞(Rn). The homogeneous anisotropic Herz space K˙p(⋅)α(⋅),q(⋅)A;ℝn associated with the dilation A is defined as the set of all f∈Llocp(·)Rn∖{0} such that

‖f‖K˙p(⋅)α(⋅),q(⋅)A;ℝn:=‖bkα(⋅)fχk k∈ℤ‖lq(⋅)(Lp⋅)<∞.

The non-homogeneous anisotropic Herz space Kp(⋅)α(⋅),q(⋅)A;ℝn associated with the dilation A consists of all f∈Llocp(·)Rn such that

‖f‖Kp(⋅)α(⋅),q(⋅)A;ℝn:=fχB0 p⋅+‖bkα(⋅)fχk k⩾1‖lq(⋅)(Lp⋅)<∞.

Clearly, K˙p(·)0,p(·)A Rn=Kp(·)0,p(·)A Rn=Lp(·)A Rn. Recall that the anisotropic Herz spaces Kp(⋅)α(⋅),qA;ℝn and K˙p(⋅)α(⋅),qA;ℝn, where q is constant, are introduced by H. Wang in [8]. A detailed discussion of the properties of these spaces may be found in [9] and [10].

By the same argument used in [6], we can establish the next result, which will be useful in the sequel.

Proposition 1.10. Let α∈L∞(Rn) and p,q∈P0(Rn). If α and q are log-Hölder continuous at

infinity, then

Kp(⋅)α(⋅),q(⋅)A;ℝn=Kp(⋅)α∞,q∞A;ℝn.

Additionally, if α and q have a log decay at the origin, then

‖f‖K˙p(⋅)α(⋅),q(⋅)A;ℝn≈(∑ k=−∞ −1‖bkα(0)fχk‖p(⋅)q(0))1/q(0)+(∑ k=0∞‖bkα ∞ fχk‖p(⋅)q ∞ )1/q∞.

Recall that the expression f≲g means that f⩽cg for some independent constant c (and non-negative functions f and g), and f≈g means f≲g≲f.

In this section, we introduce several lemmas used to prove the main theorems in sections 3 and 4. In the following, we denote by c as a generic positive constant, i.e. a constant whose value may change from line to line.

The following lemma plays an important role in the proof of the main results.

Lemma 2.1. Let p∈P(Rn) and Rk:=Bk∖Bk-1,k∈Z. If bk⩾2-n and p is log-Hölder continuous at infinity, then we have

‖χk‖p(·)≈bkp∞,

with the implicit constants independent of k.

Proof. Our proof based on an idea from [1, Lemma 2.2] where the case of the Euclidean ball was studied. First, we have

‖χk‖p(·)≈bkp∞,

which is equivalent to

‖b-kp∞χk‖p(·)≈1.

In particular, we will show that

(2.2)ϱp(⋅)(b−kp∞χk):=∫ℝn|b−kp∞χkx|p(y)dy=b−k∫Rk b k p ∞−p(y) p ∞ dy≲c,

for some constant c>0. For that, it is sufficient to prove that bkp∞-p(y)p∞ is bounded, i.e. k(p∞-p(y)p∞)logb⩽c for all y∈Rk.

Since p is log-Hölder continuous at infinity, then (2.2) is bounded by

(2.3)b−kp(y)−p∞p∞≲b−k1log(e+y)≲b−k/logy,  y∈Rk.

We can distinguish two cases as follows:

Case 1: For every integer k⩾0, due to [2, Lemma 3.2] and Definition 1.8, we deduce that

bklogλ−/logb≲y≲ϱ(x)logλ+/logb=bklogλ+/logb for k⩾0,

for all y∈Rk. This implies that (2.3) is bounded by

b-kklog(b)logλ+/logb=e-logλ+logb≲c.

Case 2: Considering the case [-nlog2logb]⩽k⩽-1, i.e. 2-n⩽bk<1, we have (2.3) bounded by

−kp(y)−p∞p∞logb ⩽p(y)−p∞log1bk ⩽ 2np+log2 ≲ c.

In either case, we obtain that (2.2) is bounded by

b-k∫Rkdy≲c.

Now, we show that bkp∞≲‖χk‖p(·). This is a consequence of Hölder's inequality and the estimate ‖χk‖p′(·)≲bkp∞′ which was already proved. In fact, we have

bkp∞=bkp∞-k∫Rnχk(y)dy⩽2b-kp∞′‖χk‖p(·)‖χk‖p′(·)≲‖χk‖p(·).

This finishes the proof.

Remark 2.4. It is known that for p∈Plog, we have

‖χB‖p(·)‖χB‖p′(·)≈∣B∣.

Also,

‖χB‖p(·)≈∣B∣1p(x), x∈B,

for small balls B⊂Rn and

‖χB‖p(·)≈∣B∣1p∞

for large balls (∣B∣⩾1), with constants only depending on the log-Hölder constant of p. See, for example, [3, Corollary 4.5.9].

The next lemma is a Hardy-type inequality which is easy to prove.

Lemma 2.5 ([5]) Let γ>1, κ>0 and 0<q⩽∞. Let εkk∈ℤ be a sequence of positive real numbers, such that

εkk∈ℤlq=I<∞.

Then, the sequences

{δk:δk=∑j⩽kγ-(k-j)κεj}k∈Zand{ηk:ηk=∑j⩾kγ-(j-k)κεj}k∈Z

belong to ℓq, and

δkk∈ℤlq+ ηkk∈ℤlq⩽cI,

with c>0 only depending on γ and q.

The following lemma presents the Hölder inequality in Lp·(Rn).

Lemma 2.6 ([3]). Let p∈P(Rn). Then, there exists a constant c such that for all f∈Lp·(Rn) and g∈Lp′⋅(ℝn),f⋅g∈L1(ℝn), and

f⋅g1⩽cfp(⋅)gp′(⋅).

Now, we establish characterizations of the spaces K˙p(⋅)α(⋅),q(⋅)A;ℝn and

Kp(⋅)α(⋅),q(⋅)A;ℝn in terms of central bloc decompositions, which will be convenient for the study of the boundedness of operators on these spaces.

Let us first recall the definition of bloc decomposition.

Definition 3.1. Let α∈L∞(Rn), be log-Hölder continuous, both at the origin and at infinity and p∈P(Rn). A function ak is said to be a central α(·),p(·)-bloc, if

(i) suppak=x∈R​n:akx≠0¯⊂Bk.

(ii) akp(⋅)⩽b−kα(0),  k<0.

(iii) akp(⋅)⩽b−kα∞,  k⩾0.

A function ak on Rn is said to be a central (α(·),p(·))-bloc of restricted type, if it satisfies the condition (iii) and suppak⊂Bk,k⩾0.

Remark 3.2. If α and p are constants, then we recover the classical case.

One of the main results of this paper will be the following theorem. It generalizes Theorem 2.3 of H. Wang [8] by taking q as a constant.

Theorem 3.3. Let α∈L∞(Rn),p∈Plog(Rn) and q∈P0(Rn). If α and q are log-Hölder continuous, both at the origin and at infinity with α(0),α∞>0, then the following two statements are equivalentes

• (i)

  f∈ K˙p(⋅)α(⋅),q(⋅)A;ℝn

• (ii)

  f can be represented by

(3.4)fx=∑k=-∞∞βkakx,  

where βk⩾0, each ak is a central (α(·),p(·))-block with support contained in Bk and

(∑ k=−∞ −1βkq(0))1q(0)+(∑ k=0∞βkq ∞)1q ∞⩽cfK˙p(⋅)α(⋅),q(⋅)A;ℝn.

Moreover, the norms ‖f‖K˙p(⋅)α(⋅),q(⋅)A;ℝn and

inf(∑ k=−∞ −1βkq(0)1q(0)+(∑ k=0∞βkq ∞ )1q ∞)

are equivalent, where the infimum is taken over all decompositions of f as in (3.4).

Proof. The idea of the proof is borrowed from [5], where the variable Herz-type Hardy spaces case is studied.

First, we show that (i) implies (ii). For every f∈ K˙p(⋅)α(⋅),q(⋅)A;ℝn, we have

fx=∑ k=−∞∞fxχkx=∑ k=−∞∞ b kα⋅fχk p⋅fxχkx b kα⋅fχk p⋅=∑ k=−∞∞βkakx,

where

βk=bkα⋅fχk p⋅ and akx=fxχkxbkα⋅fχkp⋅.

It is obvious that suppak⊂Bk and

akp(⋅)≈b−kα0 , if k⩽−1,b−kα∞ , if k⩾0.

Thus, each ak is a central (α(·),p(·))- bloc with the support Bk and

(∑−1k=−∞ βk q(0))1 q(0)+(∑∞k=0 βk q∞)1 q ∞ =(∑−1k=−∞ b kα⋅ fχk p⋅q(0))1 q(0)+(∑∞k=0 b kα⋅ fχk p⋅q∞)1 q ∞ ≈(∑−1k=−∞ bkα0q(0) fχk p⋅q(0))1 q(0)+(∑∞k=0 bkα∞q∞ fχk p⋅q∞)1 q ∞ ≈‖f‖ K˙ p(⋅) α(⋅),q(⋅) A;ℝn .

It remains to prove that (ii) implies (i). For this purpose, let f(x)=∑k=-∞∞βkakx be a decomposition of f that satisfies the hypothesis (ii) of Theorem 3.3, by the Minkowski inequality, we obtain

(3.5)fχjp⋅⩽∑ k=j∞|βk|akp⋅ for each j∈ℤ.

From this, (3.5), and Proposition 1.10, it follows that fK˙p(⋅)α(⋅),q(⋅)A;ℝn is bounded by

c(∑−1k=−∞bkα0q(0)(∑j=k∞|βj|a j p⋅)q(0))1 q(0)+c(∑∞k=0bkα∞ q∞ (∑j=k∞|βj|a j p⋅)q∞ )1 q ∞ =I1+I2.

Then we deal with I1 and I2, separatly. For I1, we divide the sum ∑j=k∞··· into two parts,

∑j=k-1···+∑j=0∞···.

I1 is bounded by I1a+I1b, where

I1a:=c(∑−1k=−∞(bkα0 ∑ j=k −1|βj|aj p⋅)q(0))1q(0)

and

I1b:=c(∑−1k=−∞(bkα0 ∑ j=0∞|βj|ajp⋅)q(0))1q(0).

Since 0<α(0)<∞, then by Lemma 2.5 (with γ=bα(0)>1), we get

I1a⩽c(∑−1k=−∞( ∑ j=k −1|βj|b−(j−k)α(0))q(0))1q(0)⩽c(∑−1k=−∞βkq(0))1q(0).

By Hölder's inequality in ℓ1 with 1q∞+1q∞′=1 and since α(0),α∞>0, we obtain

I1b⩽c(∑−1k=−∞ bkα0q(0)(∑j=0∞|βj|b −jα ∞ )q(0))1 q(0)⩽c(∑−1k=−∞ bkα0q(0))1 q(0)(∑ j=0∞|βj|q∞ )1 q ∞ (∑ j=0∞ b −j α∞ q∞′ )1 q ∞′ ⩽c(∑ j=0∞|βj|q∞ )1 q ∞ ⩽c‖f‖K˙ p(⋅) α(⋅),q(⋅) A;ℝn .

Thus, we have the desired estimate for I1.

Next, we deal with I2. We have

I2=(∑∞k=0(bkα∞ ∑ j=k ∞|βj| aj p⋅)q∞)1q∞⩽(∑∞k=0( ∑ j=k ∞|βj|b−(j−k)α∞)q∞)1q∞.

Since 0<α∞<∞, then by Lemma 2.5 (with γ=bα∞>1), we deduce that

I2⩽(∑∞k=0βkq∞)1q∞⩽ (∑−1k=−∞βkq(0))1q(0)+(∑∞k=0βkq∞)1q∞.

This finishs the estimation of I2 and the proof of Theorem 3.3.

Remark 3.6. A non-homogeneous counterpart of Theorem 3.3 is available. Since Kp(⋅)α(⋅),q(⋅)A;ℝn=Kp(⋅)α∞,q∞A;ℝn, its proof is an immediate consequence of [8, Theorem 2.5].

The next result concerns the boundedness, on anisotropic variable Herz spaces, of some sublinear operators T satisfying the size condition

(4.1)∣Tf(x)∣≲∫Rn∣f(y)∣ϱ(x-y)dy, x∉suppf

for integrable and compactly supported functions f.

Theorem 4.2. Let α∈L∞(ℝn),p∈P(ℝn),q∈P0(ℝn), and if α,p and q are log-Hölder continuous, both at the origin and at infinity such

0<α0<1−1/p(0) and 0<α∞<1−1/p∞.

Then every sublinear operator T satisfying (4.1) which is bounded on Lp(⋅)(ℝn) is also bounded on K˙p(⋅)α(⋅),q(⋅)A;ℝn and Kp(⋅)α(⋅),q(⋅)A;ℝn, respectively.

Proof. It suffices to prove that T is bounded on K˙p(⋅)α(⋅),q(⋅)A;ℝn. The non-homogeneous case can be proved similarly. We must show that

TfK˙p(⋅)α(⋅),q(⋅)A;ℝn⩽cfK˙p(⋅)α(⋅),q(⋅)A;ℝn

for all f∈K˙p(⋅)α(⋅),q(⋅)A;ℝn. Thanks to Theorem 3.3, it holds that

f=∑i=-∞∞βiai

where βi⩾0 and ai's are α(⋅),p(⋅)- bloc with suppai⊆Bi. Hence, we obtain

TfK˙ p(⋅) α(⋅),q(⋅) A;ℝn≈(∑ k=−∞ −1bkα(0)q(0)( ∑i=−∞∞ βiTa i⋅χk p(⋅))q(0))1/q(0)+(∑ k=0∞bkα∞ q∞ ( ∑i=−∞∞ βiTa i⋅χk p(⋅)) q∞ )1/q∞≲(∑ k=−∞ −1bkα(0)q(0)( ∑i=−∞k−θ−1 βiTa i⋅χk p(⋅))q(0))1/q(0)+(∑ k=−∞ −1bkα(0)q(0)( ∑i=k−θ∞ βiTa i⋅χk p(⋅))q(0))1/q(0)+(∑ k=0∞bkα∞ q∞ ( ∑i=−∞k−θ−1 βiTa i⋅χk p(⋅)) q∞ )1/q∞+(∑ k=0∞bkα∞ q∞ ( ∑i=k−θ∞ βiTa i⋅χk p(⋅)) q∞ )1/q∞:=J1+J2+J3+J4.

First, we estimate J1. Since

σx⩽bθσx-y+σy,

and taking x∈Rk,y∈Bi with k⩾i+θ+1, then x∈Bi+θ+1\Bi+θ, and we get

σ(x−y)⩾b−θσ(x)−σ(y)=b−θσ(x)−bi−1=b−θσ(x)−b−θ−1σ(x)=b−θ(1−1b)σ(x).

The condition (4.1) gives

Taix≲∫Bi a i y ϱ(x)dy⩽ cb−k∫Bi a i y dy.

By Lemma 2.6 and the condition (ii) in Definition 3.1, we get

Taix≲b−k aip(⋅) χ Bi p′(⋅)⩽cb−k−i(α0−1+1/p(0)),

which implies that

Tai⋅χkp(⋅)⩽cb−k−i(α0−1+1/p(0))‖χk‖p(⋅)⩽cbk(−1+1/p(0))−i(α0−1+1/p(0)).

By Lemma 2.5 (with γ=b-α(0)+1-1/p(0)>1), we have

J1≲(∑k=−∞−1(∑i=−∞k−θ−1 βi b− k−i(−α0+1−1/p(0)))q(0))1/q(0)≲( ∑ k=−∞−1βk q(0))1/q(0)⩽c‖f‖ K˙ p(⋅)α(⋅),q(⋅) A;ℝn .

To estimate J2, we distinguish two cases, k-θ<0 and k-θ⩾0. Here we assume that k-θ<0. The other case will follow in the same way.

We divide the sum ∑i=k-θ∞··· into two parts

∑i=k-θ-1···+∑i=0∞···,

then J2 is bounded by J2a+J2b, where

J2a:=(∑k=−∞−1b kα(0)q(0)( ∑ i=k−θ −1 βi T a i ⋅χk p(⋅))q(0))1/q(0)J2b:=(∑k=−∞−1b kα(0)q(0)( ∑ i=0∞ βi T a i ⋅χk p(⋅))q(0))1/q(0).

For J2a, the numbers k and i are negatives numbers. Then, by the Lp(·)(Rn)-boundedness of T, Definition 3.1 and Lemma 2.5 (with γ=bα0>1), we deduce that

J2a≲(∑k=−∞−1b kα(0)q(0)( ∑ i=k−θ −1 βi a i p(⋅))q(0))1/q(0)≲(∑k=−∞−1(∑i=k−θ−1βi b−(i−k)α(0))q(0))1/q(0)≲( ∑ k=−∞−1 βkq(0) )1/q(0)⩽c‖f‖K˙ p(⋅)α(⋅),q(⋅)A;ℝn .

For J2b, we have k ⩽-1 and i⩾0. By the Lp(·)(Rn)-boundedness of T and Definition 3.1, we have

J2b:=(∑k=−∞−1 b kα(0)q(0)( ∑ i=0∞ βi Ta i ⋅χk p(⋅))q(0))1/q(0)≲(∑k=−∞−1 b kα(0)q(0)( ∑ i=0∞ βi a i p(⋅))q(0))1/q(0)≲(∑k=−∞−1 b kα(0)q(0)( ∑ i=0∞ βi b−iα ∞ )q(0))1/q(0).

By Hölder's inequality in ℓ1 with 1q∞+1q∞′=1 and since α(0),α∞>0, we obtain

J2b≲(∑k=−∞−1 b kα(0)q(0))1/q(0)(∑i=0∞ βi q ∞ )1/q∞(∑i=0∞ b −iα ∞ q ∞′ )1/q∞′≲(∑i=0∞ βi q ∞ )1/q∞⩽c‖f‖K˙ p(⋅)α(⋅),q(⋅)A;ℝn .

Next, we estimate J3. We distinguish two cases, k-θ-1⩾0 and k-θ-1<0. Here we assume that k-θ-1⩾0. The other case follows similarly.

Let us decompose the sum ∑i=-∞k-θ-1··· into two parts

∑i=-∞-1···+∑i=0k-θ-1···.

Then J3 is bounded by J3a+J3b, where

J3a:=∑ k=0∞bkα ∞q ∞∑ i=−∞ −1βiT ai⋅χkp(⋅)q ∞1/q∞,J3b:=∑ k=0∞bkα ∞q ∞∑ i=0 k−θ−1βiT ai⋅χkp(⋅)q ∞1/q∞.

For J3a, we have k⩾0 and i⩽-1. By the condition (ii) in Definition 3.1 and Lemma 2.5, we obtain

Tai⋅χkp(⋅)≲b−k−i(α0−1+1/p0))‖χk‖p(⋅)≲bk(−1+1/p∞)−i(α0−1+1/p0),

which gives

J3a≲(∑k=0∞bk(α∞-1+1/p∞)q∞(∑i=-∞-1βib-i(α0-1+1/p0))​q∞)​1/q∞.

Thanks to Hölder's inequality in ℓ1, with 1q(0)+1q′(0)=1, we easily obtain

J3a≲(∑k=0∞b k(α ∞−1+1 p ∞)q ∞)1q∞ (∑i=−∞−1 βi q(0) )1q(0)(∑i=−∞−1b −i(α0−1+1 p0)q′(0))1 q′ (0)≲(∑i=−∞−1 βi q(0) )1/q(0)⩽c‖f‖K˙ p(⋅)α(⋅),q(⋅)A;ℝn .

Concerning J3b, where k and i are non-negatives numbers, we have by the condition (iii) in Definition 3.1 and Lemma 2.5

Tai⋅χkp(⋅)≲b−k−i(α∞−1+1/p∞))‖χk‖p(⋅)≲bk(−1+1/p∞)−i(α∞−1+1/p∞),

which gives

J3b:=(∑k=0∞ b kα ∞q ∞( ∑ i=0 k−θ−1 βi Ta i⋅χkp(⋅))q ∞ )1/q∞≲(∑k=0∞(∑i=0k−θ−1βi b−(k−i)(−α ∞ +1−1/p ∞ ))q∞)1/q∞,

by Lemma 2.5 (with γ=b-α∞+1-1/p∞>1), we obtain

J3b≲(∑ k=0∞βkq ∞)1/q∞⩽c‖f‖K˙p(⋅)α(⋅),q(⋅)A;ℝn.

Finally, we estimate J4. In this case k and i are non-negatives numbers, then by the Lp(·)(Rn)-boundedness of T, the condition (iii) in Definition 3.1 and Lemma 2.5 (with γ=bα∞>1), we easily obtain that

J4≲(∑k=0∞b kα ∞ q ∞( ∑ i=k−θ ∞ βi a ip(⋅)) q ∞ )1/q∞≲(∑k=0∞(∑i=k−θ∞ βi b− i−kα ∞ )q∞)1/q∞≲(∑k=0∞βk q ∞)1/q∞⩽c‖f‖K˙ p(⋅)α(⋅),q(⋅)A;ℝn .

Combing the estimations of J1,J2,J3 and J4, we finish the proof of Theorem 4.2.

Remark 4.3. We would like to mention that if q(·) is constant, then the statements corresponding to Theorem 4.2 can be found in Theorem 3.1 of [8].