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

Authors: Baohua Dong; Jingshi Xu
Journal: Journal of Function Spaces and Applications
 (2012)

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## 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.


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## Body

## 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 Kovác˘ik 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 K˙p(·)α,q(ℝn) and Kp(·)α,q(ℝn) 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 K˙p(·)α,q(ℝn) and Kp(·)α,q(ℝn) 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, K˙p(·)α,q(ℝn)Bβs and Kp(·)α,q(ℝn)Bβs and K˙p(·)α,q(ℝn)Fβs and Kp(·)α,q(ℝn)Fβs, 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 K˙p(·)α(·),q(ℝn) and Kp(·)α(·),q(ℝn) 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 Section2 we will establish the boundedness of vector-valued Hardy-Littlewood maximal operator in spaces K˙p(·)α(·),q(ℝn) and Kp(·)α(·),q(ℝn). 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 χS the characteristics function for a measurable set S⊂ℝn. 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.
LetE be a measurable set in ℝn with |E|>0. Let p(·):E→[1,∞) be a measurable function. Denote
(2.1)Lp(·)(E)≔{fismeasurableonE:ρp(·)(fλ)<∞forsomeconstantλ>0},
where ρp(·)(f)≔∫E‍|f(x)|p(x)dx, and
(2.2)∥f∥Lp(·)(E)≔inf{λ>0:ρp(·)(fλ)⩽1}.
Then Lp(·)(E) is a Banach space with the norm ∥·∥Lp(·)(E).LetLloc1(ℝn) be the collection of all locally integrable functions on ℝn. Given a function f∈Lloc1(ℝn), the Hardy-Littlewood maximal operator ℳ is defined by
(2.3)ℳf(x)≔supr>0r-n∫B(x,r)‍|f(y)|dy,∀x∈ℝn,
where B(x,r)≔{y∈ℝn:|x-y|<r}. We also use the following notation: p-≔essinf{p(x):x∈ℝn} and p+≔esssup{p(x):x∈ℝn}. The set 𝒫(ℝn) consists of all p(·) satisfying p->1 and p+<∞.  ℬ(ℝn) is the set of p(·)∈𝒫(ℝn) satisfying the condition that ℳ is bounded on Lp(·)(ℝn). It is well known that if p(·)∈𝒫(ℝn) then p(·)∈ℬ(ℝn) is equivalent to p′(·)∈ℬ(ℝn), where p′(·) is the conjugate exponent to p(·), that means p′(·)=p(·)/(p(·)-1); see [12]. For more about the set ℬ(ℝn), one can see [11, 12, 31, 32]. Moreover, we define 𝒫0(ℝn) to be the set of measurable functions p on ℝn with the range in (0,∞) such that p-≔essinf{p(x):x∈ℝn}>0 and p+≔esssup{p(x):x∈ℝn}<∞. Given p(·)∈𝒫0(ℝn), one can define the space Lp(·)(ℝn) as in Definition 2.1. This is equivalent to defining it to be the set of all functions f such that |f|p0∈Lp(·)/p0(ℝn), where 0<p0<p- and p(·)/p0∈𝒫(ℝn). We also define a quasi-norm on this space by ∥f∥Lp(·)(ℝn)=∥|f|p0∥Lp(·)/p0(ℝn)1/p0.Lemma 2.2 (see [25]).
Letp(·)∈ℬ(ℝn). Then there exist 0<δ1,δ2<1 depending only on p(·) and n such that for balls B in ℝn and all measurable subsets S⊂B,
(2.4)∥χS∥Lp(·)(ℝn)∥χB∥Lp(·)(ℝn)⩽C(|S||B|)δ1,∥χS∥Lp′(·)(ℝn)∥χB∥Lp′(·)(ℝn)⩽C(|S||B|)δ2.Lemma 2.3 (see [25]).
Letp(·)∈ℬ(ℝn). Then there exists a positive constant C>0 such that for balls B in ℝn,
(2.5)1|B|∥χB∥Lp(·)(ℝn)∥χB∥Lp′(·)(ℝn)⩽C.Lemma 2.4 (see [11]).
Letp(·)∈ℬ(ℝn) and 1<r<∞, then there exists a positive constant C such that for all sequences {fj}j=1∞ of locally integrable functions on ℝn,
(2.6)∥(∑j=1∞‍|ℳfj|r)1/r∥Lp(·)⩽C∥(∑j=1∞‍|fj|r)1/r∥Lp(·).For giving the definition of the Herz spaces with variable exponent, let us introduce the following notations. Letk∈ℤ.Bk≔{x∈ℝn:|x|⩽2k}.  Rk≔Bk∖Bk-1.  χk≔χRk.ℕ
0 denotes the set of all nonnegative integers. For m∈ℕ0, we denote χ~m≔χRm if m≥1 and χ~0≔χB0.Definition 2.5.
Let0<q⩽∞,p(·)∈𝒫(ℝn) and α(·):ℝn→ℝ with α(·)∈L∞(ℝn).(i)
The homogeneous Herz spaceK˙p(·)α(·),q(ℝn) is defined by
(2.7)K˙p(·)α(·),q(ℝn)≔{f∈Llocp(·)(ℝn∖0):∥f∥K˙p(·)α(·),q(ℝn)<∞},where(2.8)∥f∥K˙p(·)α(·),q(ℝn)≔{∑k=-∞∞‍∥2α(·)kfχk∥Lp(·)(ℝn)q}1/q.(ii)
The nonhomogeneous Herz spaceKp(·)α(·),q(ℝn) is defined by
(2.9)Kp(·)α(·),q(ℝn)≔{f∈Llocp(·)(ℝn):∥f∥Kp(·)α(·),q(ℝn)<∞},where(2.10)∥f∥Kp(·)α(·),q(ℝn)≔{∑m=0∞‍∥2α(·)mfχ~m∥Lp(·)(ℝn)q}1/q.Definition 2.6.
A functionα(·):ℝn→ℝ is called locally log-Hölder continuous on ℝn if there exists clog>0 such that
(2.11)|α(x)-α(y)|⩽cloglog(e+1/|x-y|),
for all x,y∈ℝn. If |α(x)-α(0)|⩽clog/log(e+1/|x|) for all x∈ℝn, 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 clog>0 such that
(2.12)|α(x)-α∞|⩽cloglog(e+|x|),
for all x∈ℝn.Byℓq,q∈(0,∞], we denote the discrete Lebesgue space equipped with the usual quasi-norm. Let us denote
(2.13)∥{gk}∥ℓ>q(Lp(·))≔(∑k=0∞‍∥gk∥p(·)q)1/q,∥{gk}∥ℓ<q(Lp(·))≔(∑k=-∞-1‍∥gk∥p(·)q)1/q,
for sequences {gk}k∈ℤ of measurable functions (with the usual modification when q=∞.)Proposition 2.7.
Letα(·)∈L∞(ℝn),p(·)∈𝒫(ℝn) and q∈(0,∞]. If α(·) satisfies log-Hölder decay condition at infinity, then
(2.14)Kp(·)α(·),q(ℝn)=Kp(·)α∞,q(ℝn).
Additionally, ifα(·) is log-Hölder continuous at the origin, then
(2.15)∥f∥K˙p(·)α(·),q(ℝn)≈∥{2kα(0)fχk}∥ℓ<q(Lp(·))+∥{2kα∞fχk}∥ℓ>q(Lp(·)).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.
Letp(·)∈ℬ(ℝn),0<q<∞,α(·):ℝn→ℝ with α(·)∈L∞(ℝn) 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 {fj}j=1∞ of locally integrable functions on ℝn such that
(2.16)∥(∑k=1∞‍|ℳfk|r)1/r∥K˙p(·)α(·),q(ℝn)⩽C∥(∑k=1∞‍|fk|r)1/r∥K˙p(·)α(·),q(ℝn),∥(∑k=1∞‍|ℳfk|r)1/r∥Kp(·)α(·),q(ℝn)⩽C∥(∑k=1∞‍|fk|r)1/r∥Kp(·)α(·),q(ℝn).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{fj}j=1∞ of locally measurable functions on ℝn such that (∑k=1∞‍|fk|r)1/r∈K˙p(·)α(·),q(ℝn).
Firstly, by Proposition2.7 we have
(2.17)∥(∑j=1∞‍|ℳfj|r)1/r∥K˙p(·)α(·),q(ℝn)≈∥{2kα(0)(∑j=1∞‍|ℳfj|r)1/rχk}∥ℓ<q(Lp(·))+∥{2kα∞(∑j=1∞‍|ℳfj|r)1/rχk}∥ℓ>q(Lp(·))≔A<+A>.
To continue, we let fji=χifj. Now we estimate A< by Minkowski inequality as follows:
(2.18)A<≔{∑k=-∞-1‍2α(0)qk∥χk(∑j=1∞‍|ℳfj|r)1/r∥Lp(·)q}1/q={∑k=-∞-1‍2α(0)qk∥χk(∑j=1∞‍|ℳ∑i=-∞∞‍fji|r)1/r∥Lp(·)q}1/q⩽{∑k=-∞-1‍2α(0)qk∥χk∑i=-∞∞‍(∑j=1∞‍|ℳfji|r)1/r∥Lp(·)q}1/q≲{∑k=-∞-1‍2α(0)qk∥χk∑i=-∞k-2‍(∑j=1∞‍|ℳfji|r)1/r∥Lp(·)q}1/q+{∑k=-∞-1‍2α(0)qk∥χk∑i=k-1k+1‍(∑j=1∞‍|ℳfji|r)1/r∥Lp(·)q}1/q+{∑k=-∞-1‍2α(0)qk∥χk∑i=k+2∞‍(∑j=1∞‍|ℳfji|r)1/r∥Lp(·)q}1/q≔I<+II<+III<.
By the same way we considerA>,
(2.19)A>≲{∑k=0∞‍2α∞qk∥χk∑i=-∞k-2‍(∑j=1∞‍|ℳfji|r)1/r∥Lp(·)q}1/q+{∑k=0∞‍2α∞qk∥χk∑i=k-1k+1‍(∑j=1∞‍|ℳfji|r)1/r∥Lp(·)q}1/q+{∑k=0∞‍2α∞qk∥χk∑i=k+2∞‍(∑j=1∞‍|ℳfji|r)1/r∥Lp(·)q}1/q≔I>+II>+III>.
So we get(2.20)∥(∑j=1∞‍|ℳfj|r)1/r∥K˙p(·)α(·),q(ℝn)≲I+II+III,
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
(2.21)II<≲{∑k=-∞-1‍2α(0)qk∑i=k-1k+1‍∥(∑j=1∞‍|fji|r)1/r∥Lp(·)q}1/q≲{∑k=-∞-1‍2α(0)qk∑i=k-1k+1‍∥χi(∑j=1∞‍|fj|r)1/r∥Lp(·)q}1/q≲{∑k=-∞-1‍2α(0)qk∥χk-1(∑j=1∞‍|fj|r)1/r∥Lp(·)q}1/q+{∑k=-∞-1‍2α(0)qk∥χk(∑j=1∞‍|fj|r)1/r∥Lp(·)q}1/q+{∑k=-∞-1‍2α(0)qk∥χk+1(∑j=1∞‍|fj|r)1/r∥Lp(·)q}1/q≲{∑k=-∞-2‍2α(0)qk∥χk(∑j=1∞‍|fj|r)1/r∥Lp(·)q}1/q+∥{2α(0)k(∑j=1∞‍|fj|r)1/rχk}∥ℓ<q(Lp(·))+{∑k=-∞0‍2α(0)qk∥χk(∑j=1∞‍|fj|r)1/r∥Lp(·)q}1/q≲∥{2α(0)k(∑j=1∞‍|fj|r)1/rχk}∥ℓ<q(Lp(·)).
Similarly, we have
(2.22)II>≲{∑k=-1∞‍2α∞qk∥χk(∑j=1∞‍|fj|r)1/r∥Lp(·)q}1/q+∥{2α∞k(∑j=1∞‍|fj|r)1/rχk}∥ℓ>q(Lp(·))+{∑k=1∞‍2α∞qk∥χk(∑j=1∞‍|fj|r)1/r∥Lp(·)q}1/q≲∥{2α∞k(∑j=1∞‍|fj|r)1/rχk}∥ℓ>q(Lp(·)).
Thus
(2.23)II≔II<+II>≲∥(∑j=1∞‍|fj|r)1/r∥K˙p(·)α(·),q.Step  2. We consider I. We recall that the Hardy-Littlewood maximal operator satisfies the size condition
(2.24)ℳf(x)⩽C∫ℝn‍|x-y|-n|f(y)|dy,
for all f∈Lloc1(ℝn) with compact support and a.e x∉suppf.
Therefore, for for alli⩽k-2,x∈Rk,1<r<∞, by the size condition and the generalized Minkowski inequality, we obtain
(2.25)(∑j=1∞‍ℳr(fji)(x))1/r≲[∑j=1∞‍(2-kn∫ℝn‍|fji(y)|dy)r]1/r≲2-kn∫ℝn‍(∑j=1∞‍|fji|r)1/rdy.
By Hölder's inequality we get(2.26)I<≲{∑k=-∞-1‍2α(0)qk∥χk∑i=-∞k-2‍2-kn∫ℝn‍(∑j=1∞‍|fji|r)1/rdy∥Lp(·)q}1/q≈{∑k=-∞-1‍2α(0)qk∥χk∥Lp(·)q(∑i=-∞k-2‍2-kn∫ℝn‍(∑j=1∞‍|fji|r)1/rdy)q}1/q≲{∑k=-∞-1‍2α(0)qk∥χk∥Lp(·)q×(∑i=-∞k-2‍2-kn∥(∑j=1∞‍|fj|r)1/rχi∥Lp(·)∥χi∥Lp′(·))q}1/q.
On the other hand, by using Lemmas2.2 and 2.3, we have
(2.27)2-kn∥χk∥Lp(·)∥χi∥Lp′(·)≲2-kn∥χBi∥Lp′(·)|Bk|∥χBk∥Lp′(·)-1≲2nδ2(i-k).
We put (2.27) into (2.26) and get
(2.28)I<≲{∑k=-∞-1‍2α(0)kq(∑i=-∞k-2‍∥(∑j=1∞‍|fj|r)1/rχi∥Lp(·)2nδ2(i-k))q}1/q≈{∑k=-∞-1‍(∑i=-∞k-2‍2α(0)k∥(∑j=1∞‍|fj|r)1/rχi∥Lp(·)2nδ2(i-k))q}1/q≈{∑k=-∞-1‍(∑i=-∞k-2‍2α(0)i∥(∑j=1∞‍|fj|r)1/rχi∥Lp(·)2b(i-k))q}1/q,
here b=nδ2-α(0)>0.
For1<q<∞, we can make (2.28) further calculation by Hölder's inequality
(2.29)I<≲{∑k=-∞-1‍(∑i=-∞k-2‍2α(0)iq∥(∑j=1∞‍|fj|r)1/rχi∥Lp(·)q2bq(i-k)/2)(∑i=-∞k-2‍2bq′(i-k)/2)q/q′}1/q≲{∑k=-∞-1‍∑i=-∞k-2‍2α(0)iq∥(∑j=1∞‍|fj|r)1/rχi∥Lp(·)q2bq(i-k)/2}1/q≈{∑i=-∞-3‍2α(0)iq∥(∑j=1∞‍|fj|r)1/rχi∥Lp(·)q∑k=i+2-1‍2bq(i-k)/2}1/q≲{∑i=-∞-3‍2α(0)iq∥(∑j=1∞‍|fj|r)1/rχi∥Lp(·)q}1/q≲∥{2α(0)k(∑j=1∞‍|fj|r)1/rχk}∥ℓ<q(Lp(·)).
For 0<q⩽1, we have
(2.30)I<≲{∑k=-∞-1‍∑i=-∞k-2‍2α(0)iq∥(∑j=1∞‍|fj|r)1/rχi∥Lp(·)q2bq(i-k)}1/q≈{∑i=-∞-3‍2α(0)iq∥(∑j=1∞‍|fj|r)1/rχi∥Lp(·)q∑k=i+2-1‍2bq(i-k)}1/q≲{∑i=-∞-3‍2α(0)iq∥(∑j=1∞‍|fj|r)1/rχi∥Lp(·)q}1/q≲∥{2α(0)k(∑j=1∞‍|fj|r)1/rχk}∥ℓ<q(Lp(·)).
Similarly, we have(2.31)I>≲{∑k=0∞‍(∑i=-∞k-2‍2α∞i∥(∑j=1∞‍|fj|r)1/rχi∥Lp(·)2b(i-k))q}1/q,
here b=nδ2-α∞>0.
For1<q<∞, we have
(2.32)I>≲{∑k=0∞‍∑i=-∞k-2‍2α∞iq∥(∑j=1∞‍|fj|r)1/rχi∥Lp(·)q2bq(i-k)/2}1/q≈{∑i=-∞-2‍2α∞iq∥(∑j=1∞‍|fj|r)1/rχi∥Lp(·)q∑k=0∞‍2bq(i-k)/2+∑i=-1∞‍2α∞iq∥(∑j=1∞‍|fj|r)1/rχi∥Lp(·)q∑k=i+2∞‍2bq(i-k)/2}1/q≲{∑i=-∞-2‍2(α∞+nδ2)iq/2∥(∑j=1∞‍|fj|r)1/rχi∥Lp(·)q+∑i=-1∞‍2α∞iq∥(∑j=1∞‍|fj|r)1/rχi∥Lp(·)q}1/q≲{∑i=-∞-2‍2α(0)iq+((α∞-2α(0)+nδ2)iq)/2∥(∑j=1∞‍|fj|r)1/rχi∥Lp(·)q+∑i=-1∞‍2α∞iq∥(∑j=1∞‍|fj|r)1/rχi∥Lp(·)q}1/q≲{∑i=-∞-2‍2α(0)iq∥(∑j=1∞‍|fj|r)1/rχi∥Lp(·)q+∑i=-1∞‍2α∞iq∥(∑j=1∞‍|fj|r)1/rχi∥Lp(·)q}1/q≲∥{2α(0)k(∑j=1∞‍|fj|r)1/rχk}∥ℓ<q(Lp(·))+∥{2α∞k(∑j=1∞‍|fj|r)1/rχk}∥ℓ>q(Lp(·)).
For 0<q⩽1, we have
(2.33)I>≲{∑k=0∞‍∑i=-∞k-2‍2α∞iq∥(∑j=1∞‍|fj|r)1/rχi∥Lp(·)q2bq(i-k)}1/q≈{∑i=-∞-2‍2α∞iq∥(∑j=1∞‍|fj|r)1/rχi∥Lp(·)q∑k=0∞‍2bq(i-k)+∑i=-1∞‍2α∞iq∥(∑j=1∞‍|fj|r)1/rχi∥Lp(·)q∑k=i+2∞‍2bq(i-k)}1/q≲{∑i=-∞-2‍2nδ2iq∥(∑j=1∞‍|fj|r)1/rχi∥Lp(·)q+∑i=-1∞‍2α∞iq∥(∑j=1∞‍|fj|r)1/rχi∥Lp(·)q}1/q≲{∑i=-∞-2‍2α(0)iq+(nδ2-α(0))iq∥(∑j=1∞‍|fj|r)1/rχi∥Lp(·)q+∑i=-1∞‍2α∞iq∥(∑j=1∞‍|fj|r)1/rχi∥Lp(·)q}1/q≲{∑i=-∞-2‍2α(0)iq∥(∑j=1∞‍|fj|r)1/rχi∥Lp(·)q+∑i=-1∞‍2α∞iq∥(∑j=1∞‍|fj|r)1/rχi∥Lp(·)q}1/q≲∥{2α(0)k(∑j=1∞‍|fj|r)1/rχk}∥ℓ<q(Lp(·))+∥{2α∞k(∑j=1∞‍|fj|r)1/rχk}∥ℓ>q(Lp(·)).
Thus
(2.34)I≔I<+I>≲∥(∑j=1∞‍|fj|r)1/r∥K˙p(·)α,q.Step  3. We consider III.
For for alli≥k+2,x∈Rk, by size condition and generalized Minkowski inequality, we have
(2.35)(∑j=1∞‍Mr(fji)(x))1/r≲[∑j=1∞‍(2-in∫ℝn‍|fji(y)|dy)r]1/r≈2-in(∑j=1∞‍(∫ℝn‍|fji(y)|dy)r)1/r≲2-in∫ℝn‍(∑j=1∞‍|fji|r)1/rdy.
By Hölder's inequality we get(2.36)III<≲{∑k=-∞-1‍2α(0)qk∥χk∑i=k+2∞‍2-in∫ℝn‍(∑j=1∞‍|fji|r)1/rdy∥Lp(·)q}1/q≈{∑k=-∞-1‍2α(0)qk∥χk∥Lp(·)q(∑i=k+2∞‍2-in∫ℝn‍(∑j=1∞‍|fji|r)1/rdy)q}1/q≲{∑k=-∞-1‍2α(0)qk∥χk∥Lp(·)q×(∑i=k+2∞‍2-in∥(∑j=1∞‍|fj|r)1/rχi∥Lp(·)∥χi∥Lp′(·))q}1/q.
On the other hand, by using Lemmas2.2 and 2.3, we have
(2.37)2-in∥χk∥Lp(·)∥χi∥Lp′(·)⩽2-in∥χBk∥Lp(·)∥χBi∥Lp′(·)≲2-in∥χBk∥Lp(·)|Bi|∥χBi∥Lp(·)-1≲2nδ1(k-i).
We put (2.37) into (2.36) and get
(2.38)III<≲{∑k=-∞-1‍2α(0)kq(∑i=k+2∞‍∥(∑j=1∞‍|fj|r)1/rχi∥Lp(·)2nδ1(k-i))q}1/q≈{∑k=-∞-1‍(∑i=k+2∞‍2α(0)k∥(∑j=1∞‍|fj|r)1/rχi∥Lp(·)2nδ1(k-i))q}1/q≈{∑k=-∞-1‍(∑i=k+2∞‍2α(0)i∥(∑j=1∞‍|fj|r)1/rχi∥Lp(·)2b(k-i))q}1/q,
here b=nδ1+α(0)>0.
Similar to Step  2, we can make (2.38) further calculation and obtain
(2.39)III<≲∥{2α(0)k(∑j=1∞‍|fj|r)1/rχk}∥ℓ<q(Lp(·))+∥{2α∞k(∑j=1∞‍|fj|r)1/rχk}∥ℓ>q(Lp(·)).
Similarly, we have(2.40)III>≲{∑k=0∞‍(∑i=k+2∞‍2α∞i∥(∑j=1∞‍|fj|r)1/rχi∥Lp(·)2b(k-i))q}1/q,
here b=nδ1+α∞>0. By the same argument followed (2.28), we have
(2.41)III>≲∥{2α∞k(∑j=1∞‍|fj|r)1/rχk}∥ℓ>q(Lp(·)).
Thus(2.42)III≔III<+III>≲∥(∑j=1∞‍|fj|r)1/r∥K˙p(·)α(·),q.
We get what we want by putting I, II, and III into (2.20).We remark that there is an analogue of Theorem2.8 for sublinear operators.Theorem 2.9.
Letp(·)∈ℬ(ℝn),0<q<∞ and α∈L∞(ℝn) 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 Lp(·)(ℝn) satisfying size condition
(2.43)|Tf(x)|⩽C∫ℝn‍|x-y|-n|f(y)|dy,
for all f∈Lloc1(ℝn) with compact support and a.e. x∉suppf, and for any 1<r<∞, there exists a positive constant C such that for all sequences {fj}j=1∞ of locally integrable functions on ℝn,
(2.44)∥(∑j=1∞‍|Tfj|r)1/r∥Lp(·)⩽C∥(∑j=1∞‍|fj|r)1/r∥Lp(·).
Then
(2.45)∥(∑k=1∞‍|Tfk|r)1/r∥K˙p(·)α(·),q(ℝn)⩽C∥(∑k=1∞‍|fk|r)1/r∥K˙p(·)α(·),q(ℝn),∥(∑k=1∞‍|Tfk|r)1/r∥Kp(·)α(·),q(ℝn)⩽C∥(∑k=1∞‍|fk|r)1/r∥Kp(·)α(·),q(ℝn),
for all sequences {fj}j=1∞ of locally integrable functions on ℝn, where C is independent of {fj}j=1∞.


## 3. The Quasi-Norm Characterizations

Let𝒮(ℝn) be the Schwartz space on ℝn,𝒮′(ℝn) be its dual space on ℝn. For φ∈𝒮(ℝn),  φ^ denotes its Fourier transform, and φ∨ denotes its inverse Fourier transform. Take φ0∈𝒮(ℝn) with φ0(x)≥0 and
(3.1)φ0(x)={1,|x|⩽1,0,|x|≥2.Now defineφ(x)=φ0(x)-φ0(2x) and set φj(x)=φ(2-jx) for all j∈ℕ. Let ℕ0=ℕ∪{0}. Then {φj}j∈ℕ0 is a resolution of unity, that means ∑j=0∞‍φj(x)=1 for all x∈ℝn.Now, we introduce the Herz-type Besov spaces and Triebel-Lizorkin spaces with variable exponents as following.Definition 3.1.
Let{φj}j∈ℕ0 be a resolution of unity as above, s∈ℝ,0<β,q≤∞,p(·)∈𝒫0(ℝn) and α(·):ℝn→ℝ with α(·)∈L∞(ℝn).
(i) The set(3.2){f∈𝒮′(ℝn):∥{2sjφj∨*f}j=0∞∥ℓβ(Kp(·)α(·),q)<∞}
is named to the Herz-type Besov space with variable exponents and denoted by Kp(·)α(·),qBβs(ℝn). The quasi-norm of f in this space is denoted by
(3.3)∥f∥Kp(·)α(·),qBβs(ℝn)≔∥{2sjφj∨*f}j=0∞∥ℓβ(Kp(·)α(·),q)
(ii) Forp+<∞, the set
(3.4){f∈𝒮′(ℝn):∥{2sjφj∨*f}j=0∞∥Kp(·)α(·),q(ℓβ)<∞}
is named to the Herz-type Triebel-Lizorkin space with variable exponents and denoted by Kp(·)α(·),qFβs(ℝn). The quasi-norm of f in this space is denoted by
(3.5)∥f∥Kp(·)α(·),qFβs(ℝn)≔∥{2sjφj∨*f}j=0∞∥Kp(·)α(·),q(ℓβ).Hereℓβ(Kp(·)α(·),q) and Kp(·)α(·),q(ℓβ) are the spaces of all sequences {gj} of measurable functions on ℝn with finite quasi-norms
(3.6)∥{gj}∥ℓβ(Kp(·)α(·),q)≔(∑j=0∞‍∥gj∥Kp(·)α(·),qβ)1/β,∥{gj}j=0∞∥Kp(·)α(·),q(ℓβ)≔∥(∑j=0∞‍|gj|β)1/β∥Kp(·)α(·),q.The spacesK˙p(·)α(·),qBβs(ℝn) and K˙p(·)α(·),qFβs(ℝn) are defined similarly by replacing the norm Kp(·)α(·),q with K˙p(·)α(·),q. To make these spaces definite, we need to show them independent of the choice of the resolution of unity {φj}j∈ℕ0. To this aim we need more notation.LetΨ0,Ψ∈𝒮(ℝn),ϵ>0, integer S≥-1 be such that
(3.7)|Ψ0^(ξ)|>0on{|ξ|<2ɛ},(3.8)|Ψ^(ξ)|>0on{ɛ2<|ξ|<2ɛ},(3.9)DτΨ^(0)=0∀|τ|⩽S.
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 {Ψk}k∈ℤ⊂𝒮(ℝn), a tempered distribution f∈𝒮′(ℝn) and a positive number a>0, we define the system of maximal functions
(3.10)(Ψk*)af(x)≔supy∈ℝn|Ψk*f(x+y)|(1+2k|y|)a,x∈ℝn,k∈ℤ.
Since Ψk*f(y) makes sense pointwise everything is well defined. We will often use dilates Ψk(x)=2knΨ(2kx) of a fixed function Ψ∈𝒮(ℝn), where Ψ0(x) might be given by a separate function. Also continuous dilates are needed. Let Ψt(·)≔t-nΨ(t-1·). We define (Ψt*)af(x) by
(3.11)(Ψt*)af(x)≔supy∈ℝn|Ψt*f(x+y)|(1+|y|/t)ax∈ℝn,t>0.Now we have equivalent quasi-norms for these new spaces.Theorem 3.2.
Lets<S+1,0<β,q⩽∞,p(·)∈𝒫0(ℝn) for p(·)/p0∈ℬ(ℝn) with p0<min(p-,1), α(·):ℝn→ℝ with α(·)∈L∞(ℝn) 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(ℝn) be given by (3.7), (3.8), and (3.9).
(i) Fora>n/p0, then the space Kp(·)α(·),qBβs can be characterized by
(3.12)Kp(·)α(·),qBβs(ℝn)={f∈𝒮′(ℝn):∥f∥Kp(·)α(·),qBβs(ℝn)(i)<∞},i=1,…,4,
where
(3.13)∥f∥Kp(·)α(·),qBβs(ℝn)(1)≔∥Φ0*f∥Kp(·)α(·),q+(∫01‍t-sβ∥(Φt*f)(·)∥Kp(·)α(·),q(ℝn)βdtt)1/β,∥f∥Kp(·)α(·),qBβs(ℝn)(2)≔∥(Φ0*)af∥Kp(·)α(·),q+(∫01‍t-sβ∥(Φt*)af∥Kp(·)α(·),q(ℝn)βdtt)1/β,∥f∥Kp(·)α(·),qBβs(ℝn)(3)≔(∑k=0∞‍2skβ∥(Φk*)af∥Kp(·)α(·),q(ℝn)β)1/β,∥f∥Kp(·)α(·),qBβs(ℝn)(4)≔(∑k=0∞‍2skβ∥Φk*f∥Kp(·)α(·),q(ℝn)β)1/β.
Furthermore, {∥·∥Kp(·)α(·),qBβs(ℝn)(i)}i=14 are equivalent.
(ii) Ifp0<β, then for a>n/p0 the space Kp(·)α(·),qFβs can be characterized by
(3.14)Kp(·)α(·),qFβs(ℝn)={f∈𝒮′(ℝn):∥f∥Kp(·)α(·),qFβs(ℝn)(i)<∞},i=1,…,5,
where
(3.15)∥f∥Kp(·)α(·),qFβs(ℝn)(1)≔∥Φ0*f∥Kp(·)α(·),q+∥(∫01‍t-sβ|Φt*f(·)|βdtt)1/β∥Kp(·)α(·),q(ℝn),(3.16)∥f∥Kp(·)α(·),qFβs(ℝn)(2)≔∥(Φ0*)af∥Kp(·)α(·),q+∥(∫01‍[t-s(Φt*)af]βdtt)1/β∥Kp(·)α(·),q(ℝn),(3.17)∥f∥Kp(·)α(·),qFβs(ℝn)(3)≔∥Φ0*f∥Kp(·)α(·),q(ℝn)+∥(∫01‍t-sβ∫|z|<t‍|(Φt*f)(·+z)|βdzdttn+1)1/β∥Kp(·)α(·),q(ℝn),(3.18)∥f∥Kp(·)α(·),qFβs(ℝn)(4)≔∥(∑k=0∞‍[2ksβ(Φk*)af]β)1/β∥Kp(·)α(·),q(ℝn),(3.19)∥f∥Kp(·)α(·),qFβs(ℝn)(5)≔∥(∑k=0∞‍2ksβ|Φk*f|β)1/β∥Kp(·)α(·),q.
Furthermore, {∥·∥Kp(·)α(·),qFβs(ℝn)(i)}i=15 are equivalent.Theorem3.2 is the generalization of Theorem 1 in [28] for the Herz type space Kp(·)α,q 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]).
Letμ,ν∈𝒮(ℝn),-1⩽M∈ℤ,
(3.20)Dτμ^(0)=0∀|τ|⩽M.
Then for any N>0 there is a constant CN so that
(3.21)supz∈ℝn|μt*ν(z)|(1+|z|)N⩽CNtM+1,
where μt(x)=t-nμ(x/t) for all 0<t⩽2.Lemma 3.4 (see [36]).
Let0<q⩽∞,δ>0. For any sequence {gj}0∞ of nonnegative numbers denote
(3.22)Gj=∑k=0∞‍2-|k-j|δgk.
Then
(3.23)∥{Gj}0∞∥ℓq⩽C∥{gj}0∞∥ℓq,
holds, where C is constant only depend on q,δ.Lemma 3.5.
Let0<β⩽∞,δ>0,0<q⩽∞, p(·)∈𝒫0(ℝn) and α(·):ℝn→ℝ with α(·)∈L∞(ℝn). For any sequence {gj}0∞ of nonnegative measurable functions on ℝn denote
(3.24)Gj(x)=∑k=0∞‍2-|k-j|δgk(x),x∈ℝn.
Then
(3.25)∥{Gj}0∞∥Kp(·)α(·),q(ℓβ)⩽C1∥{gj}0∞∥Kp(·)α(·),q(ℓβ),(3.26)∥{Gj}0∞∥ℓβ(Kp(·)α(·),q)⩽C2∥{gj}0∞∥ℓβ(Kp(·)α(·),q),
hold with some constants C1=C1(q,δ) and C2=C2(p(·),q,δ).Proof.
By Lemma3.4, (3.25) follows immediately from (3.23). Now we prove (3.26).
Letp(·)∈𝒫0(ℝn). Here we can divide it into two cases.
Case  1 (
p
-
≥
1
,
q
≥
1
). Since ∥·∥Kp(·)α(·),q is a norm, we have
(3.27)∥Gj∥Kp(·)α(·),q⩽∑k=0∞‍2-|k-j|δ∥gk∥Kp(·)α(·),q.
By Lemma 3.4, we obtain (3.26).
Case  2. If q<1, take p0<min(p-,q). Then
(3.28)∥Gj∥Kp(·)α(·),qp0=∥|Gj|p0∥Kp(·)/p0α(·)p0,q/p0⩽∥∑k=0∞‍2-|k-j|p0δ|gk|p0∥Kp(·)/p0α(·)p0,q/p0⩽∑k=0∞‍2-|k-j|p0δ∥|gk|p0∥Kp(·)/p0α(·)p0,q/p0.
By Lemma 3.4, we obtain
(3.29)∥{Gj}∥ℓβ(Kp(·)α(·),q)p0=∥{|Gj|p0}∥ℓβ/p0(Kp(·)/p0α(·)p0,q/p0)≲∥{|gk|p0}∥ℓβ/p0(Kp(·)/p0α(·)p0,q/p0)=C∥{gk}∥ℓβ(Kp(·)α(·),q)p0.
In the last inequality, we used (3.26) has been proved for space Kp(·)/p0α(·)p0,q/p0. Raising to power 1/p0, we obtain (3.26).Lemma 3.6 (Theorem 6 in [37]).
Let{φj}j∈ℕ0 is a resolution of unity and let R∈ℕ. Then there exist functions θ0,θ∈𝒮(ℝn) satisfy
(3.30)suppθ0,suppθ⊆{x∈ℝn:|x|⩽1},|θ0^(ξ)|>0on{|ξ|<2ϵ},|θ^(ξ)|>0on{ϵ2<|ξ|<2ϵ},∫ℝn‍xγθ(x)dx=0for0<|γ|⩽R
such that
(3.31)θ0^(ξ)ψ0^(ξ)+∑j=1∞‍θ^(2-jξ)ψ^(2-jξ)=1,∀ξ∈ℝn,
where the functions ψ0,ψ∈𝒮(ℝn) are defined via ψ0^(ξ)=φ0(ξ)/θ0^(ξ) and ψ^(ξ)=φ1(2ξ)/θ^(ξ).Proof of Theorem3.2.
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′(ℝn)(3.32)∥f∥Kp(·)α(·),qFβs(2)≲∥f∥Kp(·)α(·),qFβs(1)≲∥f∥Kp(·)α(·),qFβs(2).
From Lemmas3.3 and 3.6, we have that, see [35], for r<min{p-,β},N∈ℕ, there exists positive constant C for f∈𝒮(ℝn),
(3.33)(∫12‍|2ls(Φ2-lt*f)a(x)|βdtt)r/β⩽C∑k∈l+ℕ0‍2(l-k)(Nr-n+rs)2krs×ℳ[(∫12‍|((Φk)t*f)(·)|βdtt)r/β](x).
Take nown/a<r=p0<min{p-,β},N>max{0,-s}+a and put δ=N+s-d/r>0. We obtain for l∈ℕ(3.34)(∫12‍|2ls(Φ2-lt*f)a(x)|βdtt)r/β≲∑k∈l+ℕ0‍2-δr|l-k|2krsℳ[(∫12‍|((Φk)t*f)(·)|βdtt)r/β](x).
Now we apply Lemma3.5 in Kp(·)/rrα(·),q/r(ℓβ/r) which yields
(3.35)∥{(∫12‍|2ls(Φ2-lt*f)a(x)|βdtt)r/β}l∈ℕ∥Kp(·)/rrα(·),q/r(ℓβ/r)≲∥{ℳ[(∫12‍|2ks((Φk)t*f)(·)|βdtt)r/β]}l∈ℕ∥Kp(·)/rrα(·),q/r(ℓβ/r).
Next, using Theorem2.8, we can obtain
(3.36)∥{(∫12‍|2ls(Φ2-lt*f)a(x)|βdtt)r/β}l∈ℕ∥Kp(·)/rrα(·),q/r(ℓβ/r)≲∥{ℳ[(∫12‍|2ks((Φk)t*f)(·)|βdtt)r/β]}l∈ℕ∥Kp(·)/rrα(·),q/r(ℓβ/r)≲∥{(∫12‍|2ks((Φk)t*f)(·)|βdtt)r/β}k∈ℕ∥Kp(·)/rrα(·),q/r(ℓβ/r)=∥{(∫12‍|2ks((Φk)t*f)(·)|βdtt)1/β}k∈ℕ∥Kp(·)α(·),q(ℓβ)r.
Hence, we obtain(3.37)∥(∫01‍|λ-s(Φλ*f)a(·)|βdλλ)1/β∥Kp(·)α(·),q≈∥(∑l=1∞‍∫12‍|2ls(Φ2-lt*f)a(·)|βdtt)1/β∥Kp(·)α(·),q≲∥{(∫12‍|2lsΦ2-lt*f(·)|βdtt)1/β}l∈ℕ∥Kp(·)α(·),q(ℓβ)~∥(∫01‍|λ-sΦλ*f(·)|βdλλ)1/β∥Kp(·)α(·),q.
This proves ∥f∥Kp(·)α(·),qFβs(2)≲∥f∥Kp(·)α(·),qFβs(1). Since the reverse inequality is trivial, this finishes Step  1.
Step  2. Let Ψ0,Ψ∈𝒮'(ℝn) be functions satisfying (3.25).
First, we are going to prove for allf∈𝒮'(ℝn)(3.38)∥f∥Kp(·)α(·),qFβs(ℝn,Ψ)(4)≲∥f∥Kp(·)α(·),qFβs(ℝn,Φ)(2).
Again from Lemmas3.6 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(ℝn),
(3.39)2ls(Ψl*f)a(x)⩽C∑k∈ℕ0‍2-|k-l|δ2ks(Φ2-kt*f)a(x),
for all x∈ℝn and all t∈[1,2].
Suppose first thatβ≥1. Then we take on both sides (∫12‍|·|βdt/t)1/β, which gives
(3.40)2ls(Ψl*f)a(x)≲∑k∈ℕ0‍2-|k-l|δ2ks(∫12‍|(Φ2-kt*f)a(x)|βdtt)1/β.
Applying Lemma 3.5 we obtain that
(3.41)∥{2ls(Ψl*f)a}l∈ℕ∥Kp(·)α(·),q(ℓβ)≲∥(∑k=1∞‍2ksβ|(Φ2-kt*f)a(x)|βdtt)1/β∥Kp(·)α(·),q,
which gives the desired result.
In caseβ<1 we argue as follows. The quantity (∫12‍|·|βdt/t)1/β is no longer a norm. This gives
(3.42)(2ls(Ψl*f)a(x))β≲∑k∈ℕ0‍2-β|k-l|δ2ksβ∫12‍|(Φ2-kt*f)a(x)|βdtt.
Notice that the right-hand side is nothing more than a convolution (γ*α(·))ℓ of the sequences
(3.43)γk=2-|k|δβ,α(·)k=2ksβ∫12‍|(Φ2-kt*f)a(x)|βdtt.
Now we apply the ℓ1-norm to both sides and get for all x∈ℝn(3.44)∥2ls(Ψl*f)a(x)∥ℓββ⩽∥γ∥ℓ1·∥α(·)∥ℓ1≲∑k=1∞‍2ksβ∫12‍|(Φ2-kt*f)a(x)|βdtt.
We take both sides to power(⋯)1/β and apply the Kp(·)α(·),q(ℝn)-norm. This gives (3.38).
Similarly, we obtain for allf∈𝒮′(ℝn)(3.45)∥f∥Kp(·)α(·),qFβs(ℝn,Φ)(2)≲∥f∥Kp(·)α(·),qFβs(ℝn,Ψ)(4).Step  3. Choosing t=1 in Step  1 and omitting the integration over t we see immediately
(3.46)∥f∥Kp(·)α(·),qFβs(ℝn)(5)≲∥f∥Kp(·)α(·),qFβs(ℝn)(4)≲∥f∥Kp(·)α(·),qFβs(ℝn)(5).Step  4. What remains is to show that (3.19) is equivalent to the rest.
First, let us prove that for anyf∈𝒮(ℝn)(3.47)∥f∥Kp(·)α(·),qFβs(ℝn)(2)≲∥f∥Kp(·)α(·),qFβs(ℝn)(3).
From [35], for 0<r<min{p-,β}, there exists a positive constant C for any f∈𝒮(ℝn)(3.48)(∫12‍|(Ψ2-lt*f)a(x)|βdtt)r/β⩽C∑k∈ℕ0‍2-kNs2(k+l)n∫ℝn‍(∫12‍∫|z|<2-(k+l)t‍|((Φk+l)t*f)(z+y)|βdz(dt/tn+1))r/β(1+2l|x-y|)ardy.
Ifar>n then we have
(3.49)gl(y)=2nl(1+2l|y|)ar∈L1(ℝn).
Thus we have
(3.50)(∫12‍|2ls(Φ2-lt*f)a(x)|βdtt)r/β≲∑k∈ℕ0‍2-kNr2kn2lsr[gl*(∫12‍∫|z|<2-(k+l)t‍|((Φk+l)t*f)(z+·)|βdzdttn+1)r/β](x).
Now we use the majorant property of the Hardy-Littlewood maximal operator in [38] and continue estimating
(3.51)(∫12‍|2ls(Φ2-lt*f)a(x)|βdtt)r/β≲∑k∈ℕ0‍2lsr2k(-Nr+n)ℳ[(∫12‍∫|z|<2-(k+l)t‍|((Φk+l)t*f)(z+·)|βdzdttn+1)r/β](x).
An index shift on the right-hand side gives
(3.52)(∫12‍|2ls(Φ2-lt*f)a(x)|βdtt)r/β≲∑k∈l+ℕ0‍2lsr2(k-l)(-Nr+n)ℳ[(∫12‍∫|z|<2-kt‍|((Φk)t*f)(z+·)|βdzdttn+1)r/β](x)=C∑k∈l+ℕ0‍2(l-k)(Nr-n+rs)2krsℳ[(∫12‍∫|z|<2-kt‍|((Φk)t*f)(z+·)|βdzdttn+1)r/β](x).
Then with analogous arguments as after (3.39), we obtain (3.47).
Second, we prove∥f∥Kp(·)α(·),qFβs(ℝn)(3)≲∥f∥Kp(·)α(·),qFβs(ℝn)(2). Since for all t>0(3.53)1tn∫|z|<t‍|(Φt*f)(x+z)|dz≲sup|z|<t|(Φt*f)(x+z)|(1+1/t|z|)a≲(Φt*f)a(x).
Therefore we can obtain what we want. The proof is complete.Remark 3.7.
One can define homogenous spacesKp(·)α(·),qF˙βs(ℝn),Kp(·)α(·),qB˙βs(ℝn),K˙p(·)α(·),qF˙βs(ℝn) and K˙p(·)α(·),qB˙βs(ℝn). Then there is an analogue of Theorem 3.2 for these spaces. We omit the detail here and leave it for interested readers.



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*Source: 384593-2012-12-25.xml*