# Weighted Grand Herz-Type Spaces and Its Applications

Authors: Xia Yu; Zongguang Liu
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, we introduce the weighted grand Herz spaces and weighted grand Herz-type Hardy spaces. The decompositional characterizations of these spaces are established. As its applications, the boundedness of some sublinear operators are established.


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## Body

## 1. Introduction

In the process of investigating the minimal hypothesis for the integrability of the Jocobian, Iwanniec and Sbordone [1] introduced grand Lebesgue spaces, denoted by LnΩ, n∈ℕ and Ω⊂ℝn being a bounded domain. Greco et al. [2] generalized the space LnΩ to Lq,θΩ with 0≤θ<∞ and 1<q<∞. During the last decades, grand Lebesgue spaces have attracted many researchers and have been considered in various aspects, such as PDE, interpolation theory, and boundedness of various operators, e.g. Hardy operator, maximal operator, singular integral operator, and fractional integral operator, see [3–7].Usually, the underlying space of grand Lebesgue spaces would be an open set with finite measure (inℝn or more generally homogeneous spaces) since that the basic idea of their definition is based on the supposition that f in grand Lebesgue space, Lq, must be integrable to the power q−ε with all ε>0. In 2011, Samko and Umarkhardzhiev [8] introduced grand Lebesgue spaces on an arbitrary set which may has infinite measure in ℝn. They also introduced a version of weighted grand Lebesgue spaces and proved that some operators bounded in a Lebesgue space with Muckenhoupt weights were also bounded in the corresponding weighted grand Lebesgue space (for bounded or unbounded Ω).Let1<q<∞ and Ω⊂ℝn be an open set. To introduce the weighted grand Lebesgue spaces in the case Ω=∞, it is quite natural to make use of weighted space; for any nonnegative weight function w, any measurable function f on Ω, we write
(1)fLqΩ,w=∫Ωfxqwxdx1/q.Definition 1 (see [8]).
Letθ>0, 1<q<∞, and β>0, w, be a weight. The grand Lebesgue space with power weights Lβq,θΩ,w is defined as the set of f with the finite norm
(2)fLβq,θΩ,w=sup0<ε<q−1εθ∫Ωfxq−εwβ,εxdx1/q−ε,where wβ,εx=wx·x−βε and x=1+x2.According to [8], the embedding
(3)Lq−ε1Ω,wβ,ε1↪Lq−ε2Ω,wβ,ε2with 0<ε1<ε2<p−1 holds if and only if ∫Ωwx/xβqdx<∞. In the case Ω is unbounded, we call a positive number β admissible for the weight w, if the condition
(4)∫ℝnwx1+xβdx<∞holds with this β. We denote the Muckenhoupt class of weights on Ω by AqΩ. It is known that
(5)w∈AqΩ⟹∫Ωwx1+xnqdx<∞.Then, all β≥n are admissible for all the weights w∈AqΩ.Supposew∈AqΩ and β≥n, with the choice ε1=0 and ε2=ε, we have
(6)fLq−εΩ,wβ,ε≤CfLqΩ,w,where C=∫Ωwx/xβqdx is independent of ε. Since sup0<ε<q−1εθ/q−ε<∞, we have
(7)LqΩ,w↪Lβq,θΩ,w.Herz spaces are arising from Herz’s study on the absolute convergence of Fourier transform [9], which is an important function space in harmonic analysis. Chen and Lau [10] and García-Cuerva [11] studied some Hardy spaces associated with nonhomogeneous Herz spaces with special indexes in n=1 and n>1, respectively. Lu et al. studied some Hardy spaces associated with homogeneous Herz spaces with special indexes, and the complete theory for the case of general indexes was established in 1995. We refer to [12] for more details. The weighted Herz spaces on ℝn were established by Lu and Yang in [13]. Let T be a sublinear operator satisfying the condition
(8)Tfx≤C∫ℝnfyx−yndy,x∈suppffor any integrable function f with compact support, in [13] such a sublinear operator was proved be bounded on K̇qα,pw1,w2, and Kqα,pw1,w2 provided T is bounded on Lqℝn, w1=1, and w2∈A1. The restriction on the weights has been relaxed to w1∈Aqw1 and w2∈Aqw1 in [14]. A certain weighted Herz-type Hardy space was introduced, its atomic decomposition was established in [15], and a boundedness theory of sublinear operators was given, where w1,w2∈A1. The atomic decomposition for the weighted Herz-type Hardy space with more general weights was established in [14], and local Caldrón-Zygmund operators of nonconvolutional type was proved to be bounded from these weighted Herz-type Hardy spaces into weighted Herz spaces.Grand variable Herz spaces using grand Lebesgue sequence spacesKq·α,p,θℝn and K̇q·α,p,θℝn were introduced by Nafis et al. in [16], and the boundedness of sublinear operators on these spaces was established in this paper. The boundedness of multilinear Calderón-Zygmund operators on these spaces was established in [17].Inspired by these work, we introduce the weighted grand Herz spaces and weighted grand Herz-type Hardy spaces which are products of weighted grand Lebesgue spaces onℝn and Herz-type spaces. The decompositional characterizations of these spaces are established. As its applications, the boundedness of some sublinear operators is established on these function spaces.We recall the definition of Muckenhoupt weight as follows. Let1<q<∞, and a weight w≥0 is a Muckenhoupt Aq weight if for any ball B,
(9)1B∫Bwxdx1B∫Bwx−1/q−1dxq−1≤C,with C a constant independent of the ball B. Further, w∈A1 if and only if there is a constant C>0 such that for any ball B,
(10)1B∫Bwxdx≤Cessinfx∈Bwx.The following properties forAq weights will be used in this paper, see [18, 19].Lemma 2.
Letw∈Aq for some 1≤q<∞ and B be any ball. Then,
(i)
For any functionf∈Lloc1ℝn,
(11)1B∫Bfxdx≤CwB−1/q∫Bfxqwxdx1/q,where C is independent of f and B.(ii)
IfE is a measurable subset of B, then there exists a δ>0 such that
(12)C1EBq≤wEwB≤C2EBδ,where C1 and C2 are independent of E and B, respectively.The following lemmas (see [8]) and notations will be used in this paper.Lemma 3.
For everyw∈Aq and β∈ℝ, there exists a δ>0 such that wβ∈Aq−ε for all 0<ε<δ, where wβx=wx·x−βε and x=1+x2.Lemma 4.
Let1<q<∞; if T is bounded on Lqℝn,w for every w∈Aq and T is bounded on Lq−ε0ℝn,w for every w∈Aq−ε0 for some ε0∈0,q−1, then T is also bounded on the weighted grand Lebesgue space Lβq,θℝn,w, where β is an arbitrary positive admissible number.Remark 5.
By the proof of Lemma4 (see [8], Theorem 5.2),
(13)TfLβq,θℝn,w≤max1,C1sup0<ε≤ε0εθ/q−εTfLq−εℝn,wβ,where C1 does not depend on ε and β is an arbitrary positive admissible number.LetE be a measurable set, E denotes the Lebesgue measure of E, and χE denotes the characteristic function. We write Bk≔x∈ℝn:x≤2k, Ck≔Bk\Bk−1, and χk≔χCk, where k∈ℤ and χ~0≔χB0, χ~k≔χk, where k∈ℕ. p′ means the conjugate exponent function, namely, 1/p+1/p′=1. A~B means C1B≤A≤C2B where C1,C2>0.


## 2. Weighted Grand Herz Spaces

Now, we give the definition of weighted grand Herz spaces.Definition 6.
Supposeθ>0, 0<α<∞, 0<p<∞, and 1<q<∞. Let w1 and w2 be nonnegative weight functions and β≥n. The homogeneous weighted grand Herz space K̇q,θ,βα,pw1,w2 on ℝn is defined by
(14)K̇q,θ,βα,pw1,w2=f∈Llocqℝn\0:fK̇q,θ,βα,pw1,w2<∞,where
(15)fK̇q,θ,βα,pw1,w2=∑k=−∞∞w1Bkαp/nfχkLβq,θℝn,w2p1/p.
The nonhomogeneous weighted grand Herz spaceKq,θ,βα,pw1,w2 on ℝn is defined by
(16)Kq,θ,βα,pw1,w2=f∈Llocqℝn:fKq,θ,βα,pw1,w2<∞,where
(17)fKq,θ,βα,pw1,w2=∑k=0∞w1Bkαp/nfχ~kLβq,θℝn,w2p1/p.It is obvious thatK̇q,θ,β0,qw1,w2=Lβq,θℝn,w2. Therefore, the weighted grand Herz spaces are natural generalization of the weighted grand Lebesgue spaces.By the embedding between weighted grand Lebesgue space and weighted Lebesgue space, we can get the following proposition immediately.Proposition 7.
Supposeθ>0, 0<α<∞, 0<p<∞, and 1<q<∞. Let w1∈Aqw1 and w2∈Aqw2 and β≥n, 1≤qw1<∞, and 1<qw2<∞. Then, we have
(18)K̇qα,pw1,w2↪K̇q,θ,βα,pw1,w2,Kqα,pw1,w2↪Kq,θ,βα,pw1,w2.In order to state our decomposition theorem, we need the following definition.Definition 8.
Suppose0<α<∞, 1≤q<∞, let w1 and w2 be nonnegative weight functions on ℝn.
(a)
A functiona on ℝn is said to be a central α,q,w1,w2-unit if
(i)
suppa⊂B0,r for some r>0(ii)
aLqw2≤w1B0,r−α/n, where B0,r=x∈ℝn:x<r(b)
A functiona on ℝn is said to be a central α,q,w1,w2-unit of restrict type if
(i)
suppa⊂B0,r for some r≥1(ii)
aLqw2≤w1B0,r−α/nRemark 9.
Ifr=2k for some k∈ℤ in Definition 8, then a central unit is called a dyadic central unit. When w1=w2=1 and α=n1−1/q, a central unit must be a block in Taibleson-Weiss’ sense (see [20]).Theorem 10.
Supposeθ>0, 0<α<∞, 1<q<∞, and 0<p<∞. Let w1∈Aqw1, 1≤qw1<∞, and w2 be a nonnegative weight function on ℝn. The following two statement are equivalent.
(i)
f∈K̇q,θ,βα,pw1,w2(ii)
f can be represented by
(19)fx=∑k=−∞∞λkakx,where each ak is a dyadic central α,q,w1,w2,β-unit with the support Bk and ∑k=−∞∞λkq<∞, w2,βx=w2x·x−βε, β>0, and 0<ε<p−1.Proof.
Firstly, we prove (i) can imply (ii). Forf∈K̇q,θ,βα,pw1,w2, write
(20)fx=∑k=−∞∞fxχkx=∑k=−∞∞w1Bkα/nfχkLq−εw2,βfxχkxw1Bkα/nfχkLq−εw2,β=∑k=−∞∞λkakx,where λk=w1Bkα/nfχkLq−εw2,β and ax=fxχkx/w1Bkα/nfχkLq−εw2,β.
It is obvious thatw2,βx is nonnegative, suppak⊂Bk, and akLq−εw2,β=w1Bk−α/n. Hence, for each k, ak is a dyadic central α,q−ε,w1,w2,β-unit with the support Bk.
(21)sup0<ε<q−1εθp/q−ε∑k=−∞∞λkp=∑k=−∞∞sup0<ε<q−1εθp/q−εw1Bkαp/nfχkLq−εw2,βp≤∑k=−∞∞w1Bkαp/nfχkLβq,θw2p≲fK̇q,θ,βα,pw1,w2p<∞.
Now, we prove (ii) can imply (i). Letfx=∑k=−∞∞λkakx be a decomposition of f satisfying the hypothesis in (ii). For each k∈ℤ, we can get
(22)fχkLq−εw2,β≤∑j=k∞λjajLq−εw2,β.
If0<q≤1, we have
(23)fK̇q,θ,βα,pw1,w2p=∑k=−∞∞w1Bkαp/nfχkLβq,θw2p=∑k=−∞∞w1Bkαp/nsup0<ε<q−1εθp/q−εfχkLq−εw2,βp≤∑k=−∞∞w1Bkαp/nsup0<ε<q−1εθp/q−ε∑j=k∞λkakLq−εw2,βp≤sup0<ε<q−1εθp/q−ε∑k=−∞∞w1Bkαp/n∑j=k∞λjpw1Bj−αp/n≤C∑k=−∞∞∑j=k∞λjpw1Bkw1Bjαp/n.
Sincew1∈Aqw1, then there exists C>0 and δ>0 such that
(24)w1Bkw1Bj≤CBkBjδ≤C2k−jnδ.
Thus,(25)fK̇q,θ,βα,pw1,w2p≤C∑j=−∞∞∑k=−∞jλjpw1Bkw1Bjαp/n≤C∑j=−∞∞λjp∑k=−∞j2k−jαpδ/n≤C∑j=−∞∞λjp<∞.
If1<q<∞, by Hölder inequality,
(26)fχkLq−εw2,βp≤∑j=k∞λjakLq−εw2,β1/2akLq−εw2,β1/2p≤∑j=k∞λjpakLq−εw2,βp/2∑j=k∞akLq−εw2,βp′/2p/p′≤∑j=k∞λjpw1Bj−αp/2n∑j=k∞w1Bj−αp′/2np/p′.
Sincew1∈Aqw1, then there exists C>0 and δ>0 such that
(27)∑j=k∞w1Bj−αp′/2n≤C∑j=k∞w1BkBkBj−δ−αp′/2n≤Cw1Bk−αp′/2n∑j=k∞2k−jαpδ/2≤Cw1Bk−αp′/2n.
Thus,(28)fK̇q,θ,βα,pw1,w2p=∑k=−∞∞w1Bkαp/nfχkLβq,θw2p=∑k=−∞∞w1Bkαp/nsup0<ε<q−1εθp/q−εfχkLq−εw2,βp≤sup0<ε<q−1εθp/q−ε∑k=−∞∞w1Bkαp/n∑j=k∞λjpw1Bj−αp/n≤C∑k=−∞∞∑j=k∞λjpw1Bkw1Bjαp/n≤C∑j=−∞∞∑k=−∞jλjpw1Bkw1Bjαp/n≤C∑j=−∞∞λjp∑k=−∞j2k−jαpδ/n≤C∑j=−∞∞λjp<∞.
Hence, we obtainf∈K̇q,θ,βα,pw1,w2. This completes the proof of Theorem 10.The following theorem shows the boundedness of some sublinear operators on weighted Herz spaces.Theorem 11.
Supposeθ>0, 0<α<∞, 1<q<∞, and 0<p<∞. Let w1∈Aqw1 and w2∈Aqw2, and there exists some ε0∈0,qw2−1 such that w2,β∈Aqw2−ε0, where β is an arbitrary positive admissible number. qw1 and qw2 satisfy either of the following conditions.
(i)
w1=w2, 1<qw1≤q, and −nqw1/q<αqw1<n1−qw1−ε0/q−ε0(ii)
1≤qw1<∞, 1<qw2≤q, and 0<αqw1<n1−qw2−ε0/q−ε0
If a sublinear operatorT satisfies the condition
(29)Tfx≤C∫ℝnfyx−yndy,x∈suppffor any integrable function f with compact support and T is bounded on Lβq,θw2, then T is bounded on K̇q,θ,βα,pw1,w2. Similarly, the nonhomogeneous weighted grand Herz space has the same result.Proof.
Without loss of generality, we assume thatf is nonnegative. Let fj≔fχj for j∈ℤ, and then we have f=∑j=−∞∞fj.
(30)TfK̇q,θ,βα,pw1,w2p=∑k=−∞∞w1Bkαp/nTfχkLβq,θw2p≤C∑k=−∞∞w1Bkαp/n∑j=−∞k−2TfjχkLβq,θw2p+C∑k=−∞∞w1Bkαp/n∑j=k−1k+1TfjχkLβq,θw2p+C∑k=−∞∞w1Bkαp/n∑j=k+2∞p≔I1+I2+I3.
By theLβq,θw2-boundedness of T, it is easy to obtain
(31)I2≤CfK̇q,θ,βα,pw1,w2p.
ForI1, j≤k−2, x∈Ck, y∈Cj, and then we have x−y~x.
(32)Tfjx≤C2−kn∫Cjfjydy≤C2j−knw2,βBj−1/q−εfjLq−εw2,β,where ε≤ε0. By Remark 5, Lemma 2, w1∈Aqw1, and w2,β∈Aqw2−ε0⊂Aqw2−ε for 0<ε≤ε0, we have
(33)I1=C∑k=−∞∞w1Bkαp/n∑j=−∞k−2sup0<ε<q−1εθ/q−εTfjχkLq−εw2,βp≤C∑k=−∞∞w1Bkαp/n∑j=−∞k−2sup0<ε≤ε0εθ/q−εTfjχkLq−εw2,βp≤C∑k=−∞∞w1Bkαp/n∑j=−∞k−2sup0<ε≤ε0εθ/q−ε2j−knfjLq−εw2,βw2,βBkw2,βBj1/q−εp≤C∑k=−∞∞w1Bjαp/n∑j=−∞k−2sup0<ε≤ε0εθ/q−ε2k−jqw2−ε0nq−ε−nfjLq−εw2,βw1Bkw1Bjα/np≤C∑k=−∞∞w1Bjαp/n∑j=−∞k−2sup0<ε≤ε0εθ/q−ε2k−jqw2−ε0nq−ε0−nfjLq−εw2,βw1Bkw1Bjα/np≤C∑k=−∞∞w1Bjαp/n∑j=−∞k−2sup0<ε≤ε0εθ/q−εfjLq−εw2,β2k−jαqw1+qw2−ε0nq−ε0−np.
If0<q≤1, we have
(34)I1≤C∑k=−∞∞w1Bjαp/n∑j=−∞k−2sup0<ε≤ε0εθp/q−εfjLq−εw2,βp2k−jpαqw1+qw2−ε0n/q−ε0−n≤C∑j=−∞∞w1Bjαp/nsup0<ε≤ε0εθp/q−εfjLq−εw2,βp∑k=j+2∞2k−jpαqw1+qw2−ε0n/q−ε0−n≤C∑j=−∞∞w1Bjαp/nsup0<ε≤q−1εθp/q−εfjLq−εw2,βp≤CfK̇q,θ,βα,pw1,w2p.
Ifq>1, by the Ho¨lder inequality, we have
(35)I1≤C∑k=−∞∞w1Bjαp/n∑j=−∞k−2sup0<ε≤ε0εθp/q−εfjLq−εw2,βp2k−jp/2αqw1+qw2−ε0n/q−ε0−n×∑j=−∞k−22k−jp′/2αqw1+qw2−ε0n/q−ε0−np/p′≤C∑j=−∞∞w1Bjαp/nsup0<ε≤ε0εθp/q−εfjLq−εw2,βp∑k=j+2∞2k−jp/2αqw1+qw2−ε0n/q−ε0−n≤C∑j=−∞∞w1Bjαp/nsup0<ε≤q−1εθp/q−εfjLq−εw2,βp≤CfK̇q,θ,βα,pw1,w2p.
Finally, we estimateI3. In this case, j≥k+2, x∈Ck, and y∈Cj, we have x−y~y.
(36)Tfjx≤C2−jn∫Cjfjydy≤Cw2,βBj−1/q−εfjLq−εw2,β,where ε≤ε0. By Remark 5, Lemma 2, w1∈Aqw1 and w2,β∈Aqw2−ε0⊂Aqw2−ε for 0<ε≤ε0, we have
(37)I3=C∑k=−∞∞w1Bkαp/n∑j=k+2∞sup0<ε<q−1εθ/q−εTfjχkLq−εw2,βp≤C∑k=−∞∞w1Bkαp/n∑j=k+2∞sup0<ε≤ε0εθ/q−εTfjχkLq−εw2,βp≤C∑k=−∞∞w1Bkαp/n∑j=k+2∞sup0<ε≤ε0εθ/q−εfjLq−εw2,βw2,βBkw2,βBj1/q−εp≤C∑k=−∞∞w1Bjαp/n∑j=k+2∞sup0<ε≤ε0εθ/q−εfjLq−εw2,βw1Bkw1Bjα/nw2,βBkw2,βBj1/q−εp≤C∑k=−∞∞w1Bjαp/n∑j=k+2∞sup0<ε≤ε0εθ/q−ε2k−jqw2−ε0n/qfjLq−εw2,βw1Bkw1Bjα/np≤C∑k=−∞∞w1Bjαp/n∑j=k+2∞sup0<ε≤ε0εθ/q−εfjLq−εw2,β2k−jαqw1+qw2−ε0nqp.
By the same argument asI1, we obtain
(38)I3≤CfK̇q,θ,βα,pw1,w2p.
Combining the estimates ofI1, I2, and I3, we complete the proof of Theorem 11.Corollary 12.
Let1<q<∞; by Lemma 4, Hardy-Littlewood maximal operator M and Calderón-Zygmund singular integral operator T are bounded on the weighted grand Lebesgue space Lβq,θℝn,w, w∈Aq, for every admissible β>0. Then, by Theorem 11, M and T are bounded on K̇q,θ,βα,pw1,w2 and Kq,θ,βα,pw1,w2.


## 3. Weighted Grand Herz-Type Hardy Spaces

We begin with recalling some definitions. Fors∈ℝ, let s denote the greatest integer ≤s when s>0 and s=0 when s<0. Define
(39)Aα,qqw1,qw2ℝn=ϕ∈Sℝn:supγ≤αqw1+nqw2/q−1+11+xαqw1+nqw2/q−1+n+1Dγϕx≤1,where Sℝn is the space of Schwartz functions, γ=γ1,⋯,γn∈ℕn and Dγ=∂/∂x1γ1⋯∂/∂xnγn. Moreover, we define
(40)ϕAα,qqw1,qw2ℝn=supx∈ℝnsupγ≤αqw1+nqw2/q−1+11+xαqw1+nqw2/q−1+n+1Dγϕx.LetS′ℝn be the space of Schwartz distributions. For s∈S′ℝn, we define
(41)Gα,qqw1,qw2fx=supϕ∈Aα,qqw1,qw2ℝnsupx−y<tf∗ϕty,where ϕty=t−nϕy/t for t>0. Gα,qqw1,qw2f is usually called the grand maximal of f, see [19].Now, we give the definition of the weighted grand Herz-type Hardy spaces.Definition 13.
Supposeθ>0, 0<α<∞, 0<p<∞, and 1<q<∞. Let w1∈Aqw1 and w2∈Aqw2, and 1≤qw1,qw2<∞ and β≥n. The homogeneous weighted grand Herz-type Hardy space ḢKq,θ,βα,pw1,w2 on ℝn is defined by
(42)ḢKq,θ,βα,pw1,w2=f∈S′ℝn:Gα,qqw1,qw2f∈K̇q,θ,βα,pw1,w2,fḢKq,θ,βα,pw1,w2=Gα,qqw1,qw2fK̇q,θ,βα,pw1,w2.
The nonhomogeneous weighted grand Herz-type Hardy spaceKq,θ,βα,pw1,w2 on ℝn is defined by
(43)HKq,θ,βα,pw1,w2=f∈S′ℝn:Gα,qqw1,qw2f∈Kq,θ,βα,pw1,w2,fHKq,θ,βα,pw1,w2=Gα,qqw1,qw2fKq,θ,βα,pw1,w2.By the embedding between weighted grand Herz spaces and weighted Herz spaces, we can get the following proposition immediately.Proposition 14.
Supposeθ>0, 0<α<∞, 0<p<∞, and 1<q<∞. Let w1∈Aqw1 and w2∈Aqw2 and β≥n, 1≤qw1<∞, and 1<qw2<∞. Then, we have
(44)ḢKqα,pw1,w2↪ḢKq,θ,βα,pw1,w2,HKqα,pw1,w2↪HKq,θ,βα,pw1,w2.Applying Theorem11, we can obtain the following relation between the weighted grand Herz spaces and the weighted grand Herz-type Hardy spaces.Theorem 15.
Supposeθ>0, 0<α<∞, 1<q<∞, and 0<p<∞. Let w1∈Aqw1 and w2∈Aqw2, and there exists some ε0∈0,qw2−1 such that w2,β∈Aqw2−ε0, where β is an arbitrary positive admissible number. qw1 and qw2 satisfy either of the following conditions.
(i)
w1=w2, 1<qw1≤q, and −nqw1/q<αqw1<n1−qw1−ε0/q−ε0(ii)
1≤qw1<∞, 1<qw2≤q, and 0<αqw1<n1−qw2−ε0/q−ε0
Then,ḢKq,θ,βα,pw1,w2=K̇q,θ,βα,pw1,w2 and HKq,θ,βα,pw1,w2=Kq,θ,βα,pw1,w2.Thus, we are interested in the caseαqw1≥n1−qw2−ε0/q−ε0 for the weighted grand Herz-type Hardy spaces. We can establish their atomic decomposition.Definition 16.
Letw1∈Aqw1 and w2∈Aqw2, 1≤qw1<∞, 1<qw2<∞, 1<q<∞, n1−qw2−ε0/q−ε0≤αqw1<∞, and the nonnegative integer s=nqw2−ε0/q−ε0−1+αqw1. There exists some ε0∈0,qw2−1 such that w2,β∈Aqw2−ε0, β>0. A function a is said to be a central α,q;w1,w2,β-atom, if it satisfies
(i)
suppa⊂B0,r for some r>0(ii)
aLq−εw2,β≤w1B0,r−α/n for 0≤ε≤ε0(iii)
∫ℝnaxxγdx=0, γ≤sTheorem 17.
Supposeθ>0, 0<α<∞, 1<q<∞, and 0<p<∞. Let w1∈Aqw1 and w2∈Aqw2. There exists some ε0∈0,qw2−1 such that w2,β∈Aqw2−ε0, where β>0, 1≤qw1<∞, 1<qw2≤q<∞, and n1−qw2−ε0/q−ε0≤αqw1<∞. Then, f∈ḢKq,θ,βα,pw1,w2 if and only if f=∑k=−∞∞λkak holds in S′ℝn, where ax is a central α,q;w1,w2,β-atom supported in Bk and ∑k=−∞∞λkp<∞. Moreover,
(45)fḢKq,θ,βα,pw1,w2~inf∑k=−∞∞λkp1/p,where the infimum is taken over all above decompositions of f. Similarly, the nonhomogeneous weighted grand Herz space has the same result.Proof.
Firstly, we prove the sufficiency. If0<p≤1, it suffices to verify that
(46)Gα,qqw1,qw2aK̇q,θ,βα,pw1,w2≤Cholds for any central α,q;w1,w2,β-atom a, where C is a constant independent of a. Let suppa⊂Bk0 for some k∈ℤ. We write
(47)1B∫Bwxdx1B∫Bwx−1/q−1dxq−1≤C.
Sincew1∈Aqw1 and w2,β∈Aqw2−ε0⊂Aqw2−ε for 0<ε≤ε0, combining with Remark 5, we have
(48)I1=∑k=−∞k0+3w1Bkαp/nsup0<ε<q−1εθ/q−εGα,qqw1,qw2aχkLq−εw2,βp≤C∑k=−∞k0+3w1Bkαp/nsup0<ε≤ε0εθ/q−εGα,qqw1,qw2aχkLq−εw2,βp≤C∑k=−∞k0+3w1Bkαp/nsup0<ε≤ε0εθ/q−εaLq−εw2,βp≤Csup0<ε≤ε0εθp/q−ε∑k=−∞k0+3w1Bkw1Bk0αp/n≤C∑k=−∞k0+32k−k0αpδ≤C.
In order to estimateI2, suppose ϕ∈Aα,qqw1,qw2ℝn and m+n≤nqw2−ε0/q−ε0−αqw1<m+n+1 for some m∈ℤ+. By the similar method in [15], let Pm be the m-order Taylor expansion for ϕ; if x−y<t, we have
(49)a∗ϕty=t−n∫azϕy−zt−Pmytdz≤C∫azzm+11x−y+y−θzm+n+1dz≤C2k0m+n+1xm+n+1∫azdz≤C2k0m+n+1xm+n+1w1Bk0−α/nw2,βBk0−1/q−ε,where θ∈0,1 and x∈Ck, k≥k0+4. Therefore, we have
(50)Gα,qqw1,qw2ax≤C2k0m+n+1xm+n+1w1Bk0−α/nw2,βBk0−1/q−εfor x∈Ck and k≥k0+4. Since w1∈Aqw1 and w2,β∈Aqw2−ε0⊂Aqw2−ε for 0<ε≤ε0, combining with Remark 5, we have
(51)I2=∑k=k0+4∞w1Bkαp/nsup0<ε<q−1εθ/q−εGα,qqw1,qw2aχkLq−εw2,βp≤C∑k=k0+4∞w1Bkαp/nsup0<ε≤ε0εθ/q−εGα,qqw1,qw2aχkLq−εw2,βp≤Csup0<ε≤ε0εθp/q−ε∑k=k0+4∞2k0−km+n+1pw1Bkw1Bk0αp/nw2,βBkw2,βBk0p/q−ε≤Csup0<ε≤ε0εθp/q−ε∑k=k0+4∞2k0−km+n+1−nqw2−ε0q−ε0−αqw1p≤C.
This proves the sufficiency in the case0<p≤1. If 1<p<∞, we have
(52)Gα,qqw1,qw2aK̇q,θ,βα,pw1,w2p≤C∑k=−∞∞w1Bkαp/n∑j=−∞k−1λjGα,qqw1,qw2ajχkLβq,θw2p+C∑k=−∞∞w1Bkαp/n∑j=k∞λjGα,qqw1,qw2ajχkLβq,θw2p≔E1+E2.
Firstly, we estimateE2. Since w1∈Aqw1 and w2,β∈Aqw2−ε0⊂Aqw2−ε for 0<ε≤ε0, combining with Remark 5, we have
(53)E2=C∑k=−∞∞w1Bkαp/n∑j=k∞λjsup0<ε<q−1εθp/q−εGα,qqw1,qw2ajχkLq−εw2,βp≤C∑k=−∞∞w1Bkαp/n∑j=k∞λjsup0<ε≤ε0εθp/q−εGα,qqw1,qw2ajχkLq−εw2,βp≤C∑k=−∞∞w1Bkαp/n∑j=k∞λjsup0<ε≤ε0εθp/q−εajLq−εw2,βp≤Csup0<ε≤ε0εθp/q−ε∑k=−∞∞w1Bkαp/n∑j=k∞λjw1Bj−α/np≤C∑k=−∞∞w1Bkαp/n∑j=k∞λjw1Bj−αp/2n∑j=k∞w1Bj−αp′/2np/p′≤C∑k=−∞∞∑j=k∞λjpw1Bkw1Bjαp/2n≤C∑j=−∞∞λjp∑k=−∞j2k−jαpδ/2≤C∑j=−∞∞λjp<∞.
Similar to the estimation ofI2, we have
(54)E1≤Csup0<ε≤ε0εθp/q−ε∑k=−∞∞∑j=−∞k−1λj2j−km+n+1w1Bkw1Bjα/nw2,βBkw2,βBj1/q−εp≤Csup0<ε≤ε0εθp/q−ε∑k=−∞∞∑j=−∞k−1λj2j−km+n+1−αqw1−nqw2−ε0q−εp≤Csup0<ε≤ε0εθp/q−ε∑k=−∞∞∑j=−∞k−1λj2j−km+n+1−αqw1−nqw2−ε0q−ε0p≤C∑k=−∞∞∑j=−∞k−1λjp2j−kp/2m+n+1−αqw1−nqw2−ε0q−ε0×∑j=−∞k−12j−kp′/2m+n+1−αqw1−nqw2−ε0q−ε0p/p′≤C∑j=−∞∞λjp∑k=j+1∞2j−kp/2m+n+1−αqw1−nqw2−ε0q−ε0≤C∑j=−∞∞λjp<∞.
This finish the proof of the sufficiency. Then, by a similar proof to that of Theorem 1 in [15], we can prove the necessity and f=∑k=−∞∞λjaj in S′ℝn.Now, we have the following boundedness theorem.Theorem 18.
Supposeθ>0, 0<α<∞, 1<q<∞, and 0<p<∞. Let w1∈Aqw1 and w2∈Aqw2, 1≤qw1<∞, and 1<qw2≤q<∞. There exist some ε0∈0,qw2−1 such that w2,β∈Aqw2−ε0, where β is an arbitrary positive admissible number. n1−qw2−ε0/q−ε0≤αqw1<n1−qw2−ε0/q−ε0+s+n+1 and the nonnegative integer s=nqw2−ε0/q−ε0−1+αqw1.
Suppose that sublinear operatorT satisfies the condition
(55)Tfx≲Cdiamsuppfs+1x−n+s+1f1,distx,suppf≥x2,for any integrable function f with compact support in B0,r and ∫ℝnfxxγdx=0, γ≤s. If T is bounded on Lβq,θw2, then T maps ḢKq,θ,βα,pw1,w2 into K̇q,θ,βα,pw1,w2. Similarly, the nonhomogeneous weighted grand Herz space has the same result.Proof.
By Theorem17, we have f=∑j=−∞∞λjajx, where aj is a central α,q;w1,w2,β-atom with the support Bj and
(56)fḢKq,θ,βα,pw1,w2~inf∑j=−∞∞λjp1/p.
Then, we have(57)TfK̇q,θ,βα,pw1,w2p=∑k=−∞∞w1Bkαp/nTfχkLβq,θw2p≤C∑k=−∞∞w1Bkαp/n∑j=−∞k−2λjTbjχkLβq,θw2p+C∑k=−∞∞w1Bkαp/n∑j=k−1∞λjTbjχkLβq,θw2p≔F1+F2.
By Remark5, Lemma 2, w1∈Aqw1 and w2,β∈Aqw2−ε0⊂Aqw2−ε for 0<ε≤ε0, we have
(58)F2=∑k=−∞∞w1Bkαp/n∑j=k−1∞sup0<ε<q−1εθ/q−ελjTbjχkLq−εw2,βp≤C∑k=−∞∞w1Bkαp/n∑j=k−1∞sup0<ε≤ε0εθ/q−ελjTbjχkLq−εw2,βp≤C∑k=−∞∞w1Bkαp/n∑j=k−1∞sup0<ε≤ε0εθ/q−ελjbjLq−εw2,βp≤Csup0<ε≤ε0εθp/q−ε∑k=−∞∞∑j=k−1∞λjw1Bkw1Bjα/np≤C∑k=−∞∞∑j=k−1∞λj2k−jαδp.
If0<q≤1, we have
(59)F2≤C∑j=−∞∞λjp∑k=−∞j+12k−jαpδ≤C∑j=−∞∞λjp.
Ifq>1, by the Ho¨lder inequality, we have
(60)F2≤C∑k=−∞∞∑j=k−1∞λjp2k−jαδp/2∑j=k−1∞2k−jαδp′/2p/p′≤C∑j=−∞∞λjp∑k=−∞j+12k−jαδp/2≤C∑j=−∞∞λjp.
Then, we estimateF1. Since, w1∈Aqw1 and w2,β∈Aqw2−ε0⊂Aqw2−ε for 0<ε≤ε0, we have
(61)TbjχkLq−εw2,β=∫CkTbjxq−εw2,βxdx1/q−ε≤C∫Ckx−n+s+1q−ε2js+1∫Bjbjxdyq−εw2,βxdx1/q−ε≤C2j−ks+n+1w1Bk−α/nw2,βBkw2,βBj1/q−ε.
It follows that(62)F1≤C∑k=−∞∞w1Bkαp/n∑j=−∞k−2sup0<ε≤ε0εθ/q−ελjTbjχkLq−εw2,βp≤Csup0<ε≤ε0εθ/q−ε∑k=−∞∞∑j=−∞k−2λj2j−ks+n+1w1Bkw1Bjα/nw2,βBkw2,βBj1/q−εp≤C∑k=−∞∞∑j=−∞k−2λj2j−ks+n+1−qw2−ε0n/q−ε0−αqw1p.
Sincen1−qw2−ε0/q−ε0≤αqw1<n1−qw2−ε0/q−ε0+s+n+1, by the same argument as F2, we have
(63)F1≤C∑j=−∞∞λjp.
Thus,(64)TfK̇q,θ,βα,pw1,w2≤CfḢKq,θ,βα,pw1,w2.



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