# Generalized Hausdorff Operators onK̇α,qβ,pℝ and HK̇α,qβ,p,Nℝ in the Dunkl Settings

Authors: Faouaz Saadi; Othman Tyr; Radouan Daher
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In the present paper, we obtain some new results, and we generalize some known results for the Hausdorff operators. We have studied the generalized Hausdorff operatorsHα,φ on the Dunkl-type homogeneous weighted Herz spaces K̇α,qβ,pℝ and Dunkl Herz-type Hardy spaces HK̇α,qβ,p,Nℝ. We have determined simple sufficient conditions for these operators to be bounded on these spaces. As applications, we provide necessary and sufficient conditions for generalized Cesàro operator to be bounded on K̇α,qβ,pℝ and Hardy inequality for K̇α,qβ,pℝ.


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## Body

## 1. Introduction and Preliminaries

We recall that the Fourier transformf^ of a (complex-valued) function f in L1ℝ is defined as
(1)f^t≔∫ℝfxe−itxdx,t∈ℝ.The Hausdorff operatorH generated by a function φ in L1ℝ as introduced in [1] can be defined both directly and via the Fourier transform. The latter reads as follows:
(2)Hφf∧t≔∫ℝf^txφxdx,t∈ℝ,where f is also in L1ℝ. The existence of such a function Hf in L1ℝ is established in [1]. The theory of Hausdorff operators, while dating in a sense back to Hurwitz and Silverman [2] in 1917 with summability of number series, now becomes a notable ingredient in modern harmonic analysis and has received an extensive attention in recent years. To save the length of this article, we refer the reader to the survey article [3] for its background and historical developments.Dunkl’s theory generalizes classical Fourier analysis onℝd. This theory began twenty years ago with Dunkl’s seminal work in [4]. It was later developed by many mathematicians. On the real line, the Dunkl operators Dα are differential-difference operators associated with the reflection group ℤ2 on ℝ. An important motivation to study Dunkl operators originates from their relevance for the analysis of quantum many-body systems of the Calogero-Moser-Sutherland type. These describe algebraically integrable systems in one dimension and have gained considerable interest in mathematical physics (see [5]).Letμα be the measure on ℝ, given by
(3)dμαx=x2α+1dx.We denote byLαpℝ, 1≤p≤∞, the space of measurable functions on ℝ such that
(4)∥f∥Lαp≔∫ℝfxpdμαx1/p<∞,if1≤p<∞,(5)∥f∥Lα∞≔esssupx∈ℝ∣fx∣<∞.The Dunkl-Hausdorff operatorHα (see [6–9]) acting on Lα1ℝ generated by a function φ belonging to L1ℝ is defined directly as:
(6)Hα,φfx≔∫0∞φtt2α+2fxtdt,f∈Lα1ℝ,x∈ℝ,and for all function f in Lα1ℝ, the Dunkl-Hausdorff operator Hα verifies
(7)FαHαft=∫ℝFαftxφxdx,t∈ℝ,where Fα is the Dunkl transform. When α=−1/2, the operator Hα,φ is the direct definition of the Hausdorff operator associated with Fourier transform defined in (2)
(8)Hφfx=∫0∞φttfxtdt,from which several well-known operators can be deduced for suitable choices of φ, e.g., for φt=1/tχ1,∞t, the operator Hφ reduces to the standard Hardy averaging operator
(9)Hfx=1x∫0xftdt,while for φt=χ0,1t, it reduces to the adjoint of Hardy averaging operator
(10)H∗fx=∫x∞fttdt.Chen et al. [10] established boundedness of the classical Hausdorff operators in Herz type spaces, which are a natural generalization of the Lebesgue spaces Lp. Gasmi et al. [11] introduced a new weighted Herz space associated with the Dunkl operators on ℝ. They also characterize the corresponding Herz-type Hardy spaces by atomic decomposition. Motivated by this result concerning Herz spaces (see also [12–14] and reference therein), this paper is aimed at extending these results to the context of Dunkl theory. We investigate the Dunkl-Hausdorff operators on the Dunkl-type homogeneous weighted Herz spaces K̇α,qβ,pℝ and Dunkl Herz-type Hardy space HK̇α,qβ,p,Nℝ in the spirit of those in [10]. As applications, we provide necessary and sufficient conditions for Dunkl-Cesàro operator and sufficient conditions for Dunkl-Hardy operator to be bounded on the homogeneous weighted Herz space K̇α,qβ,pℝ.This paper is organized as follows: in Section1, we have presented some definitions and fundamental results from Dunkl’s analysis. In Sections 2 and 3, we have presented and proven our main results.Forα≥−1/2, the Dunkl differential-difference operator is defined as (see [4])
(11)Dαfx=dfdxx+2α+1x·fx−f−x2,f∈C1ℝ.Forλ∈ℂ, the initial problem
(12)Dαfx=λfx,f0=1,x∈ℝ,has a unique solution Eαλ. (called the Dunkl kernel) given by:
(13)Eαz=jαiz+z2α+1jα+1iz,z∈ℂ,where jα is the normalized Bessel function of the first kind (with order α) defined on ℂ by
(14)jαz=Γα+1∑n=0∞−1nz/22nn!Γn+α+1.The integral representation ofEα is given by
(15)Eαiλx=Γα+1πΓα+1/2∫−111−t1−t2α−1/2e−iλxtdt.The Dunkl transformFα is defined for f∈Lα1ℝ by:
(16)Fαfx=1cα∫ℝEα−ixyfydμαy,cα=2α+1Γα+1.This transform satisfies the following properties:(i)
For allf∈Lα1ℝ such that Fαf∈Lα1ℝ, we have the inversion formula(17)fx=1cα∫ℝEαiλxFαfλdμαλ,a.e.,x∈ℝ.(ii)
For allf∈Sℝ (the usual Schwartz space)(18)FαDαfx=ixFαfx.For anyx,y,z∈ℝ, we consider:
(19)wαx,y,z=Γα+122α−1πΓα+1/21−σx,y,z+σz,x,y+σz,y,xqαx,y,z,where
(20)σx,y,z==x2+y2−z22xy,ifxy≠0,0otherwise,and for x,y,z>0,
(21)qαx,y,z=x+y2−z2z2−x−y2α−1/2xyz2αX∣x−y∣,x+yz.We further have (see [15])
(22)∫0+∞qαx,y,zdμαz=1and∫ℝ∣wαx,y,z∣dμαz≤2.In the sequel, we consider the signed measureγx,y on ℝ given by
(23)dγx,yz=wαx,y,zdμαz,ifx,y∈ℝ∗,dδyz,ifx=0,dδxz,ify=0.Forx,y∈ℝ and a continuous function f on ℝ, we put
(24)τxfy=∫ℝfzdγx,yz,which is called the Dunkl translation operator. The Dunkl translation operator has the following properties:
(i)
Forx,y∈ℝ and a continuous function f on ℝ, we have(25)τxfy=τyfx.(ii)
(Product formula) For allx,y,z∈ℝ(26)τxEαiyz=EαixyEαiyz.(iii)
For allx,y∈ℝ and f∈Lα1ℝ, we have(27)Fατxfy=EαixyFαfy.The Dunkl convolution of two functionsf, g on ℝ is defined by the relation
(28)f∗αgx=∫ℝτxf−ygydμαy.Letf∈S′ℝ and ϕ∈Sℝ such that ∫ℝϕxdμαx=1, we have
(29)limt⟶0f∗αϕt=f,inS′ℝwhere ϕt is the dilation of ϕ given by
(30)ϕtx≔t−2α+1ϕxt,x∈ℝ.For allN∈ℕ, we denote by FN the subset of Sℝ constituted by all those ϕ∈Sℝ such that suppϕ⊂−1,1, and for all m,n∈ℕ such that m,n≤N, we have
(31)ρm,nϕ≔supx∈ℝ1+∣x∣mDαnϕx≤1.Moreover, the system of seminormsρm,nm,n∈N generates the topology of Sℝ (see [16]).Letf∈S′ℝ and N∈ℕ. The α-grand maximal function of N-order Gα,Nf of f is defined by
(32)Gα,Nfx≔supt>0,ϕ∈FNϕt∗αfx,x∈ℝ.Theα-grand maximal function Gα,N is a bounded continuous operator from Lαpℝ into itself, for every p∈1,∞, provided thatN>2α+1 (see [11]).


## 2. Boundedness ofHα,φ on the Homogeneous Weighted Herz Space K̇α,qβ,p

Letβ∈ℝ, 0<p<+∞, and 1≤q<+∞. The homogeneous weighted Herz space K̇α,qβ,p is the space constituted by all the functions f∈Lαqℝloc, such that
(33)∥f∥K̇α,qβ,pℝ≔∑k=−∞+∞22α+1βkp∥fχk∥Lαqℝp1/p<+∞,where χk is the characteristic function of the set
(34)Ak=x∈ℝ;2k−1≤x≤2k,fork∈ℤ,and Lαqℝloc is the space Llocqℝ,x2α+1dx.Note thatK̇α,qβ,0ℝ=Lαqℝ. The main result of this subsection is the following theorem.Theorem 1.
Letα≥−1/2, β∈ℝ, 1<p<+∞, 1≤q<+∞, and φ a measurable function on ℝ such that
(35)Cq,α,β,φ=∫0∞∣φt∣t2α+1β−1+1/qdt<∞.
Then, the Dunkl-Hausdorff operatorHα,φ is bounded from K̇α,qβ,pℝ to itself, i.e.,
(36)∥Hα,φf∥K̇α,qβ,pℝ<~Cq,α,β,φ∥f∥K̇α,qβ,pℝ.Proof.
Note that for anyt>0, there exists an integer number l=lt satisfying 2l−1<t≤2l. The Minkowski inequality for Lαqℝ guarantees that for any k∈ℤ,
(37)∥Hα,φfχk∥Lαqℝ≤∫0∞φtt2α+11/q−1∥fχk−l∥Lαqℝ+∥fχk−l+1∥Lαqℝdt.
By applying the Minkowski inequality forlp, we obtain
(38)∥Hα,φf∥K̇α,qβ,pℝ=∑k=−∞+∞22α+1βkp∥Hα,φfχk∥Lαqℝp1/p≤∑k=−∞+∞22α+1βkp∫0∞∣φt∣t2α+11/q−1∥fχk−l∥Lαqℝ+∥fχk−l+1∥Lαqℝdtp1/p≤∫0∞φtt2α+11/q−1∑k=−∞+∞22α+1βkp∥fχk−l∥Lαqℝ1/p+∑k=−∞+∞22α+1βkp∥fχk−l+1∥Lαqℝ1/pdt.
Since2l−1<t≤2l and the definition of the Herz space K̇α,qβ,p, we estimate
(39)∑k=−∞+∞22α+1βkp∥fχk−l−1∥Lαqℝ1/p+∑k=−∞+∞22α+1βkp∥fχk−l∥Lαqℝ1/p≤22l−1α+1β+22lα+1β∥f∥K̇α,qβ,pℝ<~t2α+1β∥f∥K̇α,qβ,pℝ.
Therefore, we obtain(40)∥Hα,φf∥K̇α,qβ,pℝ<~∫0∞∣φt∣t2α+1β−1+1/qdt∥f∥K̇α,qβ,pℝ,which implies that Hα,φ is bounded from K̇α,qβ,pℝ to itself.Remark 2.
Whenα=−1/2, Theorem 1 reduce to ([10], Theorem 2.4).


### 2.1. Hardy Inequality forK̇α,qβ,pℝ

Ifφt=χ1,∞t/t, then (6) is of the following form:
(41)Hαfx=1x2α+1∫0xfξdμαξ.In this case,Hα,φ reduces to the Hardy-type averaging operator for which we deduce the following result.Corollary 3.
Letα≥−1/2, 1<p<+∞, 1≤q<+∞, 0<β<1−1/q, and φt=χ1,∞t/t. Then, the Dunkl-Hardy operator Hα,φ is bounded from K̇α,qβ,pℝ to itself and we have
(42)∥Hα,φf∥K̇α,qβ,pℝ≤12α+1β−1+1/q∥f∥K̇α,qβ,pℝ.Proof.
From Theorem1, we have
(43)Cq,α,β,φ=∫0∞∣φtt2α+1β−1+1/qdt,=∫1∞t2α+1β−1+1/q−1dt,=12α+1β−1+1/q.


### 2.2. Generalized Cesàro Operator

Ifφ is supported in the interval 0;1, then Hα,φ reduces to the generalized Cesàro operator Cα,φ defined by
(44)Cφfx≔∫01φtt2α+2fxtdt,x∈ℝ,(see [6, 17]).Corollary 4.
Letα≥−1/2, β∈ℝ×, 1<p<+∞, 1≤q<+∞, and φa nonnegative measurable function defined on 0,1. Then, the generalized Cesàro operator Cα,φ is bounded from K̇α,qβ,pℝ to itself if and only if
(45)Cq,α,φ=∫01φtt2α+1β+1/q−1dt<∞.Proof.
By Theorem1, we need just to prove the necessity part. For any ε∈0,1, we set
(46)fεx=x−2βα+1+ε+2α+1/q,if∣x∣>1,0otherwise,then for j=0,−1,−2,⋯,∥fεχj∥Lαqℝ=0, and for j∈ℕ\0, we have
(47)∥fεχj∥Lαqℝq=∫2j−1≤∣x∣≤2jx−2α+1β+ε+2α+1/qqdμαx=2∫2j−12jx−2α+1β+εq−1dx=Cε2−j2α+1β+εq.where
(48)Cε=222α+1β+εq−12α+1β+εq.
Hence, it yields(49)∑j=−∞+∞22α+1βjp∥fεχj∥Lαqℝp1/p=∑j=1+∞22α+1βjp∥fεχj∥Lαqℝp1/p=Cε1/q∑j=1+∞2−jεp1/p=Cε1/q2−ε1−2−pε1/p,thus fε∈Hp,qβ,k.
Now, it is easy to see that(50)Hα,φfεx=∫0∣x∣xt−2α+1β+ε+2α+1/qt−2α+1φtdt=x−2βα+1+ε+2α+1/q∫0xt2βα+1+ε−2α+11−1/qφtdt.∑j=−∞+∞22α+1βjp∥Hα,φfεχj∥Lαqℝp=∑j=−∞+∞22α+1βjp∫Ajx−2βα+1+ε+2α+1/q∫0xt2βα+1+ε−2α+11−1/qφtdtqdμαxp/q≥∫01t2βα+1+ε−2α+11−1/qφtdtp∑j=1+∞22α+1βjp∫Ajx−2βα+1+ε+2α+1/qqdμαxp/q.
It follows from (49) that
(51)∑j=−∞+∞22α+1βjp∥Hα,φfεχj∥Lαqℝp1/p≥Cε1/q2−ε1−2−qε1/p∫01t2βα+1+ε−2α+11−1/qφtdt,which implies when ε⟶0,
(52)∫01t2βα+1−2α+11−1/qφtdt<∞.
This completes the proof.


## 2.1. Hardy Inequality forK̇α,qβ,pℝ

Ifφt=χ1,∞t/t, then (6) is of the following form:
(41)Hαfx=1x2α+1∫0xfξdμαξ.In this case,Hα,φ reduces to the Hardy-type averaging operator for which we deduce the following result.Corollary 3.
Letα≥−1/2, 1<p<+∞, 1≤q<+∞, 0<β<1−1/q, and φt=χ1,∞t/t. Then, the Dunkl-Hardy operator Hα,φ is bounded from K̇α,qβ,pℝ to itself and we have
(42)∥Hα,φf∥K̇α,qβ,pℝ≤12α+1β−1+1/q∥f∥K̇α,qβ,pℝ.Proof.
From Theorem1, we have
(43)Cq,α,β,φ=∫0∞∣φtt2α+1β−1+1/qdt,=∫1∞t2α+1β−1+1/q−1dt,=12α+1β−1+1/q.


## 2.2. Generalized Cesàro Operator

Ifφ is supported in the interval 0;1, then Hα,φ reduces to the generalized Cesàro operator Cα,φ defined by
(44)Cφfx≔∫01φtt2α+2fxtdt,x∈ℝ,(see [6, 17]).Corollary 4.
Letα≥−1/2, β∈ℝ×, 1<p<+∞, 1≤q<+∞, and φa nonnegative measurable function defined on 0,1. Then, the generalized Cesàro operator Cα,φ is bounded from K̇α,qβ,pℝ to itself if and only if
(45)Cq,α,φ=∫01φtt2α+1β+1/q−1dt<∞.Proof.
By Theorem1, we need just to prove the necessity part. For any ε∈0,1, we set
(46)fεx=x−2βα+1+ε+2α+1/q,if∣x∣>1,0otherwise,then for j=0,−1,−2,⋯,∥fεχj∥Lαqℝ=0, and for j∈ℕ\0, we have
(47)∥fεχj∥Lαqℝq=∫2j−1≤∣x∣≤2jx−2α+1β+ε+2α+1/qqdμαx=2∫2j−12jx−2α+1β+εq−1dx=Cε2−j2α+1β+εq.where
(48)Cε=222α+1β+εq−12α+1β+εq.
Hence, it yields(49)∑j=−∞+∞22α+1βjp∥fεχj∥Lαqℝp1/p=∑j=1+∞22α+1βjp∥fεχj∥Lαqℝp1/p=Cε1/q∑j=1+∞2−jεp1/p=Cε1/q2−ε1−2−pε1/p,thus fε∈Hp,qβ,k.
Now, it is easy to see that(50)Hα,φfεx=∫0∣x∣xt−2α+1β+ε+2α+1/qt−2α+1φtdt=x−2βα+1+ε+2α+1/q∫0xt2βα+1+ε−2α+11−1/qφtdt.∑j=−∞+∞22α+1βjp∥Hα,φfεχj∥Lαqℝp=∑j=−∞+∞22α+1βjp∫Ajx−2βα+1+ε+2α+1/q∫0xt2βα+1+ε−2α+11−1/qφtdtqdμαxp/q≥∫01t2βα+1+ε−2α+11−1/qφtdtp∑j=1+∞22α+1βjp∫Ajx−2βα+1+ε+2α+1/qqdμαxp/q.
It follows from (49) that
(51)∑j=−∞+∞22α+1βjp∥Hα,φfεχj∥Lαqℝp1/p≥Cε1/q2−ε1−2−qε1/p∫01t2βα+1+ε−2α+11−1/qφtdt,which implies when ε⟶0,
(52)∫01t2βα+1−2α+11−1/qφtdt<∞.
This completes the proof.


## 3. Boundedness ofHα,φ on the Dunkl Herz-Type Hardy Space HK̇α,qβ,p,Nℝ

Definition 5.
Letα≥−1/2, N∈ℕ, β∈ℝ, 0<p<+∞, and 1≤q<+∞. The Herz-type Hardy space HK̇α,qβ,p,Nℝ is the space of distributions f∈S′ℝ such that Gα,Nf∈K̇α,qβ,pℝ. Moreover, we have
(53)∥f∥HK̇α,qβ,p,Nℝ=∥Gα,Nf∥K̇α,qβ,pℝ.
In the sequel, we are interested in the spacesHK̇α,qβ,p,Nℝ, when β≥1−1/q. Now, we turn to the atomic characterization of the space HK̇α,qβ,p,Nℝ.Definition 6.
Letα≥−1/2, 1≤q≤∞, and β≥1−1/q. A measurable function a on ℝ is called a (central) β;q-atom if it satisfies:
(1)
suppa⊂−r,r, for a certain r>0(2)
∥a∥q,α≤r−2α+1β,(3)
∫ℝaxxkdμαx=0, k=0,1,⋯,2s+1
wheres is the integer part of α+1β−1+1/q.Theorem 7.
Letα≥−1/2, 0<p<+∞, 1≤q<+∞, β≥1−1/q, and N∈ℕ; N>22s+3+α. Then, f∈HK̇α,qβ,p,Nℝ if and only if there exist, for all j∈ℕ\0, an β;q-atomajandλj∈ℂ, such that ∑j=1∞λjp<∞ and f=∑j=1∞λjaj. Moreover,
(54)∥f∥HK̇α,qβ,p,Nℝ=inf∑j=1∞λjp1/p,where the infimum is taken over all atomic decompositions of f.The main result of this subsection is the following theorem.Theorem 8.
Letα≥−1/2, 0<p⩽1<q<∞, β≥1−1/q, and N∈ℕ; N>22s+3+α.
(i)
For0<p<1, let(55)Cp,σ=∫0∞t2α+1β−1+1/qφt1+log2tσdt.
If for someσ>1−p/p,Cp≔Cp,σ<∞, then
(56)Hα,φfHK̇α,qβ,p,N<~∥f∥HK̇α,qβ,p,N.(ii)
Forp=1, let(57)C1=∫0∞t2α+1β−1+1/qφtdt.
IfC1<∞, then
(58)Hα,φfHK̇α,qβ,1,N<~∥f∥HK̇α,qβ,1,N.Proof.
By the central atomic decomposition, forf∈HK̇α,qβ,p,Nℝ, we write
(59)f=∑k=1∞λkak,where
(60)∑j=1∞λjp≃∥f∥HK̇α,qβ,p,Np.
Then, we have(61)Hα,φf=∑j=1∞λjHα,φaj.
Let us show that(62)Hα,φak=∑j∈ℤck,jak,j,where each ak,j again is a central β;q-atom and
(63)∑k∈ℤck,jp°∞.
We write(64)bk,jx=∫2j≤t≤2j+1akxtt−2α+2φtdt.
So,(65)Hα,φakx=∑j∈ℤbk,jx.
Now, we check that eachbk,j satisfies the same cancellation condition as ak.
Fori=0,1,⋯,2s+1, where s is the integer part of α+1β−1+1/q, we have
(66)∫ℝbk,jxxidμαx=∫ℝxi∫2j≤t≤2j+1akxtt−2α+2φtdt=∫2j≤t≤2j+1t−2α+2φtdt∫ℝakxtxidμαx=∫2j≤t≤2j+1t−2α+2φtdt∫ℝakxtxidμαx=∫2j≤t≤2j+1tiφtdt∫ℝakuuidμαu=0.
Also, the size ofbk,j is
(67)∥bk,j∥Lαq≤∫2j≤t≤2j+1ak·tLαqt−2α+11/q−1φtdt,then
(68)∥bk,j∥Lαq≤rk−2α+1β∫2j≤t≤2j+1t−2α+11/q−1φtdt.
If∣x∣>2j+1rk, we have
(69)∣x∣t≥2−j−1x>rk,which means akx/t=0 for all 2j≤t≤2j+1. This tells us that
(70)suppbk,j⊂B0,2j+1rk.
Now, we write(71)Hα,φak=∑k∈∞ck,jak,j,where
(72)ck,j=22j+1α+1β∫2j≤t≤2j+1t−2α+11/q−1φtdt,(73)ak,j=ck,j−1bk,j.
It is easy to check thatak,j is a central α,q atom and we have
(74)ck,j=22α+1β∫2j≤t≤2j+1t−2α+11/q−122jα+1βφtdt≤22α+1β∫2j≤t≤2j+1t−2α+11/q−1t2α+1βφtdt2j≤t=22α+1β∫2j≤t≤2j+1t2α+1β−1+1/qφtdt.
Let(75)ck,j′=∫2j≤t≤2j+1t2α+1β−1+1/qφtdt,using Holder inequality yields the following
(76)∑j∈ℤck,j′p=∑j∈ℤck,j′p1+jσp1+j−σp≤∑j∈ℤck,j′1+jσp∑j∈ℤ1+j−σp/1−p1−p≤∫ℝt2α+1β−1+1qφt1+log2tσdtp∑j∈ℤ1+j−σp/1−p1−p,since σ>1−p/p, then ∑j∈ℤ1+∣j∣−σp/1−p<∞. It follows from (75) that
(77)∑j∈ℤck,jp<~∑j∈ℤck,j′p<~∫0∞t2α+1β−1+1/qφt1+log2tσdtp=Cp,σp.
This shows(78)HΦf=∑k=1∞λkHΦak=∑k=1∞∑j∈ℤλkck,jak,j.
By the atomic decomposition, we obtain(79)Hφ,αfHK̇qα,pℝn<~∑k=1∞∑j∈ℤλkck,jp1/p<~∑k∈ℤλkp1/p<~∥f∥HK̇α,qβ,p,N,and this end the proof of i.The argument of partii can be proved in an analogous way.Remark 9.
Whenα=−1/2, Theorem 8 reduce to ([10], Theorem 2.5).We now return to the example of the generalized Cesàro operatorCα,φ.Corollary 10.
Letα≥−1/2, 0<p⩽1<q<∞, β≥1−1/q, and N∈ℕ; N>22s+3+α.
(i)
For0<p<1, let(80)Cp,σ=∫01t2α+1β−1+1/qφt1+log2tσdt.
If for someσ>1−p/p,Cp≔Cp,σ<∞, then
(81)Cα,φfHK̇α,qβ,p,N<~∥f∥HK̇α,qβ,p,N.(ii)
Forp=1, let(82)C1=∫01t2α+1β−1+1/qφtdt.
IfC1<∞, then
(83)Cα,φfHK̇α,qβ,1,N<~∥f∥HK̇α,qβ,1,N.



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*Source: 6547407-2021-08-29.xml*