# The WeightedLp and BMO Estimates for Fractional Hausdorff Operators on the Heisenberg Group

Authors: Guohua Zhang; Qianqian Li; Qingyan Wu
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In the setting of Heisenberg group, we characterize those functionsΦ, for which the fractional Hausdorff operators TΦ,β and Hausdorff operators TΦ, T˜Φ are bounded on Lp spaces with power weights, BMO space, and Hardy spaces, respectively. Meanwhile, the corresponding operator norms of TΦ and T˜Φ are worked out.


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## Body

## 1. Introduction

The Hausdorff operator has a deep root in the study of the Fourier analysis. Particularly, one-dimensional Hausdorff operator in Euclidean space is closely related to the summability of the classical Fourier series. Modern theory of Hausdorff operator started with the work of Siskakis in the complex analysis setting and with the work of Liflyand–Móricz in the Fourier transform setting (for more information about the background and the development of the Hausdorff operator one can refer to [1–5] and references therein).Fix an appropriate functionΦ on ℝ+, the one-dimensional Hausdorff operator [5, 6] is defined formally by(1)hΦfx=∫0∞Φttfxtdt,x∈ℝ.Forx>0, by a change of variables, one has(2)hΦfx=∫0∞Φx/ttftdt,x∈ℝ.Naturally, then-dimensional Hausdorff operators on ℝn are defined by(3)HΦfx=∫ℝnΦyynfxydy,H˜Φfx=∫ℝnΦx/yynfydy.It is well known that many classical operators in harmonic analysis are special cases of Hausdorff operators if one chooses suitableΦ (see, for example, [1, 6–10]), including Hardy operators and Cesàro operator [11–14]. The Hardy–Littlewood–Pólya operator and the Riemann–Liouville fractional integral can also be derived from the Hausdorff operators. Besides, if we set Φt=t−1ψt−1χ1,∞t in hΦfx and Φy=ωny−1ψy−1χ1<y<∞y in HΦfx, then hΦfx and HΦfx become the weighted Hardy–Littlewood average Uψfx defined by Xiao in [15], which is given by(4)Uψfx=∫01ftxψtdt,x∈ℝn.And, if we setΦt=ψtχ0,1t and Φy=ωn−1y−n+1ψyχ0<y<1y in hΦfx and HΦfx, respectively, then they become the weighted Cesàro average Vψfx in [15], which is defined by(5)Vψfx=∫01fxtt−nψtdt,x∈ℝn.In recent years, there are many works about the Hardy operators, weighted Hardy operators, and Hausdorff operators in the setting of the Heisenberg group (see, for example, [16–23]). It is known that the Heisenberg group plays an important role in several branches of mathematics such as harmonic analysis, representation theory, several complex analysis, partial differential equation, and quantum mechanics. In this paper, we will focus on the boundedness of Hausdorff operators on power-weighted Lebesgue spaces in the setting of the Heisenberg group ℍn, such group can be identified with ℝ2n×ℝ with the group structure (for all the notation on ℍn, we refer to Section 2).LetΦ be a locally integrable function on ℍn. The Hausdorff operators on ℍn are defined by(6)TΦfx=∫ℍnΦδyh−1xyhQfydy,T˜Φfx=∫ℍnΦyyhQfδyh−1xdy.The fractional Hausdorff operator onℍn is defined by(7)TΦ,βfx=∫ℍnΦδyh−1xyhQ−βfydy,0≤β≤Q.Remark 1.
It is clear that whenβ=0, then TΦ,βf=TΦf. And, if we choose Φy=χyh>1y/yhQ and Φy=χyh≤1y in the operator T˜Φ, respectively, we can obtain the Hardy operators and their adjoint on the Heisenberg group [22]:(8)Hfx=1xhQ∫yh<xhfydy,H∗fx=∫yh≥xhfyyhQdy.
We obtain the following weak and strong boundedness for fractional Hausdorff operators on Lebesgue spaces with power weights (for Euclidean version, one can refer to Gao et al. [9]).Theorem 1.
Let1≤p,q<∞, 0≤β<Q, γ>βp−Q, and 1/p−β/Q+γ=1/q. Let(9)K1=⋅hQ−β+γΦ⋅L∞ℍn,Kp=∫0∞Φtp/p−1tγ−βp+Q−p+1/p−1dt,p>1.
IfΦ is a radial function and Kp<∞ for p≥1, then(10)TΦ,βfLq,∞ℍn,⋅hγ≼¯fLpℍn,⋅hγ.Theorem 2.
Let1≤p,q≤∞, 0≤β<Q, and α,γ∈ℝ satisfy 1/q=1/p−β/Q+γ. If(11)KΦ,Q,p,q,γ=ωQ1/p′∫S∫0∞Φδρξsργ+Qs/q−1dρq/sdξ1/q<∞,where s satisfies 1/s=1+1/q−1/p, then(12)TΦ,βfLqℍn,⋅hγ≤KΦ,Q,p,q,γfLpℍn,⋅hα.Corollary 1.
Let1≤p,q≤∞, 0≤β<Q, and α,γ∈ℝ satisfy 1/q=1/p−β/Q+γ. For any radial function Φ, if(13)KΦ,Q,p,q,γ=∫ℍnΦxsxh−Q+γ+Qs/qdx1/s<∞,where s satisfies 1/s=1+1/q−1/p, then(14)TΦ,βfLqℍn,⋅hγ≤KΦ,Q,p,q,γfLpℍn,⋅hα.Corollary 2.
Suppose that1≤p,q≤∞ and 0≤β<Q and 1/q=1/p−β/Q. If(15)KΦ,Q,p,q=ωQ1/p′∫S∫0∞Φδρx′sρQs/q−1dρq/sdσx′1/q<∞,where s=Q/Q−β, then(16)TΦ,βfLqℍn≤KΦ,Q,p,q,γfLpℍn.
For Hausdorff operators, we get the best estimates for them on Lebesgue spaces with power weights. Chuong et al. [17] also gave the approximation of the norm of these operators on such spaces.Theorem 3.
Let1≤p≤∞, α∈ℝ, and Φ≥0. Then, TΦ is a bounded operator on Lpℍn,⋅hα if and only if(17)KΦ,Q,p,α=ωQ1/p′∫S∫0∞Φδρx′ρα+Q/p−1dρpdσx′1/p<∞.
Moreover, when (17) holds, then(18)TΦLpℍn,⋅hα⟶Lpℍn,⋅hα=KΦ,Q,p,α.Remark 2.
From Theorem3, we can see that if Φ is radial, then(19)KΦ,Q,p,α=ωQ∫0∞Φρρα+Q/p−1dρ.
And, Theorem3 reduces to Theorem 2 in [18].Theorem 4.
Let1≤p≤∞, α∈ℝ and Φ≥0. Then(i)
T˜Φ is bounded on Lpℍn,⋅hα if and only if(20)∫ℍnΦyyh−Q+Q+α/pdy<∞.Moreover, when (17) holds, then(21)T˜ΦLpℍn,⋅hα⟶Lpℍn,⋅hα=∫ℍnΦyyh−Q+Q+α/pdy.(ii)
T˜Φ is bounded on BMOℍn if and only if(22)∫ℍnΦyyh−Qdy<∞.
Moreover, when (22) holds, then(23)T˜ΦBMOℍn⟶BMOℍn=∫ℍnΦyyh−Qdy.
Since the dual space ofH1ℍn is isomorphic to BMOℍn (c.f. [24]), we can actually deduce the following results.Corollary 3.
LetΦ≥0. Then,
T
˜
Φ is bounded on H1ℍn if and only if(24)∫ℍnΦyyh−Qdy<∞.
Moreover, when (24) holds, then(25)T˜ΦH1ℍn⟶H1ℍn=∫ℍnΦyyh−Qdy.
Supposeψ:0,1⟶0,∞ is a function. For a measurable function f on ℍn, we define the weighted Hardy operator Hψ as(26)Hψfx=∫01fδtxψtdt,x∈ℍn,and define the weighted Cesàro operator Hψ∗ as(27)Hψ∗fx=∫01fδt−1xt−Qψtdt,x∈ℍn.
It is easy to check that whenΦy=ωQyh−1ψyh−1χ1<yh<∞y, then T˜Φf=Hψf. When Φy=ωQ−1yh−Q+1ψyhχ0<yh<1y, then T˜Φf=Hψ∗f. Therefore, we can get the sharp conditions for boundedness of weighted Hardy operators, and their Euclidean case is due to Xiao [15].Corollary 4.
Letψ:0,1⟶0,∞ be a function and let 1≤p≤∞, α∈ℝ. Then,(i)
Hψ is bounded on Lpℍn,⋅hα if and only if(28)∫01ψtt−Q+α/pdt<∞.Moreover, when (28) holds, the operator norm of Hψ on Lpℍn,⋅hα is given by(29)HψLpℍn,⋅hα⟶Lpℍn,⋅hα=∫01ψtt−Q+α/pdt.(ii)
Hψ∗ is bounded on Lpℍn,⋅hα if and only if(30)∫01ψtt−Q+Q+α/pdt<∞.
Moreover, when (30) holds, the operator norm of Hψ on Lpℍn,⋅hα is given by(31)Hψ∗Lpℍn,⋅hα⟶Lpℍn,⋅hα=∫01ψtt−Q+Q+α/pdt.Corollary 5.
Letψ:0,1⟶0,∞ be a function. Then,(i)
Hψ is bounded on BMOℍn if and only if(32)∫01ψtdt<∞.Moreover, when (32) holds, the operator norm of Hψ on BMOℍn is given by(33)HψBMOℍn⟶BMOℍn=∫01ψtdt.(ii)
Hψ∗ is bounded on BMOℍn if and only if(34)∫01ψtt−Qdt<∞.
Moreover, when (34) holds, the operator norm of Hψ on BMOℍn is given by(35)Hψ∗BMOℍn⟶BMOℍn=∫01ψtt−Qdt.
Also, by the duality ofH1ℍn and BMOℍn, we can obtain the sufficient and necessary conditions for boundedness of weighted Hardy and Cesàro operators on Hardy space H1ℍn.Corollary 6.
Letψ:0,1⟶0,∞ be a function. Then,(i)
Hψ is bounded on H1ℍn if and only if(36)∫01ψtdt<∞.Moreover, when (36) holds, the operator norm of Hψ on BMOℍn is given by(37)HψH1ℍn⟶H1ℍn=∫01ψtdt.(ii)
Hψ∗ is bounded on H1ℍn if and only if(38)∫01ψtt−Qdt<∞.
Moreover, when (38) holds, the operator norm of Hψ on H1ℍn is given by(39)Hψ∗H1ℍn⟶H1ℍn=∫01ψtt−Qdt.
The paper is organized as follows. In Section2, we recall the notation and definitions related to the Heisenberg group. In Section 3, we will discuss the boundedness for fractional Hausdorff operators on Lebesgue spaces with power weights, and give the proof of Theorem 1–4.


## 2. Preliminaries and Notations

Recall that the Heisenberg groupℍn is a noncommutative nilpotent Lie group, with the underlying manifold ℝ2n×ℝ, whose group law can be written as(40)x1,x2,…,x2n,x2n+1x1′,x2′,…,x2n′,x2n+1′=x1+x1′,x2+x2′,…,x2n+x2n′,x2n+1+x2n+1′+2∑j=1nxj′xn+j−xjxn+j′.By definition, we can see that the identity element onℍn is 0∈ℝ2n+1, while the reverse element of x is −x. The corresponding Lie algebra is generated by the left-invariant vector fields:(41)Xj=∂∂xj+2xn+j∂∂x2n+1,j=1,…,n,Xn+j=∂∂xn+j−2xj∂∂x2n+1,j=1,…,n,X2n+1=∂∂x2n+1.The only nontrivial commutator relations are(42)Xj,Xn+j=−4X2n+1,j=1,…,n.ℍ
n is a homogeneous group with dilations(43)δrz,t=rz,r2t,r>0.The Haar measure onℍn coincides with the usual Lebesgue measure on ℂn×ℝ. We denote the measure of any measurable set E⊂ℍn by E. Then,(44)δrE=rQE,where Q=2n+2 is called the homogeneous dimension of ℍn.The Heisenberg distance derived from the norm(45)z,th=z4+t21/4,is given by(46)dp,q=dq−1p,0=q−1ph.This distanced is left-invariant in the sense that dp,q remains unchanged when p and q are both left-translated by some fixed vector on ℍn. Furthermore, d satisfies the triangular inequality (see P. 320 in [25])(47)dp,q≤dp,x+dx,q,p,x,q∈ℍn.Forr>0 and x∈ℍn, the ball and sphere with center x and radius r on ℍn are given by(48)Bx,r=y∈ℍn:dx,y<r,Sx,r=y∈ℍn:dx,y=r,respectively. And, we have(49)Bx,r=B0,r=ΩQrQ,where(50)ΩQ=2πn+1/2Γn/2n+1ΓnΓn+1/2is the volume of the unit ball B0,1 on ℍn. The area of S0,1 on ℍn is ωQ=QΩQ [26] (for more details about Heisenberg group, see [24, 25]).LetLpℍn,⋅hα 1<p<∞ be the space of all measurable functions f on ℍn such that(51)fLpℍn,⋅hα=∫ℍnfxpxhαdx1/p<∞.The bounded mean oscillation spaceBMOℍn is defined to be the space of all locally integrable functions f on ℍn such that(52)fBMOℍn≔supB⊂ℍn1B∫Bfx−fBdx<∞,where the supremum is taken over all balls in ℍn and fB=1/B∫Bfxdx.Iff and g are measurable functions on ℍn, their convolution f∗g is defined by(53)f∗gx=∫ℍnfygy−1xdy=∫ℍnfxy−1gydy.LetSℍn denote the Schwartz class on the Heisenberg group ℍn. A measurable complex-valued function f on ℍn belongs to the Hardy space H1ℍn if and only if(54)sup0<t<∞f∗Ψt∈L1ℍn,with(55)fH1ℍn=sup0<t<∞f∗ΨtL1ℍn,where Ψ is some fixed function in Sℍn with ∫ℍnΨxdx≠0 and Ψt denotes the dilation Ψtx=t−QΨδt−1x,t>0.


## 3. Boundedness for Fractional Hausdorff Operators on the Lebesgue Spaces with Power Weights

In this section, we will give the proof of Theorem1–4.Proof of Theorem 1.
SinceΦ is radial, we can write(56)TΦ,βfx=∫ℍnΦxh/yhyhQ−βfydy.
Ifp>1, by Hölder’s inequality, we have(57)TΦ,βfx≤∫ℍnΦxhyhp′yh−p′Q−β+γ/pdy1/p′fLpℍn,⋅hγ=fLpℍn,⋅hγ∫0∞∫SΦxhrp′r−p′Q−β+γ/p+Q−1dσy′dr1/p′=fLpℍn,⋅hγωQ1/p′xh−Q/p+β−γ/p∫0∞Φtp′tp′Q−β+γ/p−Q−1dt1/p′,where 1/p+1/p′=1 and S=S0,1.
Ifp=1, we have(58)TΦ,βfx≤fL1ℍn,⋅hγ⋅h−Q+β−γΦxh⋅hL∞ℍn=fL1ℍn,⋅hγxh−Q+β−γ⋅hQ−β+γΦ⋅L∞ℍn.
LetM1=K1 and Mp=ωQ1/p′Kp1/p′, for any p≥1 and λ>0, we have(59)x∈ℍn:TΦ,βfx>λ⊂x∈ℍn:Mpxh−Q/p+β−γ/pfLpℍn,⋅hγ>λ=x∈ℍn:xhQ+γ/p−β<MpfLpℍn,⋅hγλ=x∈ℍn:xhQ+γ<MpfLpℍn,⋅hγλq.
Therefore,(60)λq∫x∈ℍn:TΦ,βfx>λxhγdx≤λq∫x∈ℍn:xhQ+γ<MpfLpℍn,⋅hγ/λqxhγdx=ωQQ+γMpfLpℍn,⋅hγq.
Consequently,(61)TΦ,βfLq,∞ℍn,⋅hγ≤ωQQ+γ1/qMpfLpℍn,⋅hγ.
This completes the proof.Proof of Theorem 2.
By changing to polar coordinates and the Fubini theorem, we have(62)TΦ,βfLqℍn,⋅hγq=∫ℍn∫ℍnΦδyh−1xyhQ−βfydyqxhγdx=∫ℍn∫0∞∫SΦδt−1xfty′tβ−1dσy′dtqxhγdx=∫0∞∫S∫0∞∫SΦδt−1xfty′tβ−1dσy′dtqργ+Q−1dσx′dρ≤∫S∫0∞∫S∫0∞Φδt−1ρx′fty′t−1ργ+Q/qtα+Q/pdttdσy′qdρρdσx′.
By Minkowski’s inequality, we obtain(63)∫0∞∫S∫0∞Φδt−1ρx′fty′t−1ργ+Q/qtα+Q/pdttdσy′qdρρ1/q≤∫S∫​0∞∫​0∞Φδt−1ρx′t−1ργ+Q/qfty′tα+Q/pdttqdρρ1/qdσy′.
Here,(64)∫0∞Φδt−1ρx′t−1ργ+Q/qfty′tα+Q/pdtt,can be regarded as a convolution in the multiplicative group ℝ+ with Haar measure dx/x. By Young’s inequality, we have(65)∫0∞∫0∞Φδt−1ρx′t−1ργ+Q/qfty′tα+Q/pdttqdρρ1/q≤∫0∞Φδρx′sργ+Qs/q−1dρ1/s∫0∞fρy′pρQ+α−1dρ1/p.
Therefore,(66)TΦ,βfLqℍn,⋅hγq≤∫S∫S∫0∞Φδρx′sργ+Qs/q−1dρ1/s×∫0∞fρy′pρQ+α−1dρ1/pdσy′qdσx′≤∫S∫0∞Φδρx′sργ+Qs/q−1dρq/sdσx′×∫S∫0∞fρy′pρQ+α−1dρ1/pdσy′q=ωQ−q/p′KΦ,Q,p,q,γq∫S∫0∞fρy′pρQ+α−1dρ1/pdσy′q≤KΦ,Q,p,q,γq∫S∫0∞fρy′pρQ+α−1dρdσy′q/p=KΦ,Q,p,q,γqfLpℍn,⋅hαq.
Theorem2 is proved.Proof of Theorem 3.
The sufficient part can be immediately deduced by choosingγ=α and p=q in Theorem 2.
Next, we will give the proof of necessary part. IfTΦ is a bounded operator on Lpℍn,⋅hα, then there exists a constant C=TΦLpℍn,⋅hα⟶Lpℍn,⋅hα>0 such that(67)TΦfLpℍn,⋅hα≤CfLpℍn,⋅hα.
For anyε>0, take(68)fεx=xh−α+Q+ε/pχxh>1x.
It is easy to check thatfε∈Lpℍn,⋅hα and(69)fεLpℍn,⋅hα=ωQ1/pε−1/p.
By changing variables, we have(70)TΦfεx=∫1∞∫SΦδr−1xr−α+Q+ε/p−1dσy′dr=ωQ∫1∞Φδr−1xhx′r−α+Q+ε/p−1dr=ωQxh−α+Q+ε/p∫0xhΦδρx′ρα+Q+ε/p−1dρ.
Thus,(71)TΦfεLpℍn,⋅hα=ωQ∫ℍnxh−Q+ε∫0xhΦδρx′ρα+Q+ε/p−1dρpdx1/p≥ωQ∫xh≥1/εxh−Q+ε∫01/εΦδρx′ρα+Q+ε/p−1dρpdx1/p=ωQ∫1/ε∞∫St−ε−1∫01/εΦδρx′ρα+Q+ε/p−1dρpdσx′dt1/p=ωQεε/pε−1/p∫S∫01/εΦδρx′ρα+Q+ε/p−1dρpdσx′1/p=ωQ1/p′fεLpℍn,⋅hαεε/p∫S∫01/εΦδρx′ρα+Q+ε/p−1dρpdσx′1/p.
Therefore,(72)ωQ1/p′∫S∫​01/εΦδρx′ρα+Q+ε/p−1dρpdσx′1/p≤Cεε/p.
Lettingε⟶0 on both sides, we get(73)KΦ,Q,p,α=ωQ1/p′∫S∫0∞Φδρx′ρα+Q/p−1dρpdσx′1/p≤C<∞.
When (17) holds, from Theorem 2, we can see that(74)TΦLpℍn,⋅hα⟶Lpℍn,⋅hα≤KΦ,Q,p,α.
On the contrary, by (73), we have(75)KΦ,Q,p,α≤TΦLpℍn,⋅hα⟶Lpℍn,⋅hα,which implies (18). Theorem 3 is proved.Proof of Theorem 4.
(i)
For anyf∈Lpℍn,⋅hα, by the Minkowski inequality, we have
(76)
T
˜
Φ
f
L
p
ℍ
n
,
⋅
h
α
=
∫
ℍ
n
∫
ℍ
n
Φ
y
y
h
Q
f
δ
y
h
−
1
x
d
y
p
x
h
α
d
x
1
/
p
≤
∫
ℍ
n
∫
ℍ
n
f
δ
y
h
−
1
x
p
x
h
α
d
x
1
/
p
Φ
y
y
h
Q
d
y
=
f
L
p
ℍ
n
,
⋅
h
α
∫
ℍ
n
Φ
y
y
h
−
Q
+
Q
+
α
/
p
d
y
.
Therefore, if (20) holds, then TΦ is bounded on Lpℍn,⋅hα and
(77)
T
˜
Φ
L
p
ℍ
n
,
⋅
h
α
⟶
L
p
ℍ
n
,
⋅
h
α
≤
∫
ℍ
n
Φ
y
y
h
−
Q
+
Q
+
α
/
p
d
y
.
On the contrary, supposeT˜Φ is bounded on Lpℍn,⋅hα, and denote M=T˜ΦLpℍn,⋅hα⟶Lpℍn,⋅hα. Like in Theorem 1.3, for any ε>0, we also take
(78)
f
ε
x
=
x
h
−
α
+
Q
+
ε
/
p
χ
x
h
>
1
x
.
Then,
(79)
T
˜
Φ
f
ε
x
=
x
h
−
α
+
Q
+
ε
/
p
∫
y
h
<
x
h
Φ
y
y
h
Q
y
h
α
+
Q
+
ε
/
p
d
y
.
Therefore,
(80)
T
˜
Φ
f
ε
L
p
ℍ
n
,
⋅
h
α
=
∫
ℍ
n
x
h
−
α
+
Q
+
ε
∫
y
h
<
x
h
Φ
y
y
h
Q
y
h
α
+
Q
+
ε
/
p
d
y
p
x
h
α
d
x
1
/
p
≥
∫
x
h
≥
1
/
ε
∫
y
h
<
1
/
ε
Φ
y
y
h
−
Q
+
α
+
Q
+
ε
/
p
d
y
p
x
h
−
Q
−
ε
d
x
1
/
p
=
∫
y
h
<
1
/
ε
Φ
y
y
h
−
Q
+
α
+
Q
+
ε
/
p
d
y
∫
x
h
≥
1
/
ε
x
h
−
Q
−
ε
d
x
1
/
p
=
ω
Q
1
/
p
ε
−
1
/
p
ε
ε
/
p
∫
y
h
<
1
/
ε
Φ
y
y
h
−
Q
+
α
+
Q
+
ε
/
p
d
y
.
The rest of the proof is the same as that of Theorem 3.
(ii)
For eacht>0 and ball Bx0,r⊂ℍn, let tBx0,r be the ball Bδtx0,tr, then tBx0,r=tQBx0,r.
Suppose (22) holds. Let f∈BMOℍn, and let B be a ball. Then, by the Fubini theorem, we have(81)T˜ΦfB=1B∫B∫ℍnΦyyhQfδyh−1xdydx=∫ℍn1B∫Bfδyh−1xdxΦyyhQdy=∫ℍn1yh−1B∫yh−1BfydyΦyyhQdy=∫ℍnΦyyhQfyh−1Bdy.
Then, using the Fubini theorem again, we get(82)1B∫BT˜Φfx−T˜ΦfBdx=1B∫B∫ℍnΦyyhQfδyh−1xdy−∫ℍnΦyyhQfyh−1Bdydx≤∫ℍn1B∫Bfδyh−1x−fyh−1BdxΦyyhQdy=∫ℍn1yh−1B∫yh−1Bfy−fyh−1BdyΦyyhQdy≤fBMOℍn∫ℍnΦyyhQdy,which implies that T˜Φ is bounded on BMOℍn.
Conversely, ifT˜Φ is bounded on BMOℍn. Choose(83)f0x=1,x2n+1>0,0,x2n+1=0,−1,x2n+1<0.
Then,f0∈BMOℍn with f0BMOℍn≠0. And,(84)T˜Φf0x=f0x∫ℍnΦyyhQdy.
Consequently,(85)∫ℍnΦyyhQdy≤T˜ΦBMOℍn⟶BMOℍn.
Moreover, when (22) holds, then (83) and (85) imply that(86)T˜ΦBMOℍn⟶BMOℍn=∫ℍnΦyyhQdy.
This completes the proof.



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*Source: 5247420-2020-01-04.xml*