RIESZ MEANS AND BILINEAR RIESZ MEANS ON M ETIVIER GROUPS MIN WANG1AND HUA ZHU1 Abstract. In this paper we investigate the Lp-boundedness of the Riesz means and the Lp1

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RIESZ MEANS AND BILINEAR RIESZ MEANS ON M´
ETIVIER GROUPS
MIN WANG1AND HUA ZHU1,∗
Abstract. In this paper, we investigate the Lp-boundedness of the Riesz means and the Lp1×
Lp2→Lpboundedness of the bilinear Riesz means on M´etivier groups. M´etivier groups are
generalization of Heisenberg groups and general H-type groups. Because general M´etivier groups
only satisfy the non-degeneracy condition and have high-dimensional centre, we have to use different
methods and techniques from those on Heisenberg groups and H-type groups.
1. Introduction
M´etivier groups, introduced by M´etivier [9] in his study of analytic hypoellipticity, are two-step
nilpotent Lie groups satisfying a non-degeneracy condition. They are also characterized by the
property that the quotients with respect to the hyperplanes contained in the centre are general
Heisenberg groups. H-type groups, introduced by Kaplan [5], are typical examples of M´etivier
groups, but there are many M´etivier groups which are not isomorphic to H-type groups, for exam-
ple, see [12]. Heisenberg groups are the simplest M´etivier groups. Casarino and Ciatti [2] investi-
gated the spectral resolution of the sub-Laplacian on M´etivier groups, and proved a Stein-Tomas
restriction theorem in terms of the mixed norms. So, we can use the the spectral decomposition
of the sub-Laplacian to define the Riesz means and the bilinear Riesz means on M´etivier groups.
Mauceri [8] and M¨uller [11] obtained the same results on the Lp-boundedness of the Riesz means
on Heisenberg groups by using different methods. We proved the Lp1×Lp2→Lpboundedness
of the bilinear Riesz means on Heisenberg groups [6]. To extend these results to general M´etivier
groups, we notice that there are two essential difficulties: one is that Heisenberg groups have one
dimensional centre but the centre dimension of M´etivier groups is in general bigger than one; the
other is that on H-type groups, the non-degeneracy condition becomes a better orthogonality con-
dition, which is not true on general M´etivier groups. Therefore, in this paper, we shall introduce
some different techniques from those used in [8] or [11] and [6] to obtain the Lp- boundedness of
the Riesz means and Lp1×Lp2→Lpboundedness of the bilinear Riesz means on M´etivier groups.
This paper is organized as follows. In Section 2, we recall the spectral decomposition of the
sub-Laplacian on Metivier groups and define the Riesz means and bilinear Riesz means. In Section
3, we give the pointwise estimates for the kernel of the Riesz means and the kerner of the bilinear
Riesz means. In Section 4, we present the Lp-boundedness of the Riesz means. In the rest of this
paper, we study the Lp1×Lp2→Lpboundedness of the bilinear Riesz means, which for the case
of 1 ≤p1, p2≤2 in Section 5 and for some particular cases in Section 6. In Section 7, we outline
the bilinear interpolation method and obtain the results in other cases.
2010 Mathematics Subject Classification. Primary 43A80, 43A85, 43A45.
Key words and phrases. M´etivier groups, Riesz means, bilinear Riesz means, restriction theorem.
1 These two authors contributed to this work equally and should be regarded as co-first authors.
∗This author is supported by “Young top-notch talent” Program concerning teaching faculty development of
universities affiliated to Beijing Municipal Government(2018-2020).
1
arXiv:2210.14673v1 [math.FA] 26 Oct 2022
2 MIN WANG1AND HUA ZHU1,∗
2. Preliminaries
We first recall M´etivier groups. Let gbe a real finite dimensional two step nilpotent Lie algebra,
equipped with an inner product h·,·i. Then
g=g1⊕g2,
where [g1,g1] = g2, [g1,g2]=[g2,g2] = 0 and dim g1=d, dim g2=m. Let {X1, X2,··· , Xd}be an
orthonormal basis of g1,{U1, U2,··· , Um}be an orthonormal basis of g2.
Define for µ∈g∗
2the skew symmetric form ωµon g1by
ωµ(X, Y ) = µ([X, Y ])
and the corrsponding metrix
(Jµ)jk =ωµ(Xj, Xk).
We say that the corresponding connected, simply connected Lie group Gof gis a M´etivier group (M-
type group) if ωµis non-degenerate for all µ6= 0. Under this non-degenerate hypothesis, Jµis a d×d
skew symmetric, invertible metrix for all µ6= 0. It follows that d= 2nis even. Since the exponential
mapping is a bijection, we shall parametrise the element g= exp P2n
i=1 xiXi+Pm
j=1 ujUj∈Gby
(x, u) where x= (x1,··· , x2n)∈R2n,u= (u1,··· , um)∈Rm. By the Baker-Campbell-Hausdorff
formula, the group multiplication is given by
(x, u)·(y, v) = (x+y, u +v+1
2[x, y]),
where [x, y] = (hx, J1yi,hx, J2yi,··· ,hx, Jmyi)∈Rm,Jkis 2n×2nskew-symmetric, non-degenerate
metrix for any k= 1,··· , m. The identity element of Gis (0,0) and the inversion of (x, u) is denoted
by (x, u)−1= (−x, −u).
A homogeneous structure on Gis obtained by defining the dilations δt(x, u)=(tx, t2u), t > 0.
The homogeneous dimension of Gis
Q= 2n+ 2m.
The Haar measure on Gcoincides with the Lebesgue measure on gdenoted by dxdu. It is easy to
verify that the Jacobian determinant of the dilations δt,t > 0 is constant, equal to tQ. Define a
homogeneous norm of degree one under the dilations {δt,t > 0}on Gby
|ω|=|(x, u)|=1
16 |x|4+|u|21
4, ω = (x, u)∈G.
This norm satisfies the triangle inequality |ω·ω0|≤|ω|+|ω0|and leads to a left-invariant distance
d(ω, ω0) = ω−1·ω0.
For any f, g ∈L1(G), their convolution is defined by
f∗g(x, u) = ZG
f((x, u)·(y, v)−1)g(y, v)dydv.
The µ-twisted convolution of two suitable functions or distributions on g1is defined by
φ×µψ(x) = Zg1
φ(x−y)ψ(y)ei
2µ([x,y])dµy,
where dµy=pdet Jµdy. Given f∈L1(G), we define the Fourier transform with respect to the
central variables by
fµ(x) = Zg2
f(x, u)eiµ(u)du.
RIESZ MEANS AND BILINEAR RIESZ MEANS ON M´
ETIVIER GROUPS 3
It is easy to verify that
(f∗g)µ=fµ×µgµ.
For any X∈g1, we obtain a left-invariant vector field Xon Gdefined by
Xf(g) = d
dtf(g·(tX))|t=0, f∈C∞(G), g∈G.
Then, the orthonormal basis of g1,{X1, X2,··· , X2n}, is associated to the left-variant vector field:
Xj=∂
∂xj
+1
2
m
X
k=1 hx, Jkeji∂
∂uk
, ej= (0,··· ,1j,0,··· ,02n)T, j = 1,2,··· ,2n.
The sub-Laplacian on Gis the left-invariant hypoelliptic operator defined by
L=−
2n
X
j=1
X2
j=−∆x−
m
X
k=1 xTJk,∇x∂
∂uk−1
4
n
X
k,l=1 xTJk, xTJl∂2
∂up∂uq
,
where
∆x=
2n
X
j=1
∂2
∂x2
j
,∇x=∂
∂x1
,∂
∂x2
,··· ,∂
∂x2nT
.
For any f∈S(G) and j= 1,··· , n, the Fourier transform of Xjfwith respect ot the central
variables is given by
(Xjf)µ(x) = Zg2
∂f
∂xj
(x, u)eiµ(u)du +1
2
m
X
k=1 hx, JkejiZg2
∂f
∂uk
(x, u)eiµ(u)du
=∂
∂xj−i
2hx, Jµejifµ(x).
Setting
Xµ
j=∂
∂xj−i
2hx, Jµeji,
we have that
(Xjf)µ=Xµ
jfµ.
Define
∆µ=−
2n
X
j=1 Xµ
j2.
It follows that
(Lf)µ= ∆µfµ.
Next, we let Sm−1={η∈g?
2:|η|= 1}denote the unit sphere of Rm∼
=g?
2. Since that for any
fixed η∈Sm−1, the corresponding metrix Jηis skew symmetric and non-degenerate, then there
exists a 2n×2ninvertible matrix Aηsuch that
Jη=AT
ηJ2nAη
where J2n=0nIn
−In0. Clearly, (det Aη)2= det Jη.Since det Jη, which is a polynomial function
of the components of η, never vanishes on Sm−1, then there exists a positive constant Ksuch that
for any η∈Sm−1
(2.1) 1
K≤ |det Aη|2=|det Jη| ≤ K.
4 MIN WANG1AND HUA ZHU1,∗
Let (Z1,··· , Z2n)=(X1,··· , X2n)A−1
η.{Z1,··· , Z2n}is an orthonormal basis of g1such that the
corresponding metrix of ωηis J2n, i.e.
ωη(Zj, Zk)=(J2n)jk.
The new coordinates of the element in g1in term of the basis {Z1,··· , Z2n}is z=Aηx. For any
λ > 0, we have that
(Zjf)λη (z) = ∂
∂zj−iλ
2hz, J2nejifλη (z).
Set
Zλη
j=∂
∂zj−iλ
2hz, J2neji.
More explicitly,
Zλη
j=∂
∂zj
+iλ
2zn+j,Zλη
j+n=∂
∂zj+n−iλ
2zj,j= 1,··· , n.
It follows that
Lλη =−
2n
X
j=1 Zλη
j2=−
2n
X
j=1
∂2
∂z2
j−iλ
n
X
j=1 zn+j
∂
∂zj−zj
∂
∂zn+j+λ2
4
2n
X
j=1
z2
j
=−∆z−iλ
n
X
j=1 zn+j
∂
∂zj−zj
∂
∂zn+j+λ2
4|z|2,
and
(Lf)λη =Lληfλη.
Notice that Lλη coincides with the the λ-scaled special Hermite operator Lλon g1. The λ-twisted
convolution of f, g ∈S(g1) in terms of the coordinates zis given by
φ×λψ(z) = φ×λη ψ(z)
=Zg1
φ(z−w)ψ(w)ei
2λη([z,w])dw
=Zg1
φ(z−w)ψ(w)ei
2λPn
j=1 zn+jwj−zjwn+jdw.
Next, we give the spectral decomposition of the sub-Laplacian Lon G. We have to recall some
facts about the special Hermite expansion. The Hermite polynomials Hkon Ris defined by
Hk(t)=(−1)kdk
dxke−t2et2, k = 0,1,2,··· .
Let
hk(t) = 2kk!√π−1
2Hk(t)e−1
2t2, k = 0,1,2,··· .
For any multiindex α, the Hermite functions Φαon Rnis defined by
Φα(y) =
n
Y
j=1
hαj(yj), y = (y1,···yn)∈Rn.
For each pair of multiindices α,β, the special Hermite function Φαβ on Cnis given by
Φαβ(z) = (2π)−n
2ZRn
e2πix·ξΦαξ+y
2Φβξ−y
2dξ, z =x+iy ∈Cn.
RIESZ MEANS AND BILINEAR RIESZ MEANS ON M´
ETIVIER GROUPS 5
The special Hermite functions form a complete orthonormal system for L2(Cn). Since that g1'Cn,
then for any g∈L2(g1) we have the special Hermite expansion
g(z) = X
α∈NnX
β∈Nnhf, Φαβ iΦαβ(z).
We define the Laguerre functions on Cnby
ϕk(z) = Ln−1
k1
2|z|2e−1
4|z|2, k = 0,1,2,··· ,
where
Lν
k(x) = 1
k!
dk
dxke−xxk+νexx−ν, k = 0,1,2,··· , ν > −1,
is the Laguerre polynomial on Rof type νand degree k. It follows that
X
|α|=k
Φαα(z) = (2π)−n
2ϕk(z),
and the special Hermite expansion can be written as
g(z) = (2π)−n
∞
X
k=0
g×λ=1 ϕk(z).
Set ϕλ
k(z) = ϕk(√λz) for any λ > 0. By a dilation argument, we have the expansion
(2.2) g(z) = λ
2πn∞
X
k=0
g×λϕλ
k(z),
where g×λϕλ
kis the eigenfunction of the operator Lλwith the eigenvalue λ(2k+n).
Let gη=g◦A−1
η. (2.2) yields that
(2.3) g(x) = gη(Aηx) = gη(z) = λ
2πn∞
X
k=0
gη×λϕλ
k(z) = λ
2πn∞
X
k=0 gη×λϕλ
k◦Aη(x).
So, we can get the following results:
Proposition 1. [2] Let η∈Sm−1and λ∈R.For any g∈S(g1),Πλη
kg=gη×λϕλ
k◦
Aηis the eigenfunction of the operator ∆λη with the eigenvalue λ(2k+n˙
), and e−iλη(u)Πλη
kg=
e−iλη(u)gη×λϕλ
k◦Aηis the eigenfunction of the sub-Laplacian Lon Gwith the eigenvalue
λ(2k+n).
Proof. Since that (Xλη
1,··· , Xλη
2n)=(Zλη
1,··· , Zλη
2n)Aη, then
∆λη (g◦Aη) = Lλg◦Aη.
Notice that Πλη
kg=gη×λϕλ
kis the eigenfunction of Lλwith the eigenvalue λ(2k+n). We have
∆λη gη×λϕλ
k◦Aη=Lλη(gη×λϕλ
k)◦Aη=λ(2k+n)(gη×λϕλ
k)◦Aη.
This implies (gη×λϕλ
k)◦Aηis the eigenfunction of the operator ∆λη with the eigenvalue λ(2k+n).
Since that for any f∈S(G), (Lf)λη = ∆ληfλη, then we have that
Le−iλη(u)Πλη
kgλη = ∆λη e−iλη(u)Πλη
kgλη =e−iλη(u)λη ∆λη Πλη
kg
摘要：

RIESZMEANSANDBILINEARRIESZMEANSONMETIVIERGROUPSMINWANG1ANDHUAZHU1;Abstract.Inthispaper,weinvestigatetheLp-boundednessoftheRieszmeansandtheLp1Lp2!LpboundednessofthebilinearRieszmeansonMetiviergroups.MetiviergroupsaregeneralizationofHeisenberggroupsandgeneralH-typegroups.BecausegeneralMetiviergr...

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