Bilinear Riesz means on the Heisenberg group
Heping Liu Min Wang
math.FA 43A80 (Primary), 22E30, 42B15, 15A15 (Secondary)

Bilinear Riesz means on the Heisenberg group]Bilinear Riesz means on
the Heisenberg group

School of Mathematical Sciences, Peking University, Beijing, 100871,
P. R. China, 
hpliu@math.pku.edu.cn

School of Mathematical Sciences, Peking University, Beijing, 100871,
P. R. China, 
wangmin09150102@163.com

[2010]Primary 43A80; Secondary 22E30, 42B15, 15A15



In this article, we investigate the bilinear Riesz means S^α
associated to the sublaplacian on the Heisenberg group. We prove that the
operator S^α is bounded from L^p_1× L^p_2 into L^p for 1≤ p_1, p_2≤∞ and 1/p=1/p_1+1/p_2 when α is large than a suitable smoothness index α (p_1,p_2).
There are some essential differences between the Euclidean space and the
Heisenberg group for studying the bilinear Riesz means problem. We make use
of some special techniques to obtain a lower index α (p_1,p_2).


[
    Min Wang
    December 10, 2017
=====================







§ INTRODUCTION


The bilinear Bochner-Riesz means problem originates from the study of the
summability of the product of two n-dimensional Fourier series. This leads
to the study of the L^p_1× L^p_2→ L^p boundedness
of the bilinear Bochner-Riesz multiplier

    B^α(f,g)(x)=∫∫_|ξ| ^2+|η| ^2≤ 1 (1-|ξ| ^2-|η| ^2)^αf(ξ ) g(η )e^2π
    ix· (ξ +η )dξ dη .

Here x∈ℝ^n, f, g are functions on ℝ^n and f, g are their Fourier transforms. Bernicot et al.
<cit.> gave a comprehensive study on the L^p_1×
L^p_2→ L^p boundedness of the operator B^α.
Inspired by their work, we investigate the corresponding problem on the
Heisenberg group.

Strichartz <cit.> developed the harmonic analysis on the
Heisenberg group as the spectral theory of the sublaplacian. We can define
the bilinear Riesz means in terms of the spectral decomposition of the
sublaplacian. Let

    ℒf=∫_0^∞λ P_λf dμ (λ ).

be the spectral decomposition of the sublaplacian ℒ. The
bilinear Riesz means associated to the sublaplacian ℒ is defined
by

    S^α(f,g)=∫_0^∞∫_0^∞( 1- λ
    _1-λ _2) _+^αP_λ _1fP_λ
    _2g dμ (λ _1)dμ (λ _2).

As same as the Euclidean case, we hope to obtain a lower smoothness index α (p_1,p_2) such that the operator S^α is bounded from L^p_1× L^p_2 into L^p for 1≤ p_1, p_2≤∞
and 1/p=1/p_1+1/p_2 when α > α (p_1,p_2).

There are some essential differences between the Euclidean space and the
Heisenberg group for studying the bilinear Riesz means problem. Firstly, the
kernel of the bilinear Bochner-Riesz operator B^α on ℝ
^n coincides with the kernel of the Bochner-Riesz operator on ℝ
^2n. The pointwise estimate of this kernel is well known and gives a
basic result: B^α is bounded from L^p_1× L^p_2 into L^p when α > n- 1/2, which is optimal in case of (p_1,
p_2, p)= (1,1, 1/2). It is not the
case for the Heisenberg group because the product of two Heisenberg groups
is not a Heisenberg group. We will give a pointwise estimate for the kernel
of the bilinear Riesz means S^α, which is similar to the known
estimate for the kernel of the Riesz means on the Heisenberg group but very
worse than the corresponding estimate for bilinear Bochner-Riesz operator B^α. Such a pointwise estimate only gives a very rough result for
smoothness index α (p_1,p_2). We don't know if there exist a
better pointwise estimate even for the Riesz means on the Heisenberg group
(cf. <cit.> or <cit.>). So we have to develop a new technique to
obtain a better result about index α (p_1,p_2), for example, in
case of (p_1, p_2, p)= (∞, ∞, ∞). Secondly, on Euclidean
space, the Fourier transform of the product of two functions is the
convolution of Fourier transforms of two functions because the dual of the
Euclidean space is itself. As a consequence, the L^2× L^2→ L^2 boundedness holds for suitable bilinear Fourier
multiplier, which play an important role for giving the estimate in case of (p_1, p_2, p)= (2, ∞, 2). But this convenience false on the Heisenberg
group. Finally, the restriction estimate for the sublaplacian is very
different from that for the Laplacian on the Euclidean space because the
Heisenberg group has the center of dimension one. Fefferman <cit.>
pointed out that the restriction theorem apply to the study of the
boundedness of Bochner-Riesz means. Stein's earlier result in <cit.>
was improved by using the restriction theorem. Mauceri <cit.>
investigated the L^p boundedness of Riesz means on the Heisenberg group.
Mauceri's result corresponds with Stein's earlier result. Müller <cit.> construct a counter-example to show that the usual norm estimate of
restriction operators holds only in the trivial case p=1. Müller <cit.> gave a new proof of Mauceri's result by using a revised restriction
estimate but didn't improve Mauceri's result. As a result of above reasons,
our techniques are very different from that in <cit.>.

This article is organized as follows. In the next section, we state some
basic facts about the Heisenberg group, and summarize our full results in
the main theorem. In Section 3, we give the pointwise estimate for the
kernel of S^α. We prove the L^p_1× L^p_2→
L^p boundedness of S^α in the case of 1 ≤ p_1, p_2≤
2 in Section 4 and for some particular triples of points (p_1, p_2, p)
in Section 5. The boundedness in other cases are derived from the results in
Section 4 and Section 5 by using of bilinear interpolation method. We
outline this argument in Appendix for reader's convenience.

0.5 cm



§ PRELIMINARIES


First we recall some basic facts about the Heisenberg group. These facts are
familiar and easy to find in many references. Let ℍ^n denote
the Heisenberg group whose underlying manifold is ℂ^n×ℝ and the group law is given by

    (z,t)(w,s)=(z+w,t+s+1/2Im(z·w)).

The Haar measure on ℍ^n coincides with the Lebesgue measure on ℂ^n×ℝ. A homogeneous structure on ℍ^n
is given by the non-isotropic dilations δ_r(z,t)=(rz,r^2t). We
define a homogeneous norm on ℍ^n by

    | x | = ( 1/16| z| ^4+
    t^2)^1/4,    x =(z,t)∈ℍ^n.

This norm satisfies the triangle inequality and leads to a left-invariant
distance d(x,y) = | x^-1y |. The ball of radius r
centered at x is

    B(x,r)= { y∈ℍ^n: | x^-1y | <r }.

The Haar measure dx satisfies dδ _r(x)=r^Qdx where Q=2n+2 is
the homogeneous dimension of ℍ^n. If f and g are functions
on ℍ^n, their convolution is defined by

    (f ∗ g)(x)= ∫_ℍ^n f( xy^-1)g(y)  dy,  
    x,y ∈ℍ^n.

For each λ∈ℝ^∗ and f∈𝒮(ℍ^n), the
inverse Fourier transform of f in variable t is defined by

    f^λ(z)=∫_-∞^∞e^iλ tf(z,t)  dt.

An easy calculation shows that

    (f∗ g)^λ(z)=∫_ℂ^nf^λ(z-ω
    )g^λ (ω )e^i/2λIm(z·ω)  dω ,   z,ω∈ℂ^n.

Thus, we are led to the convolution of the form

    f∗ _λg=∫_ℂ^nf(z-ω )g(ω ) e^i/2λIm(z·ω)  dω ,

which are called the λ-twisted convolution.

The sublaplacian ℒ is defined by

    ℒ= -∑_j=1^n(X_j^2+Y_j^2)

where

    X_j   =   ∂/∂ x_j+1/2y_j∂/∂ t,    j=1,2,⋯ ,n, 
    
    Y_j   =   ∂/∂ y_j-1/2x_j∂/∂ t,    j=1,2,⋯ ,n,

are left invariant vector fields on ℍ^n. Up to a constant
multiple, ℒ is the unique left invariant, rotation invariant
differential operator that is homogeneous of degree two. Therefore, it is
regarded as the counterpart of the Laplacian on ℝ^n. The
sublaplacian ℒ is a positive and essentially self-adjoint
operator. In the following, we state the spectral decomposition of ℒ (cf. <cit.>).

Let φ _k be the Laguerre functions on ℂ^n given by

    φ _k(z)=L_k^n-1(1/2| z| ^2)
    e^-1/4| z| ^2,

where L_k^n-1 are the Laguerre polynomials of type n-1 defined on ℝ by

    L_k^n-1(t)e^-tt^n-1=1/k!( d/dt)
    ^k(e^-tt^k+n-1).

Define functions

    e_k^λ(z,t)=e^-iλ tφ _k^λ(z)=e^-iλ
    tφ _k(√(|λ|)z),  λ∈ℝ^∗.

For f∈ L^2(ℍ^n), we have the expansion

    f(z,t)=∑_k=0^∞∫_-∞^∞f∗ e_k^λ(z,t) dμ (λ )

where dμ (λ )=(2π )^-n-1|λ|
^ndλ is the Plancherel measure for ℍ^n. Each f∗
e_k^λ is the eigenfunction of ℒ with eigenvalue (2k+n)|λ|. We also have the Plancherel formula

    ‖ f‖ _2^2=(2π )^-2n-1∑_k=0^∞∫_-∞^∞∫_ℂ^n| f^λ∗
    _λφ _k^λ(z)| ^2λ
    ^2n dzdλ .

Defining

    e_k^λ(z,t)=e_k^λ/2k+n(z,t),

we can rewrite the decomposition (<ref>) as

    f(z,t)=∫_-∞^∞∑_k=0^∞(2k+n)^-n-1f∗e_k^λ(z,t) dμ (λ ).

Let

    P_λf(z,t)=∑_k=0^∞(2k+n)^-n-1f∗ (e
    _k^λ+e_k^-λ)(z,t).

Then (<ref>) can be written as

    f(z,t)=∫_0^∞P_λf(z,t) dμ (λ ).

It is clear that P_λf is an eigenfunction of the ℒ
with eigenvalue λ and we have the spectral decomposition

    ℒf=∫_0^∞λ P_λf dμ (λ ).


Now we define the bilinear Riesz means associated to the sublaplacian ℒ for f,g∈𝒮(ℍ^n) by

    S_R^α(f,g)=∫_0^∞∫_0^∞( 1-λ _1+λ _2/R) _+^αP_λ
    _1fP_λ _2g dμ (λ _1)dμ (λ _2).

It is easy to see that

    S_R^α(f,g)(x)=∫_ℍ^n∫_ℍ^nf(xω
    _1^-1)g(xω _2^-1)S_R^α(ω _1,ω
    _2) dω _1dω _2,

where the kernel is given by

    S_R^α((z_1,t_1),(z_2,t_2))=∑_k=0^∞∑_l=0^∞∫_-∞^∞∫_-∞^∞( 1-(2k+n)|λ _1| +(2l+n)|λ _2|/R) _+^α
                                              ×
    e_k^λ _1(z_1,t_1)e_l^λ _2(z_2,t_2) dμ
    (λ _1)dμ (λ _2).

Because

    S_R^α((z_1,t_1),(z_2,t_2))=R^QS_1^α((√(R)z_1,Rt_1),(√(R)z_2,Rt_2)).

By a dilation argument, the L^p_1× L^p_2→ L^p
boundedness of S_R^α is deduced from the L^p_1×
L^p_2→ L^p boundedness of S_1^α as 1/p=1/p_1+1/p_2. We will concentrate on the operator S_1^α
and write S^α instead of S_1^α.

Our full results are summarized in the following theorem.

Main Theorem. Let 1≤ p_1,p_2≤∞ and 1/p=1/p_1+1/p_2. 

(1) (region I) For 2≤ p_1,p_2≤∞ and p≥ 2, if
α >Q( 1-1/p) - 1/2, then S^α is
bounded from L^p_1(ℍ^n)× L^p_2 (ℍ^n) to
L^p(ℍ^n). 

(2) (region II) For 2≤ p_1,p_2≤∞ and 1≤ p≤
2, if α >(Q-1)( 1-1/p), then S^α is
bounded from L^p_1(ℍ^n)× L^p_2(ℍ^n) to L^p(ℍ^n). 

(3) (region III) For 1≤ p_1≤ 2≤ p_2≤∞ and p≥ 1, if α >Q( 1/2-1/p_2) - ( 1-
1/p), then S^α is bounded from L^p_1(ℍ
^n)× L^p_2(ℍ^n) to L^p(ℍ^n); For 1≤ p_2≤ 2≤ p_1≤∞ and p≥ 1, if α >Q(
1/2-1/p_1) - ( 1-1/p), then S^α is bounded from L^p_1 (ℍ^n)× L^p_2(
ℍ^n) to L^p(ℍ^n). 

(4) (region IV) For 1≤ p_1≤ 2≤ p_2≤∞ and p≤ 1 , if α >Q( 1/p_1-1/2), then S^α is bounded from L^p_1(ℍ^n)× L^p_2(
ℍ ^n) to L^p(ℍ^n); For 1≤ p_2≤ 2≤
p_1≤∞ and p≤ 1, if α >Q( 1/p_2-1
/2), then S^α is bounded from L^p_1(ℍ
^n)× L^p_2(ℍ^n) to L^p(ℍ^n). 

(5) (region V) For 1≤ p_1, p_2≤ 2, if α >Q(
1 /p-1), then S^α is bounded from L^p_1(
ℍ ^n)× L^p_2(ℍ^n) to L^p(ℍ^n).



    < g r a p h i c s >




§ POINTWISE ESTIMATE FOR THE KERNEL


Note that S^α((z_1,t_1),(z_2,t_2)) is bi-radial
with respect to z_1 and z_2, which means S^α(
(z_1,t_1),(z_2,t_2)) depends only on |
z_1| ,| z_2| ,t_1,t_2. If F(
(z_1,t_1),(z_2,t_2)) is bi-radial with respect to z_1 and z_2, we also write it as F((r_1,t_1),(r_2,t_2)) for
convenience where r_1=| z_1|, r_2=|
z_2|.

Suppose F((z_1,t_1),(z_2,t_2)) is bi-radial with respect
to z_1 and z_2, we define

    R_k,l(λ _1,λ_2,F)=2^(1-n)k!/(k+n-1)!
    2^(1-n)l!/(l+n-1)!
                          ×∫_0^∞∫_0^∞
    F^λ_1,λ _2(r_1,r_2) φ _k^λ
    _1(r_1)φ_l^λ_2(r_2)r_1^2n-1r_2^2n-1 
    dr_1dr_2,

where

    F^λ _1,λ _2(r_1,r_2)=∫_-∞^∞∫_-∞^∞ e^iλ_1t_1 e^iλ_2t_2 F(
    (r_1,t_1),(r_2,t_2))  dt_1dt_2.

Taking use of the Laguerre transform, we have

    F((r_1,t_1),(r_2,t_2))=∑_k=0^∞∑_l=0^∞∫_-∞^∞∫_-∞^∞e^-i( λ_1t_1+λ_2t_2)
    R_k,l(λ_1,λ_2,F) φ_k^λ_1(r_1) φ
    _l^λ_2(r_2)  dμ (λ _1)dμ (λ _2).

Especially, if

    ∫_-∞^∞∫_-∞^∞(∑_k=0^∞∑_l=0^∞| R_k,l(λ
    _1,λ_2,F)|(k+n-1)!/k!(l+n-1)!/l!) 
    dμ (λ _1)dμ (λ _2)< ∞,

then F is bounded, because

    ‖φ _k‖ _∞= c_n(k+n-1)!/k!.



If α >4m-1 where m is a positive integer, then for
any ω _1=(z_1,t_1), ω _2=(z_2,t_2) ∈ℍ
^n.

    | S^α(ω _1,ω _2)|≤
    C(1+|ω _1| )^-2m(1+|ω_2| )^-2m.




  Set F((z_1,t_1),(z_2,t_2))=S^α(
(z_1,t_1),(z_2,t_2)). If we can show that

    ∫_-∞^∞∫_-∞^∞(∑_k=0^∞∑_l=0^∞| R_k,l(λ _1,λ
    _2,(it_1-1/4| z_1| ^2)^m(it_2-1
    /4| z_2| ^2)^mF)|
                          ×(k+n-1)!/k!(l+n-1)!/l!) dμ (λ _1)dμ (λ _2)<∞ ,

then (it_1-1/4| z_1| ^2)^m(it_2-
1 /4| z_2| ^2)^mS^α(
(z_1,t_1),(z_2,t_2)) is bounded. It follows that

    | S^α(ω _1,ω _2)|≤
    C(1+|ω _1| )^-2m(1+|ω
    _2| )^-2m

and Theorem <ref> is proved.

It is clear that R_k,l(λ _1,λ _2,F)=(
1-(2k+n)|λ _1| -(2l+n)|λ
_2|)_+^α. We first calculate R_k,l(
λ _1,λ _2,(it_1-1/4| z_1|
^2)(it_2-1/4| z_2| ^2)F)
and assume that λ _1,λ _2>0. From (<ref>), we have

    R_k,l(λ _1,λ _2,it_1· it_2F)=2^(1-n)k!/
    (k+n-1)!2^(1-n)l!/(l+n-1)!
                      ×∫_0^∞∫_0^∞(
    it_1· it_2F) ^λ _1,λ _2(r_1,r_2)φ
    _k^λ _1(r_1)φ _l^λ
    _2(r_2)r_1^2n-1r_2^2n-1 dr_1dr_2.

Since that

    ( it_1· it_2F) ^λ _1,λ _2(r_1,r_2)=
    ∂/∂λ _1∂/∂λ _2F^λ _1,λ _2(r_1,r_2),

we get

    R_k,l(λ _1,λ _2,it_1· it_2F) 
       =   ∂/∂λ _1∂/∂λ
    _2R_k,l(λ _1,λ _2,F) 
          -2^(1-n)k!/(k+n-1)!2^(1-n)l!/(l+n-1)!∫_0^∞∫_0^∞∂/∂λ _1F^λ
    _1,λ _2(r_1,r_2)φ _k^λ _1(r_1)(
    ∂/∂λ _2φ _l^λ
    _2(r_2)) r_1^2n-1r_2^2n-1 dr_1dr_2
          -2^(1-n)k!/(k+n-1)!2^(1-n)l!/(l+n-1)!∫_0^∞∫_0^∞∂/∂λ _2F^λ
    _1,λ _2(r_1,r_2)( ∂/∂λ _1φ _k^λ _1(r_1)) φ _l^λ
    _2(r_2)r_1^2n-1r_2^2n-1 dr_1dr_2
          -2^(1-n)k!/(k+n-1)!2^(1-n)l!/(l+n-1)!∫_0^∞∫_0^∞F^λ _1,λ _2(r_1,r_2)( ∂/∂λ _1φ _k^λ _1(r_1))
    ( ∂/∂λ _2φ _l^λ
    _2(r_2)) r_1^2n-1r_2^2n-1 dr_1dr_2.

Noticing the fact (see <cit.>, p. 92)

    ∂/∂λφ _k^λ(r)=k/λφ _k^λ(r)-(k+n-1)/λφ
    _k-1^λ(r)-1/4r^2φ _k^λ(r),

it then follows that

    R_k,l(λ _1,λ _2,it_1· it_2F) 
       =   ∂/∂λ _1∂/∂λ
    _2R_k,l(λ _1,λ _2,F)-R_k,l( λ _1,λ
    _2,1/4| z_1| ^2| z_2|
    ^2) 
          +l/λ _2∂/∂λ _1(
    R_k,l-1(λ _1,λ _2,F)-R_k,l(λ _1,λ
    _2,F)) 
          +k/λ _1∂/∂λ _2(
    R_k-1,l(λ _1,λ _2,F)-R_k,l(λ _1,λ
    _2,F)) 
          +k/λ _1l/λ _2( R_k-1,l(λ
    _1,λ _2,F)+R_k,l-1(λ _1,λ _2,F)-R_k,l(λ
    _1,λ _2,F)-R_k-1,l-1(λ _1,λ _2,F)) 
          +∂/∂λ _1R_k,l(λ _1,λ _2,
    1/4| z_2| ^2F)+∂/∂λ _2R_k,l(λ _1,λ _2,1/4|
    z_1| ^2F) 
          +k/λ _1( R_k,l(λ _1,λ _2,1/4| z_2| ^2F)-R_k-1,l(λ _1,λ _2,
    1/4| z_2| ^2F)) 
          +l/λ _2( R_k,l(λ _1,λ _2,1/4| z_1| ^2F)-R_k,l-1(λ _1,λ _2,
    1/4| z_1| ^2F)) .

Similarly, since that

    ( -it_11/4| z_2| ^2F) ^λ
    _1,λ _2(r_1,r_2)    =   -∂/∂λ _1( 1/4| z_2| ^2F) ^λ
    _1,λ _2(r_1,r_2), 
    ( -it_21/4| z_1| ^2F) ^λ
    _1,λ _2(r_1,r_2)    =   -∂/∂λ _2( 1/4| z_1| ^2F) ^λ
    _1,λ _2(r_1,r_2),

we have

    R_k,l(λ _1,λ _2,-it_11/4|
    z_2| ^2F)    =   -∂/∂λ _1
    R_k,l(λ _1,λ _2,1/4| z_2|
    ^2F)-R_k,l( λ _1,λ _2,1/4|
    z_1| ^2| z_2| ^2) 
          +k/λ _1( R_k,l(λ _1,λ _2,1/4| z_2| ^2F)-R_k-1,l(λ _1,λ _2,
    1/4| z_2| ^2F)) ,

and

    R_k,l(λ _1,λ _2,-it_21/4|
    z_1| ^2F)    =   -∂/∂λ _2
    R_k,l(λ _1,λ _2,1/4| z_1|
    ^2F)-R_k,l( λ _1,λ _2,1/4|
    z_1| ^2| z_2| ^2) 
          +l/λ _2( R_k,l(λ _1,λ _2,1/4| z_1| ^2F)-R_k,l-1(λ _1,λ _2,
    1/4| z_1| ^2F)) .

Therefore,

    R_k,l(λ _1,λ _2,( it_1-1/4|
    z_1| ^2) ( it_2-1/4|
    z_2| ^2) F) 
       =   ∂/∂λ _1∂/∂λ
    _2R_k,l(λ _1,λ _2,F)-2R_k,l( λ
    _1,λ _2,1/4| z_1| ^21/4| z_2| ^2) 
          +l/λ _2∂/∂λ _1(
    R_k,l-1(λ _1,λ _2,F)-R_k,l(λ _1,λ
    _2,F)) 
          +k/λ _1∂/∂λ _2(
    R_k-1,l(λ _1,λ _2,F)-R_k,l(λ _1,λ
    _2,F)) 
          +k/λ _1l/λ _2( R_k-1,l(λ
    _1,λ _2,F)+R_k,l-1(λ _1,λ _2,F)-R_k,l(λ
    _1,λ _2,F)-R_k-1,l-1(λ _1,λ _2,F)) 
          +2k/λ _1( R_k,l(λ _1,λ _2,1/4
    | z_2| ^2F)-R_k-1,l(λ _1,λ _2,
    1/4| z_2| ^2F)) 
          +2l/λ _2( R_k,l(λ _1,λ _2,1/4
    | z_1| ^2F)-R_k,l-1(λ _1,λ _2,
    1/4| z_1| ^2F)) .


Let σ =(2k+n)λ _1+(2l+n)λ _2 so that R_k,l(λ
_1,λ _2,F)=ψ (σ ) where

    ψ (σ )=(1-σ )_+^α.

Define

    ψ _1(σ )=( it_1-1/4| z_1|
    ^2) ( it_2-1/4| z_2| ^2)
    F.

It can be written as

    ψ _1(σ )    =   ∂/∂λ _1∂/∂λ _2ψ (σ )-l/λ _2∂/∂λ _1( ψ (σ )-ψ (σ -2λ
    _2)) -k/λ _1∂/∂λ _2( ψ (σ )-ψ (σ -2λ _1)) 
          -kl/λ _1λ _2( ψ (σ )-ψ (σ
    -2λ _2)-ψ (σ -2λ _1)+ψ (σ -2λ
    _1-2λ _2)) 
          +C_1k/λ _1( ψ (σ )-ψ (σ -2λ
    _1) +C_2l/λ _2( ψ (σ )-ψ (σ
    -2λ _2)) -C_3ψ (σ ).

The function ψ (σ ) has two properties: the first one is that ψ (σ ) is supported in a set of the form 0<(2k+n)λ
_1+(2l+n)λ _2≤ c for large k and l, and the second is

    ∫_0^∞∫_0^∞|ψ( (2k+n)λ
    _1+(2l+n)λ _2) | dμ (λ _1)dμ
    (λ _2)≤ C(2k+n)^-n-1(2l+n)^-n-1

We claim that the function ψ _1(σ ) also satisfies the same
conditions. The first property is obvious. To verify that ψ _1(σ
) satisfies the second property (<ref>), we rewrite ψ _1(σ )
as

    ψ _1(σ )    =   nk/2λ _1λ _2∂/∂ l∂/∂ kψ ((2k+n)λ
    _1+(2l+n)λ _2) 
          -nk/2λ _1λ _2∂/∂ l(
    ψ ((2k+n)λ _1+(2l+n)λ _2)-ψ ((2k-2+n)λ
    _1+(2l+n)λ _2)) 
          +nl/2λ _1λ _2∂/∂ k∂/∂ lψ ((2k+n)λ _1+(2l+n)λ _2) 
          -nl/2λ _1λ _2∂/∂ k(
    ψ ((2k+n)λ _1+(2l+n)λ _2)-ψ ((2k+n)λ
    _1+(2l-2+n)λ _2)) 
          -kl/λ _1λ _2∂/∂ k∂/∂ lψ ((2k+n)λ _1+(2l+n)λ _2) 
          +kl/λ _1λ _2∂/∂ k(
    ψ ((2k+n)λ _1+(2l+n)λ _2)-ψ ((2k+n)λ
    _1+(2l-2+n)λ _2)) 
          +kl/λ _1λ _2∂/∂ lψ
    ((2k+n)λ _1+(2l+n)λ _2) 
          -kl/λ _1λ _2( ψ ((2k+n)λ
    _1+(2l+n)λ _2)+ψ ((2k+n)λ _1+(2l-2+n)λ
    _2)) 
          -kl/λ _1λ _2∂/∂ lψ
    ((2k-2+n)λ _1+(2l+n)λ _2) 
          +kl/λ _1λ _2( ψ ((2k-2+n)λ
    _1+(2l+n)λ _2)-ψ ((2k-2+n)λ _1+(2l-2+n)λ
    _2)) 
          +2kl/λ _1λ _2∂/∂ l∂/∂ kψ ((2k+n)λ _1+(2l+n)λ _2) 
          -2kl/λ _1λ _2∂/∂ l(
    ψ ((2k+n)λ _1+(2l+n)λ _2)-ψ ((2k-2+n)λ
    _1+(2l+n)λ _2)) 
          +2kl/λ _1λ _2∂/∂ k∂/∂ lψ ((2k+n)λ _1+(2l+n)λ _2) 
          -2kl/λ _1λ _2∂/∂ k(
    ψ ((2k+n)λ _1+(2l+n)λ _2)-ψ ((2k+n)λ
    _1+(2l-2+n)λ _2)) 
          -C_1k/λ _1∂/∂ kψ
    ((2k+n)λ _1+(2l+n)λ _2) 
          +C_1k/λ _1( ψ ((2k+n)λ _1+(2l+n)λ
    _2)-ψ ((2k-2+n)λ _1+(2l+n)λ _2)) 
          -C_2l/λ _2∂/∂ lψ
    ((2k+n)λ _1+(2l+n)λ _2) 
          +C_2l/λ _2( ψ ((2k+n)λ _1+(2l+n)λ
    _2)-ψ ((2k+n)λ _1+(2l-2+n)λ _2)) 
          +n^2-2kl/4λ _1λ _2∂/∂ k∂/∂ lψ ((2k+n)λ _1+(2l+n)λ _2) 
          +C_1k/λ _1∂/∂ kψ
    ((2k+n)λ _1+(2l+n)λ _2)+C_2l/λ _2∂/∂ lψ ((2k+n)λ _1+(2l+n)λ _2) 
          -C_0ψ ((2k+n)λ _1+(2l+n)λ _2).

Using Taylor expansion, we have that

    ψ _1(σ )    =   4nkλ _1∫_k-1^k(t+1-k)ψ
    ^(3)((2t+n)λ _1+(2l+n)λ _2)dt 
          +4nlλ _2∫_l-1^l(s+1-l)ψ ^(3)((2k+n)λ
    _1+(2s+n)λ _2)ds 
          -16klλ _1λ
    _2∫_l-1^l∫_k-1^k(s+1-l)(t+1-k)ψ ^(4)((2t+n)λ
    _1+(2s+n)λ _2)dtds 
          +16klλ _1∫_k-1^k(t+1-k)ψ ^(3)((2t+n)λ
    _1+(2l+n)λ _2dt 
          +16klλ _2∫_l-1^l(s+1-l)ψ ^(3)((2k+n)λ
    _1+(2s+n)λ _2)ds 
          +C_1kλ _1∫_k-1^k(t+1-k)ψ ^(2)((2t+n)λ
    _1+(2l+n)λ _2)dt 
          +C_2lλ _2∫_l-1^l(s+1-l)ψ ^(2)((2k+n)λ
    _1+(2s+n)λ _2)ds 
          +(n^2-8kl)ψ ^(2)((2k+n)λ _1+(2l+n)λ _2) 
          +C_1kψ ^(1)((2k+n)λ _1+(2l+n)λ _2)+C_2lψ
    ^(1)((2k+n)λ _1+(2l+n)λ _2) 
          -C_0ψ ((2k+n)λ _1+(2l+n)λ _2).

It follows that

    ∫_0^∞∫_0^∞|ψ _1(σ
    )| dμ (λ _1)dμ (λ _2) 
       ≤   c_0∫_0^∞∫_0^∞(1-(2k+n)λ
    _1-(2l+n)λ _2)_+^α dμ (λ _1)dμ (λ
    _2) 
          +c_1∫_0^∞∫_0^∞(1-(2k+n)λ
    _1-(2l+n)λ _2)_+^α -1 dμ (λ _1)dμ
    (λ _2) 
          +c_2∫_0^∞∫_0^∞(1-(2k+n)λ
    _1-(2l+n)λ _2)_+^α -2 dμ (λ _1)dμ
    (λ _2) 
          +c_3∫_k-1^k∫_0^∞∫_0^∞λ _1(
    1-(2t+n)λ _1-(2l+n)λ _2)_+^α -2 dμ (λ
    _1)dμ (λ _2)dt 
          +c_4∫_l-1^l∫_0^∞∫_0^∞λ _2(
    1-(2k+n)λ _1-(2s+n)λ _2)_+^α -2 dμ (λ
    _1)dμ (λ _2)ds 
          +c_5∫_k-1^k∫_0^∞∫_0^∞λ _1(
    1-(2t+n)λ _1-(2l+n)λ _2)_+^α -3 dμ (λ
    _1)dμ (λ _2)dt 
          +c_6∫_l-1^l∫_0^∞∫_0^∞λ _2(
    1-(2k+n)λ _1-(2s+n)λ _2)_+^α -3 dμ (λ
    _1)dμ (λ _2)ds 
          +c_7∫_k-1^k∫_l-1^l∫_0^∞∫_0^∞λ _1λ _2(1-(2t+n)λ _1-(2s+n)λ _2)
    _+^α -4 dμ (λ _1)dμ (λ _2)dtds 
       ≤   C(2k+n)^-n-1(2l+n)^-n-1.

This proves that ψ _1(σ ) also has the second property (<ref>). An iteration of the process shows that

    R_k,l(λ _1,λ _2,(it_1-1/4
    r_1^2)^j(it_2-1/4r_2^2)^jF)=ψ _j(σ )

satisfies the condition

    ∫_0^∞∫_0^∞|ψ _j((2k+n)λ
    _1+(2l+n)λ _2)| dμ (λ _1)dμ (λ
    _2)≤ C_j(2k+n)^-n-1(2l+n)^-n-1

provided (1-λ _1-λ _2)_+^α -4 is integrable.
Thus, when j=m and α >4m-1, we have

    ∫_0^∞∫_0^∞| R_k,l(λ
    _1,λ _2,(it_1-1/4| z_1|
    ^2)^m(it_2-1/4| z_2| ^2)^mF)
    | dμ (λ _1)dμ (λ _2) 
       ≤   C_m(2k+n)^-n-1(2l+n)^-n-1,

which implies


    ∫_0^∞∫_0^∞(∑_k=0^∞∑_l=0^∞| R_k,l( λ _1,λ
    _2,(it_1-1/4| z_1| ^2)^m(it_2-1
    /4| z_2| ^2)^mF) |
                          ×(k+n-1)!/k!(l+n-1)!/l!) dμ (λ _1)dμ (λ _2)<∞ .

We can get the same result by a similar calculation when (λ
_1,λ _2) belongs to other quadrants. Then (<ref>) holds
and the proof of Theorem <ref> is completed.


As a consequence of Theorem <ref>, we have


Let 1≤ p_1,p_2≤∞ and 1/p=1/p_1+1/p_2. If
α >2Q+3, then S^α is bounded from L^p_1(ℍ
^n)× L^p_2(ℍ^n) into L^p(ℍ^n).



  We take m=Q/2+1. If α >2Q+3, we have α >4m-1.
Then, Theorem <ref> is available. By Hölder's inequality and
Young's inequality, we conclude that

    ‖ S^α(f,g)‖ _p   ≤   C‖ f‖
    _p_1‖ g‖ _p_2∫_ℍ^n(1+|ω _1| )^-2mdω _1∫_ℍ^n(1+|ω _2| )^-2mdω _2
       ≤   C‖ f‖ _p_1‖ g‖
    _p_2( ∫_1^∞t^-2m+Q-1dt) ^2
       ≤   C‖ f‖ _p_1‖ g‖ _p_2.

The proof is completed.


We note that the index in Corollary <ref> is very high. In the rest part
of this paper, we will distinguish various cases and manage to give lower
indices .



§ BOUNDEDNESS OF S^Α FOR 1≤ P_1,P_2≤ 2


In this Section, we investigate the L^p_1× L^p_2→
L^p boundedness of the bilinear Riesz means S^α for 1≤
p_1,p_2≤ 2.


 Suppose m∈ L^∞(ℝ). Define operator T_mf=∫_a^bm(λ )P_λf dμ (λ ) for 0≤ a<b
. Then, for any 1≤ p≤ 2, we have

    ‖ T_mf‖ _2≤ C‖ m‖ _∞( (b-a)b^n) ^(1/p-1/2)‖
    f‖ _p.




  Since that

    P_λf=∑_k=0^∞(2k+n)^-n-1f∗ (e
    _k^λ+e_k^-λ)(z,t),

the operator T_m can be written as

    T_mf=∫_a^bm(λ )P_λf dμ (λ )=f∗ G_m,

where the kernel is given by

    G_m(z,t)    =   ∫_a^bm(λ )∑_k=0^∞(2k+n)^-n-1(
    e_k^λ+e_k^-λ) dμ (λ )
    
       =   C∑_k=0^∞∫_a/2k+n^b/2k+n(e^-iλ
    t+e^iλ t)m((2k+n)λ )φ _k^λ(z)|λ| ^ndλ .

Applying the Plancherel theorem in the variable t, the orthogonality of φ _k^λ and the fact

    ‖φ _k^λ‖ _2≤ C |λ|^-n/2 k^n-1/2,

we get that

    ‖ G_m‖ _2^2   ≤   C∫_ℂ ^n∫_ℝ|∑_k=0^∞∫_a/2k+n^b/
    2k+n(e^-iλ t+e^iλ t)m((2k+n)λ )φ
    _k^λ(z)|λ| ^ndλ| ^2  dtdz 
       =   C∫_ℝ∫_ℂ ^n|∑_k=0^∞χ _[a/2k+n,b/ 2k+n]( |λ|
    )m((2k+n)|λ| ) φ _k^λ(z)|λ| ^n| ^2  dzdλ
       ≤   C∫_ℝ∑_k=0^∞|χ _[a/2k+n
    ,b/2k+n ](|λ| )m((2k+n)|λ| )| ^2‖φ_k^λ‖ ^2_2|λ|^2ndλ
       ≤   C( ∑_k=0^∞(2k+n)^-n-1k^n-1)
    ∫_a^b| m(λ )| ^2|λ|
    ^ndλ
       ≤   C‖ m‖ _∞^2(b-a)b^n.

This implies that

    ‖ T_mf‖ _2≤ C( (b-a)b^n) ^1/2‖ m‖ _∞‖ f‖ _1.

By interpolation with the trivial estimate

    ‖ T_mf‖ _2≤‖ m‖ _∞‖ f‖ _2,

we conclude that

    ‖ T_mf‖ _2≤ C( (b-a)b^n) ^(1/
    p -1/2)‖ m‖ _∞‖ f‖
    _p.

The proof is completed.



 Suppose that 1≤ p_1,p_2≤ 2 and 1/p=1/p_1+1/p_2. If α >Q( 1/p-1), then S^α is bounded from L^p_1(ℍ^n)× L^p_2(
ℍ ^n) into L^p(ℍ^n).



  We choose a nonnegative function φ ∈ C_0^∞(
1/2,2) satisfying ∑_-∞^∞φ (2^js)=1, s>0.
For each j≥ 0, we set function

    φ _j^α( s,t) =(1-s-t)_+^αφ(
    2^j( 1-s-t) ),

and define bilinear operator

    T_j^α(f,g)=∫_0^∞∫_0^∞φ
    _j^α( λ _1,λ _2) P_λ
    _1fP_λ _2g dμ (λ _1)dμ (λ _2).

It is obvious that

    S^α=∑_j=0^∞T_j^α,

and our result would follow if we can show that when α >Q( 
1 /p-1), there exists an ε >0 such that for each j≥ 0,

    ‖ T_j^α‖ _L^1× L^p_2→
    L^p≤ C2^-ε j.

Fixing j≥ 0. In order to prove (<ref>), we define B_j={ω
:|ω|≤ 2^j(1+γ )}⊆ℍ
^n and split the kernel K_j^α of T_j^α into four
parts:

    K_j^α=K_j^1+K_j^2+K_j^3+K_j^4,

where

    K_j^1(ω _1,ω _2)    =   K_j^α(ω _1,ω
    _2)χ _B_j(ω _1)χ _B_j(ω _2), 
    
    K_j^2(ω _1,ω _2)    =   K_j^α(ω _1,ω
    _2)χ _B_j(ω _1)χ _B_j^c(ω _2), 
    
    K_j^3(ω _1,ω _2)    =   K_j^α(ω _1,ω
    _2)χ _B_j^c(ω _1)χ _B_j(ω _2), 
    
    K_j^4(ω _1,ω _2)    =   K_j^α(ω _1,ω
    _2)χ _B_j^c(ω _1)χ _B_j^c(ω _2).

Here χ _A stands for the characteristic function of set A and γ >0 is to be fixed. Let T_j^l be the bilinear operator with
kernel K_j^l, l=1,2,3,4. Then, (<ref>) would be the consequence
of the estimates

    ‖ T_j^l‖ _L^p_1× L^p_2→
    L^p≤ C2^-ε j,   l=1,2,3,4.


We first consider T_j^4. Set R_t^l(ω ) to be the kernel of
the Riesz means ∫_0^t(1-λ/t)^lP_λ dμ
(λ ). Then,

    t→ R_t^0(ω )

is a function of bounded variation and the kernel of T_j^α can
be written as

    K_j^α(ω _1,ω _2)=∫φ _j^α(s,t)
    ∂/∂ sR_s^0(ω _1)∂/∂
    t R_t^0(ω _2) dsdt.

Intergrating by parts and using the identity

    ∂/∂ t(t^mR_t^m(ω ))
    =mt^m-1R_t^m-1(ω ),

where m is a positive integer, we get the relation

    K_j^α(ω _1,ω _2)=c_m∫(( ∂
    _s∂ _t) ^2m+2φ _j^α(s,t))s^2m+1R_s^2m+1(ω _1)t^2m+1R_t^2m+1(ω
    _2) dsdt.

Note that (see Theorem 2.5.3 in <cit.>)

    | R_t^2m+1(ω )|≤ Ct^n+1(1+t^1/2|ω| )^-2m,

This estimate, together with the bound

    |( ∂ _s∂ _t) ^2m+2φ
    _j^α|≤ C2^j(4m+4)

imply that

    | K_j^α(ω _1,ω _2)|≤
    C2^j(4m+4)(1+|ω _1| )^-2m(1+|ω
    _2| )^-2m.

So, we can use Hölder's inequality and Young's inequality to get that

    ‖ T_j^4(f,g)‖ _p   ≤   C‖ f‖
    _p_1‖ g‖ _p_22^j(4m+4)∫_|ω
    _1|≥ 2^j(1+γ )(1+|ω _1|
    )^-2mdω _1
          ×∫_|ω _2|≥ 2^j(1+γ
    )(1+|ω _2| )^-2mdω _2
       ≤   C2^j(4m+4)2^j(1+γ )(-4m+2Q)‖ f‖
    _p_1‖ g‖ _p_2.

Choosing m large enough such that

    4mγ≥ 2Q(1+γ )+4,

we have

    ‖ T_j^4‖ _L^p_1× L^p_2→
    L^p≤ C2^-ε j

for some ε >0.

Next, we consider the estimate of T_j^3. Note that K_j^α
also can be written as

    K_j^α(ω _1,ω _2)=c_m∫_0^1∫_0^1(
    ∂ _λ _1^2m+2φ _j^α(λ _1,λ
    _2)) λ _1^2m+1R_λ _1^2m+1(ω
    _1)G_λ _2(ω _2) dλ _1dμ (λ _2)

where

    G_λ _2(ω _2)=G_λ _2(z,t)=∑_k=0^∞(2k+n)^-n-1(e_k^λ _2+e_k^-λ
    _2)(z,t)

is the kernel of the projection operator P_λ _2. Then, it
follows that

    | K_j^3(ω _1,ω _2)|
       =   | K_j^α(ω _1,ω _2)χ
    _B_j^c(ω _1)χ _B_j(ω _2)|
       ≤   C∫_0^1|λ _1^2m+1R_λ
    _1^2m+1(ω _1)χ _B_j^c(ω _1)||∫_0^1( ∂ _λ _2^2m+2φ
    _j^α(λ _1,λ _2)) G_λ _2(ω
    _2)χ _B_j(ω _2) dμ (λ _2)| dλ _1
       ≤   Csup_λ _1∈ 0,1]|λ
    _1^2m+1R_λ _1^2m+1(ω _1)χ _B_j^c(ω
    _1)|∫_0^1|∫_0^1( ∂
    _λ _2^2m+2φ _j^α(λ _1,λ
    _2)) G_λ _2(ω _2)χ _B_j(ω _2) dμ
    (λ _2)| dλ _1.

We get

    ‖ T_j^3(f,g)‖ _p   ≤   C‖ f‖
    _p_1‖ g‖ _p_2∫_|ω
    _1|≥ 2^j(1+γ )sup_λ _1∈ 0,1]|λ _1^2m+1R_λ _1^2m+1(ω _1)| dω _1
          ×∫_|ω _2|≤ 2^j(1+γ
    )∫_0^1|∫_0^1( ∂ _λ
    _2^2m+2φ _j^α(λ _1,λ _2))
    G_λ _2(ω _2)χ _B_j(ω _2) dμ (λ
    _2)| dλ _1dω _2.

Applying the Plancherel theorem in the variable t, the orthogonality of φ _k^λ and (<ref>), we have that

    ∫_0^1∫_|ω _2|≤ 2^j(1+γ
    )|∫_0^1( ∂ _λ _2^2m+2φ
    _j^α(λ _1,λ _2)) G_λ _2(ω
    _2) dμ (λ _2)| dω _2dλ _1
       ≤   C2^j(1+γ )Q/2∫_0^1( ∫_ℂ
    ^n∫_ℝ|∫_0^1( ∂ _λ
    _2^2m+2φ _j^α(λ _1,λ _2))
    G_λ _2(ω _2) dμ (λ _2)|
    ^2dtdz) ^1/2dλ _1
       ≤   C2^j(1+γ )Q/2∫_0^1( ∫_ℂ
    ^n∫_ℝ|∑_k=0^∞∫_0^1/2k+ne^-iλ _2t( ∂ _λ _2^2m+2φ
    _j^α(λ _1,(2k+n)|λ _2|
    )) φ _k^λ _2(z) dμ (λ _2)|
    ^2dtdz) ^1/2dλ _1
       ≤   C2^j(1+γ )Q/2∫_0^1( ∫_ℝ∫_ℂ^n|∑_k=0^∞χ _ 0,1/2k+n
    ](λ _2)( ∂ _λ _2^2m+2φ _j^α(λ _1,(2k+n)|λ _2| )) φ
    _k^λ _2(z) | ^2dz|λ _2|
    ^2ndλ _2) ^1/2dλ _1
       =   C2^j(1+γ )Q/2∫_0^1( ∑_k=0^∞∫_0^1/2k+n|∂ _λ _2^2m+2φ
    _j^α(λ _1,(2k+n)|λ _2|
    )| ^2‖φ _k^λ ‖
    _2^2|λ _2| ^2n dλ _2) ^1/2dλ _1
       ≤   C2^j(1+γ )Q/2∫_0^1( ∑_k=0^∞(2k+n)^-n-1k^n-1∫_0^1|∂ _λ
    _2^2m+2φ _j^α(λ _1,|λ
    _2| )| ^2|λ _2|
    ^n dλ _2) ^1/2dλ _1
       ≤   C2^j(1+γ )Q/2∫_0^1( ∫_0^1|∂ _λ _2^2m+2φ _j^α(λ
    _1,|λ _2| )| ^2|λ
    _2| ^n dλ _2) ^1/2dλ _1
       ≤   C2^j(1+γ )Q/22^j(2m+2).

On the other hand, from (<ref>), we have

    sup_λ _1∈ 0,1]|λ _1^2m+1R_λ
    _1^2m+1(ω _1)|≤ C(1+|ω
    _1| )^-2m,

which implies that

    ∫_|ω _1|≥ 2^j(1+γ )sup_λ
    _1∈ 0,1]|λ _1^2m+1R_λ
    _1^2m+1(ω _1)| dω _1   ≤   C∫_|ω _1|≥ 2^j(1+γ )(1+|ω
    _1| )^-2mdω _1
       ≤   C2^j(1+γ )(-2m+Q).

Thus,

    ‖ T_j^3(f,g)‖ _p≤ C‖ f‖
    _p_1‖ g‖ _p_22^j(1+γ )Q/22^j(2m+2)2^j(1+γ )(-2m+Q),

and we can choose m large enough such that

    2mγ≥3/2Q(1+γ )+2,

which yields that

    ‖ T_j^3‖ _L^p_1× L^p_2→
    L^p≤ C2^-ε j

for some ε >0. Obviously, (<ref>) also holds for T_j^2.

Now, it remains to estimate T_j^1. For ξ∈ℍ^n, we
set B_j(ξ ,R)={ω :|ξ ^-1ω|≤
R2^j(1+γ )} with R>0, and slipt the functions f and g
into three parts respectively: f=f_1+f_2+f_3, g=g_1+g_2+g_3,
where

    f_1   =   fχ _B_j(ξ ,3/4),         g_1=gχ
    _B_j(ξ ,3/4), 
    
    f_2   =   fχ _B_j(ξ ,5/4)\ B_j(ξ ,3/4
    ),   g_2=gχ _B_j(ξ ,5/4)\ B_j(ξ ,3/
    4), 
    
    f_3   =   fχ _ℍ^n\ B_j(ξ ,5/4),     g_3=gχ _ℍ^n\ B_j(ξ ,5/4).

Assume that |ξ ^-1ω|≤1/4
2^j(1+γ ). Since that f_3 is supported on ℍ
^n\ B_j(ξ ,5/4), then f_3≠ 0 leads to

    |ξ ^-1ω _1|≥5/42^j(1+γ ).

It follows that

    |ω _1^-1ω|≥ 2^j(1+γ ).

Note that the kernel K_j^1 is supported on B_j× B_j. Hence,
T_j^1(f_3,g)=0. In the same way, T_j^1(f,g_3)=0. Since that f_2 and g_2 are supported on B_j(ξ ,5/4)\
B_j(ξ ,3/4), then f_2,g_2≠ 0 yields that

    |ω _1^-1ω|≥1/22^j(1+γ
    ) and |ω _2^-1ω|≥1/2
    2^j(1+γ ).

Repeating the proof of (<ref>), we get

    ‖ T_j^1(f_2,g_2)‖ _L^p(B_j(ξ ,1/4
    ))   ≤   C2^-ε j‖ f_2‖ _p_1‖
    g_2‖ _p_2
       ≤   C2^-ε j‖ f‖ _L^p_1(B_j(ξ ,
    5 /4))‖ g‖ _L^p_2(B_j(ξ ,5/4)).

Taking the L^p norm with respect to ξ on the both side of (<ref>) and using Hölder's inequality, we have that

    ( ∫_ℍ^n∫_B_j(ξ ,1/4)|
    T_j^1(f_2,g_2)(ω )| ^pdω dξ) ^
    1 /p
       ≤   C2^-ε j( ∫_ℍ^n∫_B_j(ξ ,
    5/4 )| f(ω )| ^p_1dω dξ) ^1/ p_1( ∫_ℍ^n∫_B_j(ξ ,5/4
    )| g(ω )| ^p_2dω dξ) ^1/
    p_2.

Changing variable and exchanging the order of integration, the left side
equals to

    ( ∫_ℍ^n∫_|ω|≤1
    /42^j(1+γ )| T_j^1(f_2,g_2)(ξω
    )| ^pdω dξ) ^1/p   =   (
    ∫_|ω|≤1/42^j(1+γ )∫_ℍ^n| T_j^1(f_2,g_2)(ξω )|
    ^pdξ dω) ^1/p
       =   ( 1/42^j(1+γ )) ^Q/p‖
    T_j^1(f_2,g_2)‖ _p,

and the right side equals to

    C2^-ε j( 5/42^j(1+γ )) ^Q/
    p_1‖ f‖ _p_1( 5/42^j(1+γ
    )) ^Q/p_2‖ g‖
    _p_2=C2^-ε j( 5/42^j(1+γ )) ^Q/p‖ f‖ _p_1‖ g‖ _p_2.

This yields that

    ‖ T_j^1(f_2,g_2)‖ _p≤ C2^-ε
    j‖ f‖ _p_1‖ g‖ _p_2.

Since f_1 is supported on B_j(ξ ,3/4), then f_1,g_2≠ 0 implies that

    |ω _1^-1ω|≤ 2^j(1+γ )
    and |ω _2^-1ω|≥1/2
    2^j(1+γ ).

We can repeat the proof of (<ref>) to get that

    ‖ T_j^1(f_1,g_2)‖ _L^p(B_j(ξ ,1/4
    ))   ≤   C2^-ε j‖ f_1‖ _p_1‖
    g_2‖ _p_2
       ≤   C2^-ε j‖ f‖ _L^p_1(B_j(ξ ,
    3 /4))‖ g‖ _L^p_2(B_j(ξ ,5/4)).

Taking the L^p norm with respect to ξ as above, we have

    ‖ T_j^1(f_1,g_2)‖ _p≤ C2^-ε
    j‖ f‖ _p_1‖ g‖ _p_2.

Obviously, (<ref>) also holds for T_j^1(f_2,g_1). Finally, we
consider T_j^1(f_1,g_1). Because f_1,g_1≠ 0 implies that

    |ω _1^-1ω|≤ 2^j(1+γ )
    and |ω _2^-1ω|≤ 2^j(1+γ ),

we have

    T_j^1(f_1,g_1)(ω )=T_j^α(f_1,g_1)(ω ).

Note that T_j^α can be written as

    T_j^α(f,g)    =   ∫_0^∞∫_0^∞φ
    _j^α( λ _1,λ _2) P_λ
    _1fP_λ _2g dμ (λ _1)dμ (λ _2) 
       =   C∫_[-1,1]^2φ _j^α( |λ
    _1| ,|λ _2|) P_|λ _1|fP_|λ _2|g dμ
    (λ _1)dμ (λ _2).

Because for any fixed s∈ -1,1], the function

    t→φ _j^α( | s|
    ,| t|)

is supported in [-1,1] and vanishes at endpoints ± 1, so we can expand
this function in Fourier series by considering a periodic extension on ℝ of period 2. Then, we have

    φ _j^α(| s| ,| t|
    )=∑_k∈ℤγ _j,k^α(s)e^iπ kt

with Fourier coefficients

    γ _j,k^α(s)=1/2∫_-1^1φ _j^α(| s| ,| t| )e^-iπ kt dt.

It is easy to see that for any 0< δ < α,

    sup_s∈ -1,1]|γ _j,k^α(s)|( 1+| k|) ^1+δ≤ C2^-j(α
    -δ).

T_j^α can be expressed by

    T_j^α(f,g)    =   C∫_[-1,1]^2φ _j^α(
    |λ _1| ,|λ _2|) P_|λ _1|fP_|λ
    _2|g dμ (λ _1)dμ (λ _2) 
       =   C∑_k∈ℤ∫_[-1,1]^2γ _j,k^α(λ
    _1)e^iπ kλ _2P_|λ _1|fP_|λ _2|g dμ (λ _1)dμ (λ
    _2) 
       =   C∑_k∈ℤ∫_-1^1γ _j,k^α(λ
    _1)P_|λ _1|f dμ (λ
    _1)∫_-1^1e^iπ kλ _2P_|λ
    _2|g dμ (λ _2).

Applying Cauchy-Schwartz's inequality and Lemma <ref>, we have

    ‖ T_j^α(f,g)‖ _1   ≤   C∑_k∈ℤ‖∫_-1^1γ _j,k^α(λ _1)P_|λ _1|f dμ (λ _1)‖ _2‖∫_-1^1e^iπ kλ _2P_|λ _2|g dμ (λ _2)‖ _2
       ≤   C∑_k∈ℤ(1+| k| )^-1-δ(
    sup_s∈ -1,1]|γ _j,k^α(s)|( 1+| k|) ^1+δ) ‖
    f‖ _p_1‖ g‖ _p_2
       ≤   C2^-j(α -δ)‖ f‖ _p_1‖
    g‖ _p_2.

Then, using Hölder's inequality and (<ref>), it follows that

    ‖ T_j^1(f_1,g_1)‖ _L^p(B_j(ξ ,1/4
    ))   ≤   2^j(1+γ )Q( 1/p-1) ‖
    T_j^1(f_1,g_1)‖ _L^1(B_j(ξ ,1/4))
       =   2^j(1+γ )Q( 1/p-1) ‖ T_j^α(f_1,g_1)‖ _L^1(B_j(ξ ,1/4))
       ≤   C2^-j(α -δ)2^j(1+γ )Q( 1/ p-1)
    ‖ f_1‖ _p_1‖ g_1‖ _p_2
       ≤   C2^-j(α -δ)2^j(1+γ )Q( 1/ p-1)
    ‖ f‖ _L^p_1(B_j(ξ ,3/4 ))‖
    g‖ _L^p_2(B_j(ξ ,3/4)).

Taking the L^p norm with respect to ξ yields that

    ‖ T_j^1(f_1,g_1)‖ _p≤ C2^-j(α
    -δ) 2^j(1+γ )Q( 1/p-1) ‖
    f‖ _p_1‖ g‖ _p_2.

Combining (<ref>), (<ref>) and (<ref>), we conclude that

    ‖ T_j^1(f,g)‖ _p≤ C2^-j(α -δ)
    2^j(1+γ )Q( 1/p-1) ‖ f‖
    _p_1‖ g‖ _p_2.

Thus, whenever α >Q( 1/p-1), we can choose γ, δ >0 such that

    α >Q(1+γ )( 1/p-1) +δ,

which implies that there exists an ε >0 such that

    ‖ T_j^1‖ _L^1× L^p_2→
    L^p≤ 2^-ε j.

The proof of Theorem <ref> is completed.




§ BOUNDEDNESS OF S^Α FOR PARTICULAR POINTS


In this section, we investigate the boundedness of S^α for some
specific triples of points (p_1,p_2,p).



 §.§ The point (1,∞ ,1)



 If α >Q/2, then S^α is bounded from L^1(ℍ^n)× L^∞(ℍ^n) to L^1(ℍ
^n).



  We keep the notations in Section 4. Note that (<ref>), (<ref>), (<ref>) and (<ref>) hold for any α >0, the proof of Theorem
<ref> is valid apart from the estimate of T_j^1(f_1,g_1).
According to (<ref>), for any 0< δ < α,

    ‖ T_j^α(f,g)‖ _1≤ C2^-j(α -δ
    )‖ f‖ _1‖ g‖ _2,

we have

    ‖ T_j^1(f_1,g_1)‖ _L^1(B_j(ξ ,1/4
    ))   =   ‖ T_j^α(f_1,g_1)‖ _L^1(B_j(ξ ,
    1 /4))
       ≤   C2^-j(α -δ)‖ f‖ _L^1(B_j(ξ ,
    3/4 ))‖ g‖ _L^2(B_j(ξ ,3/4))
       ≤   C2^-j(α -δ)2^j(1+γ )Q/2‖
    f‖ _L^1(B_j(ξ ,3/4))‖ g‖
    _L^∞(B_j(ξ ,1/4)).

It follows that

    ‖ T_j^1(f_1,g_1)‖ _1≤ C2^-j(α -δ
    )2^j(1+γ )Q/2‖ f‖ _1‖
    g‖ _∞.

Thus, when α >Q/2, we can choose γ, δ >0 such
that α >(1+γ )Q/2+δ, which yields that there
exists ε >0 such that

    ‖ T_j^1(f_1,g_1)‖ _1≤ C2^-ε
    j‖ f‖ _1‖ g‖ _∞.

The proof is completed.




 §.§ The point (∞ ,∞ ,∞ )



If α >Q- 1/2, then S^α is bounded from
L^∞( ℍ^n)× L^∞(ℍ^n) into L^∞( ℍ^n).



  We still keep the notion in Section 4. To prove this theorem, it
suffices to estimate T_j^1(f_1,g_1). Since that

    P_λf(z,t)=∑_k=0^∞(2k+n)^-n-1f∗ (e
    _k^λ+e_k^-λ)(z,t),   λ >0,

and

    f∗ e_k^λ=e^-iλ tf^λ∗ _λφ
    _k^λ,

we can write T_j^α(f,g) as

    T_j^α(f,g)(z,t)    =   ∫_0^∞∫_0^∞φ
    _j^α(λ _1,λ _2)∑_k=0^∞(2k+n)^-n-1f∗ (e_k^λ _1+e
    _k^-λ _1)(z,t) 
          ×∑_l=0^∞(2l+n)^-n-1g∗ (e_l^λ
    _2+e_l^-λ _2)(z,t) dμ (λ _1)dμ
    (λ _2) 
       =   C∫_ℝ∫_ℝ∑_k=0^∞∑_l=0^∞e^-i(λ _1+λ _2)tφ _j^α( (2k+n)|λ _1| ,(2l+n)|λ
    _2|) 
          ×( f^λ _1∗ _λ _1φ _k^λ
    _1) (z)( g^λ _2∗ _λ _2φ
    _l^λ _2) (z)|λ _1|
    ^n|λ _2| ^n dλ _1dλ _2.

Using again (<ref>), for 0<δ<1, we get

    ‖ T_j^α(f,g)‖ _∞
       ≤   C∫_ℝ∫_ℝ∑_k=0^∞∑_l=0^∞|φ _j^α((2k+n)|λ _1| , (2l+n)|λ_2|) |
          ×‖ f^λ _1∗ _λ _1φ
    _k^λ _1‖ _∞‖ g^λ _2∗
    _λ _2φ _l^λ _2‖ _∞|λ _1| ^n|λ _2|
    ^n dλ _1dλ _2
       ≤   C∫_ℝ∫_ℝ∑_k=0^∞∑_l=0^∞|φ _j^α((2k+n)|λ _1| , (2l+n)|λ_2|) |
          ×‖ f^λ _1‖ _2‖φ
    _k^λ _1‖ _2‖ g^λ _2‖
    _2‖φ _l^λ _2‖ _2|λ
    _1| ^n|λ _2| ^n dλ
    _1dλ _2
       ≤   C∫_ℝ∫_ℝ∑_k=0^∞∑_l=0^∞|φ _j^α((2k+n)|λ _1| , (2l+n)|λ_2|) ||λ _1| ^n/2k^n-1/2|λ
    _2| ^n/2l^n-1/2
          ×‖ f^λ _1‖ _2‖ g^λ
    _2‖ _2 dλ _1dλ _2
       ≤   C ( ∫_ℝ∫_ℝ( ∑_k=0^∞∑_l=0^∞|φ _j^α((2k+n)|λ _1| , (2l+n)|λ_2|) ||λ _1| ^n+δ/2k^n-1/2|λ
    _2| ^n+δ/2l^n-1/2)^2  dλ
    _1dλ _2)^1/2
          ×( ∫_|λ _1|≤ 1∫_|λ _2|≤ 1‖
    f^λ_1‖ ^2_2‖ g^λ _2‖
    ^2_2|λ _1| ^-δ|λ
    _2| ^-δ  dλ _1dλ _2)^1/2
       ≤   C ∑_k=0^∞∑_l=0^∞ k^n-1/2l^
    n-1/2( ∫_ℝ∫_ℝ|φ
    _j^α((2k+n)|λ _1| ,
    (2l+n)|λ_2|) |^2 |λ _1| ^n+δ|λ_2|
    ^n+δ  dλ _1dλ _2)^1/2
          ×( ∫_|λ _1|≤ 1‖
    f^λ_1‖ ^2_2|λ _1|
    ^-δ  dλ _1)^1/2( ∫_|λ _2|≤ 1‖ g^λ _2‖
    ^2_2|λ _2| ^-δ  dλ _2)
    ^1/2
       ≤   C ∑_k=0^∞∑_l=0^∞ k^-1-δ/2l^-1-δ/2( ∫_ℝ∫_ℝ|φ _j^α(|λ _1| ,
    |λ_2|) |^2 |λ
    _1| ^n+δ|λ_2| ^n+δ 
    dλ _1dλ _2)^1/2
          ×( ∫_|λ _1|≤ 1‖
    f^λ_1‖ ^2_2|λ _1|
    ^-δ  dλ _1)^1/2( ∫_|λ _2|≤ 1‖ g^λ _2‖
    ^2_2|λ _2| ^-δ  dλ _2)
    ^1/2
       ≤   C2^-j(α +1/2)( ∫_|λ
    _1|≤ 1‖ f^λ_1‖ ^2_2|λ _1| ^-δ  dλ _1)^1/2( ∫_|λ _2|≤ 1‖ g^λ _2‖ ^2_2|λ _2| ^-δ  dλ _2)^1/2.

Because

    T_j^1(f_1,g_1)(ω )=T_j^α(f_1,g_1)(ω ),  ω∈ B_j(ξ ,1/4),

we have

    ‖ T_j^1(f_1,g_1)‖ _L^∞(B_j(ξ 1/4))
       =   ‖ T_j^α(f_1,g_1)‖ _L^∞(B_j(ξ 1/4))
       ≤   C2^-j(α +1/2)( ∫_|λ
    _1|≤ 1‖ f_1^λ_1‖ ^2_2|λ _1| ^-δ  dλ _1)^1/2( ∫_|λ _2|≤ 1‖ g_1^λ _2‖ ^2_2|λ _2| ^-δ  dλ _2)^1/2.

Let us consider the integral about λ _1. Note that

    ‖ f_1^λ _1‖ _2   ≤   C2^nj(1+γ)‖ f_1^λ _1‖ _∞
       ≤   C2^j(1+γ )( Q/2+1) ‖ f
    ‖_L^∞(B_j(ξ ,3/4)),

we obtain

    ∫_|λ _1|≤ 1‖
    f_1^λ_1‖ ^2_2|λ _1|
    ^-δ dλ _1
       =   ∫_2^-2j(1+γ )≤|λ _1|≤
    1‖ f_1^λ_1‖ ^2_2|λ _1| ^-δ  dλ _1+ ∫_|λ
    _1|≤ 2^-2j(1+γ )‖
    f_1^λ_1‖ ^2_2|λ _1|
    ^-δ  dλ _1
       ≤   2^2δ j(1+γ )‖ f_1‖ _2^2 +
    C2^j(1+γ )(Q+2) ‖ f ‖ ^2_L^∞(B_j(ξ ,3/4))∫_|λ_1|≤
    2^-2j(1+γ )|λ _1| ^-δ 
    dλ _1
       ≤   C2^j(1+γ )(Q+2δ)‖ f ‖
    ^2_L^∞(B_j(ξ ,3/4)).

In the same way, we have

    ∫_|λ _1|≤ 1‖
    g_1^λ_2‖ ^2_2|λ _2|
    ^-δ dλ _2≤ C2^j(1+γ )(Q+2δ)‖ g ‖ ^2_L^∞(B_j(ξ ,3/4)).

From (<ref>) and above estimates, we get

    ‖ T_j^1(f_1,g_1)‖ _L^∞(B_j(ξ ,1
    /4 ))≤ C2^-j(α +1/2) 2^j(1+γ )(Q+2δ)‖ f‖ _L^∞(B_j(ξ ,3/4))‖ g‖ _L^∞(B_j(ξ ,3/4)).

It follows that

    ‖ T_j^1(f_1,g_1)‖ _L^∞≤
    C2^-j(α +1/2) 2^j(1+γ )(Q+2δ)‖ f‖ _L^∞‖ g‖ _L^∞.

Thus, whenever α >Q- 1/2, we can choose γ, δ >0
such that α >(1+γ )(Q+2δ)- 1/2, which implies there
exists an ε >0 such that

    ‖ T_j^1(f_1,g_1)‖ _L^∞≤
    C2^-ε j‖ f‖ _L^∞‖
    g‖ _L^∞.

The proof of Theorem <ref> is completed.




 §.§ The point (2,∞ ,2)



 If α >Q-1/2, then S^α is bounded
from L^2(ℍ^n)× L^∞(ℍ^n) to L^2(
ℍ^n).



  As above, it suffices to estimate T_j^1(f_1,g_1). We write T_j^α(f,g) as

    T_j^α(f,g)(z,t)    =   ∫_0^∞∫_0^∞φ
    _j^α(λ _1,λ _2)∑_k=0^∞(2k+n)^-n-1f∗ (e_k^λ _1+e
    _k^-λ _1)(z,t) 
          ×∑_l=0^∞(2l+n)^-n-1g∗ (e_l^λ
    _2+e_l^-λ _2)(z,t) dμ (λ _1)dμ
    (λ _2) 
       =   C∫_-∞^∞∫_-∞^∞ e^-i(λ
    _1+λ _2)t∑_k=0^∞∑_l=0^∞φ
    _j^α( (2k+n)|λ _1|
    ,(2l+n)|λ _2|) 
          ×( f^λ _1∗ _λ _1φ _k^λ
    _1) (z)( g^λ _2∗ _λ _2φ
    _l^λ _2) (z)|λ _1|
    ^n|λ _2| ^n dλ _1dλ _2
       =   C∫_-∞^∞e^-iλ _1t∫_-∞^∞∑_k=0^∞∑_l=0^∞φ _j^α(
    (2k+n)|λ _1-λ _2| ,(2l+n)|λ _2|) 
          ×( f^λ _1-λ _2∗ _λ _1-λ
    _2φ _k^λ _1-λ _2) (z)( g^λ
    _2∗ _λ _2φ _l^λ _2) (z)|λ _1-λ _2| ^n|λ _2|
    ^n dλ _1dλ _2.

Then, applying the Plancherel theorem in variable t, Mikowski's
inequality, the orthogonality of the special Hermite projections, (<ref>) and Plancherel formula (<ref>), we get that

    ∫_ℂ^n∫_ℝ| T_j^α(f,g)(z,t)| ^2 dtdz 
       =   C∫_ℂ^n∫_ℝ|∫_ℝe^-iλ _1t∫_ℝ∑_l=0^∞∑_k=0^∞φ _j^α( (2k+n)|λ _1-λ
    _2| ,(2l+n)|λ _2|) 
          ×( f^λ _1-λ _2∗ _λ _1-λ
    _2φ _k^λ _1-λ _2) (z)( g^λ
    _2∗ _λ _2φ _l^λ _2) (z)|λ _1-λ _2| ^n|λ _2|
    ^n dλ _2dλ _1| ^2 dtdz 
       =   C∫_ℝ∫_ℂ^n|∫_ℝ∑_l=0^∞∑_k=0^∞φ _j^α(
    (2k+n)|λ _1-λ _2| ,(2l+n)|λ _2|) 
          ×( f^λ _1-λ _2∗ _λ _1-λ
    _2φ _k^λ _1-λ _2) (z)( g^λ
    _2∗ _λ _2φ _l^λ _2) (z)|λ _1-λ _2| ^n|λ _2|
    ^ndλ _2| ^2 dzdλ _1
       ≤   C∫_ℝ∫_ℂ^n( ∫_ℝ∑_l=0^∞|∑_k=0^∞φ _j^α( (2k+n)|λ _1-λ _2|
    ,(2l+n)|λ _2|) ( f^λ
    _1-λ _2∗ _λ _1-λ _2φ _k^λ
    _1-λ _2) (z)|
          ×|( g^λ _2∗ _λ _2φ
    _l^λ _2) (z)||λ _1-λ
    _2| ^n|λ _2| ^n dλ _2) ^2 dzdλ _1
       ≤   C∫_ℝ( ∫_ℝ∑_l=0^∞‖∑_k=0^∞φ _j^α( (2k+n)|λ _1-λ _2| ,(2l+n)|λ
    _2|) ( f^λ _1-λ _2∗ _λ
    _1-λ _2φ _k^λ _1-λ _2)
    ‖ _2
          ×‖ g^λ _2∗ _λ _2φ
    _l^λ _2‖ _∞|λ _1-λ
    _2| ^n|λ _2| ^n dλ _2) ^2 dλ _1
       ≤   C∫_ℝ( ∫_ℝ∑_l=0^∞(
    ∑_k=0^∞|φ _j^α( (2k+n)|λ _1-λ _2| ,(2l+n)|λ
    _2|) | ^2‖ f^λ _1-λ
    _2∗ _λ _1-λ _2φ _k^λ _1-λ
    _2‖ _2^2) ^1/2
          ×‖ g^λ _2‖ _2‖φ
    _l^λ _2‖ _2|λ _1-λ
    _2| ^n|λ _2| ^n dλ _2) ^2 dλ _1
       ≤   C ( ∫_|λ _2|≤ 1∑_l≤1/|λ _2|‖ g^λ _2‖ ^2_2‖φ _l^λ _2‖ ^2_2|λ _2| ^2n- δ dλ_2) 
          ×( ∫_ℝ∫_ℝ∑_l=0^∞∑_k=0^∞|φ _j^α( (2k+n)|λ _1-λ _2| ,(2l+n)|λ
    _2|) | ^2
          ×‖ f^λ _1-λ _2∗ _λ
    _1-λ _2φ _k^λ _1-λ _2‖
    _2^2|λ _1-λ _2| ^2n|λ _2| ^δ dλ _2 dλ _1) 
       =    C ( ∫_|λ _2|≤ 1∑_l≤1/|λ _2|‖ g^λ _2‖ ^2_2‖φ _l^λ _2‖ ^2_2|λ _2| ^2n- δ dλ_2) 
          ×( ∑_k=0^∞∫_ℝ∑_l=0^∞∫_ℝ|φ _j^α( (2k+n)|λ _1| ,(2l+n)|λ _2|)
    | ^2‖ f^λ _1∗ _λ _1φ
    _k^λ _1‖ _2^2|λ _1|
    ^2n|λ _2| ^δ dλ _2 dλ
    _1) 
       ≤   C ( ∫_|λ _2|≤ 1‖
    g^λ _2‖ ^2_2|λ _2| ^n-
    δ( ∑_l≤1/|λ _2|
    l^n-1) dλ_2) 
          ×( ∑_k=0^∞∫_ℝ‖
    f^λ _1∗ _λ _1φ _k^λ _1‖
    _2^2|λ _1| ^2n( ∑_l=0^∞
    (2l+n)^-1- δ) ∫_ℝ|φ _j^α( (2k+n)|λ _1| , |λ
    _2|) | ^2|λ _2|
    ^δ dλ _2 dλ _1) 
       ≤   C2^-j(2α +1)( ∫_|λ _2|≤ 1‖ g^λ _2‖ ^2_2|λ
    _2| ^- δ dλ_2) ( ∑_k=0^∞∫_ℝ‖ f^λ _1∗ _λ _1φ
    _k^λ _1‖ _2^2|λ _1|
    ^2n  dλ _1) 
       ≤   C2^-j(2α +1)‖ f‖ _2^2∫_|λ _2|≤ 1‖ g^λ _2‖
    ^2_2|λ _2| ^- δ dλ_2 .

It follows from (<ref>) that

    ‖ T_j^1(f_1,g_1)‖ ^2_L^2(B_j(ξ ,1/
    4))
       =   ‖ T_j^α(f_1,g_1)‖ ^2_L^2 (B_j(ξ ,
    1/4))
       ≤   C2^-j(2α +1)‖ f_1‖ _2^2∫_|λ _2|≤ 1‖ g_1^λ
    _2‖ ^2_2|λ _2| ^-
    δ dλ_2
       ≤   C2^-j(2α +1) 2^j(1+γ )(Q+2 δ)‖ f‖ ^2_L^2(B_j(ξ ,3/4))‖ g
    ‖ ^2_L^∞(B_j(ξ ,3/4)).

Taking the L^2 norm with respect to ξ yields that

    ‖ T_j^1(f_1,g_1)‖ _2≤ C2^-j(α + 1
    /2) 2^j(1+γ )(Q/2+ δ)‖ f ‖ _2‖ g ‖ _∞.

Thus, whenever α > Q-1/2, we can choose γ, δ >0
such that α > (1+γ )(Q/2+ δ)- 1/2, which
yields that there exists ε >0 such that

    ‖ T_j^1(f_1,g_1)‖ _2≤ C2^-ε
    j‖ f‖ _2‖ g‖ _∞.

The proof of Theorem <ref> is completed.




§ APPENDIX: BILINEAR INTERPOLATION


Because p_1 and p_2 are symmetric, we have obtained, in two sections
above, the boundedness of the bilinear Riesz means S^α at some
specific triples of points (p_1,p_2,p) like

    (1,1,1/2), (2,2,1), (∞,∞,∞), (1,2,2/3), (2,1,
    2/3),(1,∞ ,1),(∞,1,1),(2,∞,2),(∞,2,2).

We can obtain the intermediate boundedness of S^α by using of the
bilinear interpolation via complex method adapted to the setting of analytic
families or real method in <cit.>. Bernicot et al. <cit.>
described how to make use of real method. The full results in Main Theorem
are proved in this way. We outline this argument for reader's convenience.

Consider a spherical decomposition of S^α as

    S^α=∑_j=0^∞2^-jαT_j,α,

where

    T_j,α(f,g)=∫_0^∞∫_0^∞φ _j,α( λ _1,λ _2) P_λ _1fP_λ
    _2gdμ (λ _1)dμ (λ _2)

and

    φ _j,α( s,t) =2^jα(1-s-t)_+^αφ (2^j( 1-s-t) ).

In the preceding sections, we have actually obtained the estimates of the
form

    ‖ T_j,α‖ _L^p_1× L^p_2→
    L^p≤ C2^jα (p_1,p_2)

at some triples of points (p_1,p_2,p). Since α (p_1,p_2)
only depends on the point (p_1,p_2,p), (<ref>) also holds for any
other T_j,α ^', i.e.,

    ‖ T_j,α ^'‖ _L^p_1×
    L^p_2→ L^p≤ C2^jα (p_1,p_2).

So, fixing j and α ^' and applying bilinear real
interpolation theorem on T_j,α ^', we can conclude that if
the point (p_1,p_2,p) satisfies

    ( 1/p_1,1/p_2,1/p) =(1-θ )(
    1/p_1^0,1/p_2^0,1/p^0) +θ( 1/p_1^1,1/p_2^1,1/p^1)

for some θ∈ (0,1) and (p_1^0,p_2^0,p^0),(p_1^1,p_2^1,p^1), we have
that

    ‖ T_j,α ^'‖ _L^p_1×
    L^p_2→ L^p≤ C2^j((1-θ )α
    (p_1^0,p_2^0)+θα (p_1^1,p_2^1)).

Define α (p_1,p_2)=(1-θ )α (p_1^0,p_2^0)+θα (p_1^1,p_2^1) and let α ^'=α. It
follows that

    ‖ S^α‖ _L^p_1× L^p_2→
    L^p≤∑_j=0^∞2^-jα‖ T_j,α‖ _L^p_1× L^p_2→ L^p≤
    C∑_j=0^∞2^-jα2^jα (p_1,p_2).

Thus, when α >α (p_1,p_2), we have ‖ S^α‖ _L^p_1× L^p_2→ L^p≤ C, i.e., S^α is bounded from L^p_1(ℍ^n)× L^p_1(
ℍ^n) to L^p(ℍ^n). 0.5 cm

Acknowledgements   The first author is supported by National
Natural Science Foundation of China (Grant No. 11371036). The second author
is supported by China Scholarship Council (Grant No. 201606010026). 

99
Bern F. Bernicot, L. Grafakos, L. Song, L. Yan, The
bilinear Bochner-Riesz problem, J. Anal. Math. 127 (2015), 179–217.

Feff C. Feffermann, Inequalities for strongly singular
convolution operators, Acta Math. 124 (1970), 9–36.

Graf L. Grafakos, L. Liu, S. Lu and F. Zhao, The
multilinear Marcinkiewicz interpolation theorem revisited: the behavior of
the constant, J. Funct. Anal. 262 (2012), no. 5, 2289–2313.

Mauc G. Mauceri, Riesz means for the eigenfunction
expansions for a class of hypoelliptic differential operators, Ann. Inst.
Fourier (Grenoble). 31 (1981), no. 4, v, 115–140.

Mull D. Müller, A restriction theorem for the
Heisenberg group, Ann. Math(2). 131 (1990), no. 3, 567–587.

Mull2 D. Müller, On Riesz means of
eigenfunction expansions for the Kohn-Laplacian, J. Reine Angew. Math. 401 (1989), 113–121.

Stein E. M. Stein, Interpolation of linear operators, Trans. Amer. Math. Soc. 83 (1956), 482–492.

Strich R. S. Strichartz, Harmonic analysis as spectral
theory of Laplacians, J. Funct. Anal. 87 (1989), no. 1, 51–148.

Strich2 R. S. Strichartz, L^p harmonic analysis and
Radon transform on the Heisenberg group, J. Funct. Anal. (96) 1991, no.2,
350–406.

Thang S. Thangavelu, Harmonic analysis on the Heisenberg group.
Progress in Mathematics, 159.


