Boundedness of the bilinear Bochner-Riesz Means in the non-Banach triangle case
Heping Liu Min Wang
math.FA 42B08 (Primary), 42B15 (Secondary)

Boundedness of the bilinear Bochner-Riesz Means] Boundedness of the
bilinear Bochner-Riesz Means

in the non-Banach triangle case

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 42B08; Secondary 42B15



In this article, we investigate the boundedness of the bilinear
Bochner-Riesz means S^α in the non-Banach triangle case. We
improve the corresponding results in <cit.> in two aspects: Our
partition of the non-Banach triangle is simpler and we obtain lower
smoothness indices α (p_1,p_2) for various cases apart from 1
≤ p_1=p_2 <2.


[
    Min Wang
    
============





§ 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 Fourier multiplier operator 

    S^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 ,

where x∈ℝ^n, f, g are functions on ℝ^n and f, g are their Fourier transforms. Of course, we hope
to obtain a smoothness index α (p_1,p_2) as low as possible such
that 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). Bernicot et al. <cit.> gave a comprehensive
study on this problem. Their results are pretty well when n=1 and in the
Banach triangle case when n ≥ 2. In this article, we investigate the L^p_1× L^p_2→ L^p boundedness of S^α
in the non-Banach triangle case, i.e. 1 ≤ p_1, p_2 ≤∞ and p<1
, when n ≥ 2. We improve the corresponding results in <cit.> in
two aspects. Firstly, our partition of the non-Banach triangle is simpler.
Secondly, we obtain lower smoothness indices α (p_1,p_2) for
various cases apart from 1 ≤ p_1=p_2 <2.

Our results are summarized in the following theorem. The pictures also
display the comparison of two partitions.


 Assume that n ≥ 2. Let 1≤ p_1, p_2≤∞
and 1/p=1/p_1+1/p_2.

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

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



    < g r a p h i c s >


Remark: Fefferman <cit.> pointed out that the restriction
estimates apply to the study of the boundedness of Bochner-Riesz means.
Stein's earlier result in <cit.> was improved by using the restriction
theorem (see <cit.>). Our results are obtained without the help of
non-trivial restriction estimates. In contrast, the results in <cit.>
depend heavily on the restriction-extension estimates. It seems still a
problem that how apply the restriction estimates to obtain a better result
about the boundedness of the bilinear Bochner-Riesz means.

0.5 cm



§ PRELIMINARIES


We state some facts about the Bessel function and the kernel of the
Bochner-Riesz operator. These facts can be found in many references, for
example, in <cit.>.

We define the restriction operators 

    R_λf(x)=∫_𝕊^n-1f(λω )e^2π
    ix·λω  dσ (ω ),   λ >0,

where dσ denote the surface measure on 𝕊^n-1. Then 

    R_1f(x)= ∫_𝕊^n-1f(ω )e^2π ix·ωdσ (ω )= f∗dσ,

It is known that 

    dσ(x)=∫_𝕊^n-1e^2π ix·ωdσ
    (ω )=2π/| x| ^n-2/2J_n-2 
    /2(2π| x| ),

where J_k is the Bessel function given by 

    J_k(r)= (r/2)^k/Γ (k+ 1/2) π^1/2∫_-1^1 e^irt (1-t^2)^k-1/2  dt,    k> -1/2.

By a dilation argument, we have 

    R_λf=f∗φ _λ,  λ >0,

where 

    φ _λ(x)=dσ( λ x) =2π/( λ| x|) ^n-2/2 J_
    n-2/2(2πλ| x| ).


The bilinear Bochner-Riesz means can be written as 

    S_R^α(f,g)(x)=∫_0^∞∫_0^∞( 1-λ _1^2+λ _2^2/R^2) _+^αR_λ
    _1f(x)R_λ _2g(x)λ _1^n-1λ _2^n-1 
    dλ _1dλ _2.

It is easy to see that 

    S_R^α(f,g)(x)=∫_ℝ^n∫_ℝ^n
    f(x-x_1) g(x-x_2) S_R^α(x_1,x_2)  dx_1dx_2,

where the kernel is given by 

    S_R^α(x_1,x_2)=∫_0^∞∫_0^∞( 1- 
    λ _1^2+λ _2^2/R^2) _+^αφ
    _λ _1(x_1)φ _λ _2(x_2)λ
    _1^n-1λ _2^n-1  dλ _1dλ _2.

Because 

    S_R^α(x_1,x_2)= R^2n S_1^α(Rx_1,Rx_2),

by a dilation argument, we know that the L^p_1×
L^p_2→ L^p boundedness of the operator S_R^α is
deduced from the L^p_1× L^p_2→ L^p boundedness of
the operator S^α:=S_1^α as 1/p=1/p_1+1/p_2.

We note that the kernel of the bilinear Bochner-Riesz operator S^α
on ℝ^n coincides with the kernel of the Bochner-Riesz operator
on ℝ^2n. This means 

    S^α(x_1,x_2)= Γ (1+ α)/π^α |
    (x_1,x_2) |^n+ α J_n+ α( 2π |(x_1,x_2)
     | ).

The Bessel function satisfies the asymptotic estimate 

    J_k(r)= O ( r^-1/2),    r →∞.

As a consequence, we have a basic result: S^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).

0.5 cm



§ PROOF OF RESULTS.



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

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




  The operator T_m can be written as 

    T_mf=f∗ G_m,

where the kernel 

    G_m(x)=∫_ℝ^nχ _ a,b] (|ξ| )m(|ξ| )e^2π i⟨ x,ξ⟩ dξ.

Applying Plancherel theorem, we have 

    ‖ G_m‖ _2^2= ∫_a≤|ξ|≤ b| m(|ξ| )| ^2  dξ≤ C ‖ m‖ _∞^2( (b-a)b^n-1).

It follows that 

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

Interpolation with the trivial estimate 

    ‖ T_mf‖ _2≤‖ m‖ _∞‖ f‖ _2,

we get 

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

Lemma <ref> is proved.



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



  First, 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^2-t^2)_+^αφ( 2^j( 1-s^2-t^2) ),

and define bilinear operator 

    T_j^α(f,g)=∫_0^∞∫_0^∞φ
    _j^α( λ _1,λ _2) R_λ
    _1fR_λ _2g λ _1^n-1λ _2^n-1  dλ
    _1dλ _2.

It is obvious that 

    S^α=∑_j=0^∞T_j^α,

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

    ‖ T_j^α‖ _L^p_1× L^p_2→
    L^p≤ 2^-ε j.

Fixing j≥ 0. In order to prove (<ref>), we define B_j={x:| x|≤ 2^j(1+γ )}⊆ℝ^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(x_1,x_2)    =   K_j^α(x_1,x_2)χ
    _B_j(x_1)χ _B_j(x_2), 
    
    K_j^2(x_1,x_2)    =   K_j^α(x_1,x_2)χ
    _B_j(x_1)χ _B_j^c(x_2), 
    
    K_j^3(x_1,x_2)    =   K_j^α(x_1,x_2)χ
    _B_j^c(x_1)χ _B_j(x_2), 
    
    K_j^4(x_1,x_2)    =   K_j^α(x_1,x_2)χ
    _B_j^c(x_1)χ _B_j^c(x_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. Note that the kernel of T_j^α is
written as 

    K_j^α(x_1,x_2)=∫_0^∞∫_0^∞φ
    _j^α(λ _1,λ _2)φ _λ
    _1(x_1)φ _λ _2(x_2) λ _1^n-1λ
    _2^n-1 dλ _1dλ _2

satisfying 

    | K_j^α(x_1,x_2)|≤ C2^-jα2^-j(1+2^-j| x_1| )^-M(1+2^-j|
    x_2| )^-M

for each M>0. So, by Hölder's inequality and Young's inequality, we
can get 

    ‖ T_j^4(f,g)‖ _p   ≤   C‖ f‖
    _p_1‖ g‖ _p_2∫_| x_1|≥ 2^j(1+γ )(1+2^-j| x_1| )^-Mdx_1
          ×∫_| x_2|≥ 2^j(1+γ
    )(1+2^-j| x_2| )^-Mdx_2
       ≤   C( 2^jM2^-j(1+γ )(M-n)) ^2‖
    f‖ _p_1‖ g‖ _p_2.

Choosing M large enough such that 

    Mγ >(1+γ )n,

it follows that 

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

for some ε >0. Similarly, for T_j^3, we have 

    ‖ T_j^3(f,g)‖ _p   ≤   C‖ f‖
    _p_1‖ g‖ _p_2∫_| x_1|≤ 2^j(1+γ )(1+2^-j| x_1| )^-Mdx_1
          ×∫_| x_2|≥ 2^j(1+γ
    )(1+2^-j| x_2| )^-Mdx_2
       ≤   C2^j(1+γ )n2^jM2^-j(1+γ )(M-n)‖ f‖
    _p_1‖ g‖ _p_2.

Choosing large M satisfying 

    Mγ >2(1+γ )n,

it follows 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 any fixed y∈ℝ ^n
, we set B_j(y,R)={x:| x-y|≤ 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(y,3/4),         g_1=gχ
    _B_j(y,3/4), 
    
    f_2   =   fχ _B_j(y,5/4)\ B_j(y,3/4),   g_2=gχ _B_j(y,5/4)\ B_j(y,3/4), 
    
    f_3   =   fχ _ℝ^n\ B_j(y,5/4),      g_3=gχ _ℝ^n\ B_j(y,5/4).

Assume that | x-y|≤1/42^j(1+γ ).
Since that f_3 is supported on ℝ^n\ B_j(y,5
/4), then f_3≠ 0 leads to 

    | x-x_1-y|≥5/42^j(1+γ ).

It follows that 

    | x_1|≥ 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, we have T_j^1(f,g_3)=0.
Since that f_2 and g_2 are supported on B_j(y,5/4
)\ B_j(y,3/4), then f_2,g_2≠ 0 yields that 

    | x_1|≥1/22^j(1+γ ) and | x_2|≥1/22^j(1+γ ).

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

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

Both side of this inequality are the fuction of y. So, taking the L^p
norm with respect to y and using Hölder's inequality, we have that 

    ( ∫_ℝ^n∫_B_j(y,1/4)|
    T_j^1(f_2,g_2)(x)| ^pdxdy) ^1/p
       ≤   C2^-ε j( ∫_ℝ^n∫_B_j(y,5/
    4)| f(x)| ^p_1dxdy) ^1/p_1(
    ∫_ℝ^n∫_B_j(y,5/4)| g(x)|
    ^p_2dxdy) ^1/p_2.

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

    ( ∫_ℝ^n∫_B_j(y,1/4)|
    T_j^1(f_2,g_2)(x)| ^pdxdy) ^1/p
       =   ( ∫_ℝ^n∫_| x|≤1/4
    2^j(1+γ )| T_j^1(f_2,g_2)(x+y)|
    ^pdxdy) ^1/p
       =   ( ∫_| x|≤1/42^j(1+γ
    )∫_ℝ^n| T_j^1(f_2,g_2)(x+y)|
    ^pdydx) ^1/p
       =   ( 1/42^j(1+γ )) ^n/p‖
    T_j^1(f_2,g_2)‖ _p.

Similarly, the right side equals to 

    C2^-ε j( 5/42^j(1+γ )) ^n/
    p_1‖ f‖ _p_1( 5/42^j(1+γ
    )) ^n/p_2‖ g‖
    _p_2=C2^-ε j( 5/42^j(1+γ )) ^n/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 that f_1 is supported on B_j(y,3/4), then f_1,g_2≠ 0 implies that 

    | x_1|≤ 2^j(1+γ ) and |
    x_2|≥1/22^j(1+γ ).

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

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

Taking the L^p norm with respect to y as above, we get that 

    ‖ 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 

    | x_1|≤ 2^j(1+γ ) and |
    x_2|≤ 2^j(1+γ ),

we have 

    T_j^1(f_1,g_1)(x)=T_j^α(f_1,g_1)(x),   x∈ B_j(y,1/4).

Note that T_j^α can be written as 

    T_j^α(f,g)    =   ∫_0^∞∫_0^∞φ
    _j^α( λ _1,λ _2) R_λ
    _1fR_λ _2g λ _1^n-1λ _2^n-1dλ
    _1dλ _2
       =   1/2∫_[-1,1]^2φ _j^α( |λ _1| ,|λ _2|)
    R_|λ _1|fR_|λ
    _2|g |λ _1| ^n-1|λ _2| ^n-1dλ _1dλ _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|) R_|λ _1|fR_|λ
    _2|g |λ _1| ^n-1|λ _2| ^n-1dλ _1dλ _2
       =   C∑_k∈ℤ∫_[-1,1]^2γ _j,k^α(λ
    _1)e^iπ kλ _2R_|λ _1|fR_|λ _2|g |λ
    _1| ^n-1|λ _2| ^n-1dλ
    _1dλ _2
       =   C∑_k∈ℤ∫_-1^1γ _j,k^α(λ
    _1)R_|λ _1|f |λ
    _1| ^n-1 dλ _1∫_-1^1e^iπ kλ
    _2R_|λ _2|g |λ
    _2| ^n-1dλ _2.

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

    ‖ T_j^α(f,g)‖ _1
       ≤   C∑_k∈ℤ‖∫_-1^1γ _j,k^α(λ _1)R_|λ _1|f |λ
    _1| ^n-1 dλ _1‖ _2‖∫_-1^1e^iπ kλ _2R_|λ _2|g |λ _2| ^n-1dλ _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.

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

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

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

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

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

    α >n(1+γ )(1/p-1)+δ ,

which implies that there exists an ε >0 such that 

    ‖ T_j^1‖ _L^p_1× L^p_2→
    L^p≤ C2^-ε j.

The proof of Theorem <ref> is completed.


As a consequence of Theorem <ref>, we can give another proof of the
estimate in case of (p_1,p_2,p)=(1, ∞, 1), which was already obtained
in <cit.>.


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



  We keep the notations in the proof of Theorem <ref>. 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(y,1/4 ))   =   ‖ T_j^α(f_1,g_1)‖ _L^1(B_j(y, 
    1 /4))
       ≤   C2^-j(α -δ)‖ f‖ _L^1(B_j(y, 
    3/4 ))‖ g‖ _L^2(B_j(y,3/4))
       ≤   C2^-j(α -δ)2^j(1+γ )n/2‖
    f‖ _L^1(B_j(y,3/4))‖ g‖
    _L^∞(B_j(y,1/4)).

It follows that 

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

Thus, when α >n/2, we can choose γ, δ >0 such
that α >(1+γ)n/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.


Now we have obtained the estimate for S^α at some specific triples
of points (p_1,p_2,p) like 

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

In fact, our new result is essentially the estimate in case of (p_1,p_2,p)= (1,2,2/3). The results in Theorem <ref> can
be obtained by using of the bilinear interpolation via complex method
adapted to the setting of analytic families or real method in <cit.>,
which was described in <cit.>.

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). 

9
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Feff C. Feffermann, Inequalities for strongly singular
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Stein E. M. Stein, Interpolation of linear operators,
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Stein2 E. M. Stein, Harmonic Analysis: Real Variable Methods,
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Princeton, NJ, 1993.


