一类具有非齐次核的Hilbert型积分不等式成立的充要条件及其应用

洪 勇, 温雅敏

(广东财经大学 统计与数学学院, 广州 510320)

1 引言与引理

设r>1,α是常数, 定义函数空间

为Hilbert型积分不等式.

Hilbert型积分不等式在分析学及算子理论中应用广泛[1]. 当积分核K(x,y)为齐次函数时, 对Hilbert型不等式的研究已有很多结果[2-19]; 但当K(x,y)为非齐次函数时, 目前研究报道相对较少. 本文研究具有非齐次核的Hilbert型积分不等式成立的充要条件及其最佳常数因子.

设K(x,y)=G(xλ1yλ2)(λ1>0,λ2>0), 则显然K(x,y)是一个非齐次函数, 且对t>0满足:

K(tx,y)=K(x,tλ1/λ2y),K(x,ty)=K(tλ2/λ1x,y).

做变换xλ1/λ2y=t, 有

类似地可得ω2(y)=y(λ2/λ1)((α+1)/p-1)W2.

2 主要结果

1) 存在常数M, 对一切f(x)∈Lp,α(0,+∞),g(y)∈Lq,β(0,+∞), 使Hilbert型不等式

(1)

则有

(2)

同时, 又有

由式(1)～(3), 得

(4)

若c<0, 对足够小的ε>0, 令

则类似地可得

(5)

(6)

对足够小的ε>0及δ>0, 取

则有

(7)

由式(6)～(8)得

令ε→0+, 得

再令δ→0+, 得

3 在算子理论中的应用

设K(x,y)非负可测,f(x)∈Lr,α(0,+∞), 定义奇异积分算子T:

(9)

则T是一个线性算子. 若存在常数M, 使得∀f(x)∈Lr,α(0,+∞), 有

‖T(f)‖r,γ≤M‖f‖r,α,

则称T是从Lr,α(0,+∞)到Lr,γ(0,+∞)的有界线性算子. 此时, 定义T的算子范数为

特别地, 当T是从Lr,α(0,+∞)到自身的有界线性算子时, 则称T是Lr,α(0,+∞)中的有界线性算子.

证明: 只需证明‖T(f)‖p,(1-p)β≤M‖f‖p,α与式(1)等价即可. 若式(1)成立, 令

则有

于是‖T(f)‖p,(1-p)β≤M‖f‖p,α.

反之, 若‖T(f)‖p,(1-p)β≤M‖f‖p,α, 则易得式(1), 因而式(1)与‖T(f)‖p,(1-p)β≤M‖f‖p,α等价. 证毕.

在定理2中, 取α=β=0, 则可得到以下推论.

则:

2) 若T是Lp(0,+∞)中的有界线性算子, 则T的范数

其中B(·,·)是Beta函数.

同理

根据推论1知定理3成立.

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