一類具有非齊次核的Hilbert型積分不等式成立的充要條件及其應(yīng)用

洪 勇, 溫雅敏

(廣東財(cái)經(jīng)大學(xué) 統(tǒng)計(jì)與數(shù)學(xué)學(xué)院, 廣州 510320)

1 引言與引理

設(shè)r>1,α是常數(shù), 定義函數(shù)空間

為Hilbert型積分不等式.

Hilbert型積分不等式在分析學(xué)及算子理論中應(yīng)用廣泛[1]. 當(dāng)積分核K(x,y)為齊次函數(shù)時(shí), 對(duì)Hilbert型不等式的研究已有很多結(jié)果[2-19]; 但當(dāng)K(x,y)為非齊次函數(shù)時(shí), 目前研究報(bào)道相對(duì)較少. 本文研究具有非齊次核的Hilbert型積分不等式成立的充要條件及其最佳常數(shù)因子.

設(shè)K(x,y)=G(xλ1yλ2)(λ1>0,λ2>0), 則顯然K(x,y)是一個(gè)非齊次函數(shù), 且對(duì)t>0滿足:

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

做變換xλ1/λ2y=t, 有

類似地可得ω2(y)=y(λ2/λ1)((α+1)/p-1)W2.

2 主要結(jié)果

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

(1)

則有

(2)

同時(shí), 又有

由式(1)～(3), 得

(4)

若c<0, 對(duì)足夠小的ε>0, 令

則類似地可得

(5)

(6)

對(duì)足夠小的ε>0及δ>0, 取

則有

(7)

由式(6)～(8)得

令ε→0+, 得

再令δ→0+, 得

3 在算子理論中的應(yīng)用

設(shè)K(x,y)非負(fù)可測(cè),f(x)∈Lr,α(0,+∞), 定義奇異積分算子T:

(9)

則T是一個(gè)線性算子. 若存在常數(shù)M, 使得?f(x)∈Lr,α(0,+∞), 有

‖T(f)‖r,γ≤M‖f‖r,α,

則稱T是從Lr,α(0,+∞)到Lr,γ(0,+∞)的有界線性算子. 此時(shí), 定義T的算子范數(shù)為

特別地, 當(dāng)T是從Lr,α(0,+∞)到自身的有界線性算子時(shí), 則稱T是Lr,α(0,+∞)中的有界線性算子.

證明: 只需證明‖T(f)‖p,(1-p)β≤M‖f‖p,α與式(1)等價(jià)即可. 若式(1)成立, 令

則有

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, 則易得式(1), 因而式(1)與‖T(f)‖p,(1-p)β≤M‖f‖p,α等價(jià). 證畢.

在定理2中, 取α=β=0, 則可得到以下推論.

則:

2) 若T是Lp(0,+∞)中的有界線性算子, 則T的范數(shù)

其中B(·,·)是Beta函數(shù).

同理

根據(jù)推論1知定理3成立.

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