黃啟亮， 楊必成

(廣東第二師范學(xué)院 數(shù)學(xué)系, 廣東 廣州 510303)

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一個(gè)聯(lián)系指數(shù)函數(shù)的全平面Hilbert積分不等式

黃啟亮， 楊必成

(廣東第二師范學(xué)院 數(shù)學(xué)系, 廣東 廣州 510303)

引入獨(dú)立參量與指數(shù)函數(shù)中間變量,應(yīng)用權(quán)函數(shù)的方法及實(shí)分析技巧,建立一個(gè)具有最佳常數(shù)因子的全平面Hilbert型積分不等式.考慮了其等價(jià)式、逆式及特殊參數(shù)的齊次與非齊次不等式;還求出了等價(jià)不等式的算子及范數(shù)表示.

權(quán)函數(shù);全平面Hilbert型積分不等式;等價(jià)式;全平面Hilbert型積分算子;范數(shù)

0　引言

(1)

(2)

(3)

1998年,文[5-6]中引入獨(dú)立參量λ∈(0,∞)及beta函數(shù),推廣式(1)為

(4)

(5)

(6)

2009年,文[22]綜述了參量化負(fù)數(shù)齊次核Hilbert型不等式的一系列研究思想. 2013年,文[23]論述了齊次與非齊次核Hilbert型積分不等式的等價(jià)聯(lián)系.

2007年,文[24]發(fā)表了如下具有最佳常數(shù)因子的全平面非齊次核Hilbert型積分不等式:

(7)

隨后,文[25-32]繼續(xù)討論了這一課題. 2009-2014年,楊在專著[21,33-37]中詳細(xì)論述了一般實(shí)數(shù)齊次核參量化Hilbert型不等式及其算子刻畫的理論.該理論凸顯了引入獨(dú)立參數(shù)及兩對共軛指數(shù)的參量化思想,且討論了Hilbert型積分算子的范數(shù)合成性質(zhì),它改進(jìn)及推廣了文[3]的理論成果;文[38]則全面綜述了近代Hilbert型不等式理論的研究思想及方法.

本文引入獨(dú)立參量及指數(shù)函數(shù)中間變量,應(yīng)用權(quán)函數(shù)的方法及實(shí)分析技巧,建立如下一個(gè)類似于式(7)的具有最佳常數(shù)因子的全平面齊次核Hilbert型積分不等式:

(8)

(λ>0).考慮了其引入獨(dú)立參量的更一般形式、等價(jià)式、逆式及特殊參量的齊次及非齊次形式;定義了全平面Hilbert型積分算子,并求出了等價(jià)不等式的算子及范數(shù)表示.

1　權(quán)函數(shù)與初始積分不等式

定義1設(shè)α,β≠0,0<σ<λ.定義如下權(quán)函數(shù):

(9)

(10)

(11)

(12)

(13)

(14)

(ii) 若0

證明 (i) 當(dāng)p>1時(shí),配方并由帶權(quán)的H?lder不等式[39]及式(9),有

(15)

由式(11)及交換積分次序的Fubini定理[40],有

(16)

再由式(12)及式(13),有式(14).

(ii) 當(dāng)0

2　具有最佳常數(shù)因子的等價(jià)不等式及逆式

(17)

(18)

(19)

證明配方,并由H?lder不等式,有

(20)

故得式(14),且它與式(17)等價(jià).

任給足夠大的n∈N (N為正整數(shù)集),定義集合Eα∶={x∈R;αx≥0},Fβ∶={y∈R;βy≤0},及

則可算得

對上式作變換 u=eλ(βy+αx),應(yīng)用交換積分次序的Fubini定理,我們有

由上面計(jì)算結(jié)果,有

(21)

(22)

及化簡得K(σ)≤k.故k=K(σ)為式(17)的最佳值.

式(14)的常數(shù)因子K(σ)必為最佳值,不然,由式(20),必導(dǎo)出式(17)的常數(shù)因子也不為最佳值的矛盾.證畢.

定理3在定理2的條件下,若把p>1改成0

故得式(14)的逆式,且它與式(17)的逆式等價(jià).

若有常數(shù)k≥K(σ),使取代式(17)的逆式的常數(shù)因子K(σ)后仍成立,任給足夠大的n∈N,則可得式(21)的逆式.

而

令n→∞,式(21)的逆式可化為式(22)的逆式,即有K(σ)≥k.故k=K(σ)為式(17)的逆式的最佳值.式(14)的逆式的常數(shù)因子必為最佳值,不然,由式(20)的逆式必導(dǎo)出式(17)的逆式的常數(shù)因子也不為最佳值矛盾.證畢.

3　等價(jià)不等式的算子刻畫

由式(14),有

(23)

定義2定義全平面Hilbert型積分算子T:Lp,φ(R)→Lp,ψ1-p(R)為：任f∈Lp,φ(R),唯一確定Tf=h∈Lp,ψ1-p(R).稱式(23)為算子T所對應(yīng)的不等式.

由定理2,式(23)的常數(shù)因子是最佳的,故得

(24)

則式(17),式(14)可改寫成如下等價(jià)的算子與范數(shù)表示式:

(Tf,g)<||T||·||f||p,φ||g||q,ψ,||Tf||p,ψ1-p<||T||·||f||p,φ.

(25)

評注在式(17)與式(14)中,令α=-1,β=1,μ=λ-σ(>0),以eλxf(x)取代f(x),則有

(26)

(27)

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A Hilbert-Type Integral Inequality in the Whole Plane Related to the Exponential Function

HUANG Qi-liang, YANG Bi-cheng

(Department of Mathematics, Guangdong University of Education, Guangzhou,Guangdong, 510303, P. R. China)

By introducing independent parameters and interval variables, applying the weight functions and using technique of real analysis, a Hilbert-type integral inequality in the whole plane related to the exponential function with a best possible constant factor is provided. The equivalent forms, the reverses, the related homogeneous homes and non-homogeneous forms with particular parameters are considered. The operator expressions with the norm for the equivalent inequalities are obtained.

weight function; Hilbert-type integral inequality in the whole plane; equivalent form; Hilbert-type integral operator in the whole plane; norm

2016-06-26

國家自然科學(xué)基金資助項(xiàng)目(61370186); 廣東第二師范學(xué)院教授博士科研專項(xiàng)基金資助項(xiàng)目(2015ARF25)

黃啟亮,男,廣西桂林人,廣東第二師范學(xué)院數(shù)學(xué)系教授.

O178

A

2095-3798(2016)05-0021-08