中文摘要：

Hilbert型奇异积分算子在分析学中有重要的作用。本文通过引入参数λ和两个实数A1，A2，在广义区间（0,b）上定义了一个带参数的核为1xλ＋yλ的Hilbert型奇异积分算子T：（Tf）（y）=∫b0f（x）xλ＋yλdx，利用权函数方法和算子理论，研究了T的有界性问题，在条件A2p＋A1q=2－λ下，得到了算子T的范数‖T‖=B1－A2pλ,1－A1qλλ。作为应用，还考虑其涉及内积的等价形式（Tf,g）≤B1－A2pλ,λ－1＋A2pλλ1pB1－A1qλ,λ－1＋A1qλλ1q‖f‖p,ω′‖g‖q,ω″。

英文摘要：

Hilbert＇s type singular integral operator plays an important role in analysis. In this paper, by introducing an independent parameterλ and two real numbers A1, A2, we define a Hilbert type singular multiple integral operator T in a general interval （0,b） as follows：（Tf）（y）=∫b0f（x）xλ＋yλdxusing the way of weight function and operator theory, the boundedness and norm of T are studied. Under condition of A2p＋A1q=2-λ we obtain the norm of Tas ‖T‖=B1-A2pλ,1-A1qλλ.As their applications,the equivalent forms with inner product considered follows：（Tf,g）≤B1-A2pλ,λ-1＋A2pλλ1pB1-A1qλ,λ-1＋A1qλλ1q‖f‖p,ω′‖g‖q,ω″.