一类离散型奇异积分算子

设｛αｋ｝∞ｋ＝－∞为正数缺项序列，满足ｉｎｆｋαｋ＋１／ｄｋ＝α＞１，Ω（ｙ′）为Ｂｅｓｏｖ空间Ｂ０，１１（Ｓｎ－１）上的函数，其中Ｓｎ－１为Ｒｎ（ｎ２）上的单位球面．本文证明：若∫Ｓｎ－１Ω（ｙ′）ｄσ（ｙ′）＝０，则离散型奇异积分ＴΩ（ｆ）（ｘ）＝∑∞ｋ＝－∞∫Ｓｎ－１ｆ（ｘ－αｋｙ′）Ω（ｙ′）ｄσ（ｙ′）和相关的极大算子ＴΩ（ｆ）（ｘ）＝ｓｕｐＮ∑∞ｋ＝Ｎ∫Ｓｎ－１ｆ（ｘ－αｋｙ′）Ω（ｙ′）ｄσ（ｙ′）均在Ｌ２（Ｒｎ）上有界．上述结果推广了Ｄｕｏａｎｄｉｋｏｅｔｘｅａ和ＲｕｂｉｏｄｅＦｒａｎｃｉａ［１］在Ｌ２情形下的一个结果

A Discrete Singular Integral Operator

Suppose that {α k} ∞ k=-∞ is a Lacunary sequence of positive numbers satisfying inf k α k+1 /α k=α>1 and that Ω( y′) is a function in the Besov space B 0,1 1(S n-1 ) where S n-1 is the unit sphere on R n (n2) . We prove that if ∫ S n-1 Ω( y′) dσ(y′)=0 then the discrete singular integral operatorT Ω f(x)=∑∞k=-∞∫ S n-1 f(x-α ky′) Ω (y′)dσ(y′)and the associated maximal operatorT * Ω f(x)= sup N∑∞k=N∫ S n-1 f(x-α ky′) Ω (y′) dσ(y′)are both bounded in the space L 2 (R n ) . The theorems in ths paper improve a result by Duoandikoetxea and Rubio de Francia in the L 2 case.