Discrete singular integral operator

Suppose that {αk}k=- infinity infinity is a Lacunary sequence of positive numbers satisfying qq αk+1/αk = α gt; 1 and that Ω(y prime ) is a function in the Besov space B10,1(Sn-1) where Sn-1 is the unit sphere on Rn(n >= 2). We prove that if integralS(n-1) Ω(y prime )dσ(y prime ) = 0 then the discrete singular integral operator TΩf(x) = qq f(x - αky prime )Ω(y prime )dσ(y prime ) and the associated maximal operator TΩ*f(x) = qq f(x - αky prime )Ω(y prime )dσ(y prime ) | are both bounded in the space L2(Rn). The theorems in this paper improve a result by Duoandikoetxea and Rubio de Francia in the L2 case.