A discrete singular integral operator

Abstract

Suppose that {α_k} ^∞_k=-∞ is a Lacunary sequance of positive numbers satisfying inf$\mathop {\inf }\limits_k \alpha _{k + 1} /\alpha _k = \alpha > 1$ and that Ω(y') is a function in the Besov spaceB ^0,1₁ (S ⁿ⁻¹) whereS ⁿ⁻¹ is the unit sphere on ℝⁿ(n≥2). We prove that if ∫S^n-1Ω(y′)dσ(y′) then the discrete singular integral operator

$$T_\Omega f\left( x \right) = \sum\limits_{k = - \infty }^\infty {\int_{S^{n - 1} } {f\left( {x - \alpha _k y'} \right)} \Omega \left( {y'} \right)d\sigma \left( {y'} \right)} $$

and the associated maximal operator

$$\left. {T_\Omega ^* f\left( x \right) = \mathop {\sup }\limits_N } \right|\sum\limits_{k = N}^\infty {\int_{S^{n - 1} } {f\left( {x - \alpha _k y'} \right)} \Omega \left( {y'} \right)d\sigma \left( {y'} \right)} $$

are both bounded in the spaceL ²(ℝⁿ)

The theorems in this paper improve a result by Duoandikoetxea and Rubio de Francia^[1] in theL ² case.