On the Boundedness of Biparameter Littlewood-Paley gλ⁎-Function

Abstract

Let m,n≥1 and let gλ1,λ2⁎ be the biparameter Littlewood-Paley gλ⁎-function defined by gλ1,λ2⁎fx = ∬R+m+1 t2/t2+x2-y2mλ2∬R+n+1 t1/t1+x1-y1nλ1×θt1,t2fy1,y22dy1dt1/t1n+1dy2dt2/t2m+11/2, λ1>1, λ2>1, where θt1,t2f is a nonconvolution kernel defined on Rm+n. In this paper we show that the biparameter Littlewood-Paley function gλ1,λ2⁎ is bounded from L2Rn+m to L2Rn+m. This is done by means of probabilistic methods and by using a new averaging identity over good double Whitney regions.