On the Boundedness of Biparameter Littlewood-Paley gλ⁎-Function. (22nd December 2016) - Cao, Mingming; Xue, Qingying
Abstract : Let m, n ≥ 1 and let g λ 1, λ 2 ⁎ be the biparameter Littlewood-Paley g λ ⁎ -function defined by g λ 1, λ 2 ⁎ f x = ∬ R + m + 1   t 2 / t 2 + x 2 - y 2 m λ 2 ∬ R + n + 1   t 1 / t 1 + x 1 - y 1 n λ 1 × θ t 1, t 2 f y 1, y 2 2 d y 1 d t 1 / t 1 n + 1 d y 2 d t 2 / t 2 m + 1 1 / 2,   λ 1 > 1,   λ 2 > 1, where θ t 1, t 2 f is a nonconvolution kernel defined on R m + n . In this paper we show that the biparameter Littlewood-Paley function g λ 1, λ 2 ⁎ is bounded from L 2 R n + m to L 2 R n + m . This is done by means of probabilistic methods and by using a new averaging identity over good double Whitney regions.