Multilinear Littlewood–Paley–Stein Operators on Non-homogeneous Spaces

Let m ≥ 2 , λ > 1 and define the multilinear Littlewood–Paley–Stein operators by g λ , μ ∗ ( f → ) ( x ) = ( ∬ R + n + 1 ( t t + | x - y | ) m λ | ∫ ( R n ) κ s t ( y , z → ) ∏ i = 1 κ f i ( z i ) d μ ( z 1 ) ⋯ d μ ( z κ ) | 2 d μ ( y ) d t t m + 1 ) 1 / 2 . In this paper, our main aim is to investigate the boundedness of g λ , μ ∗ on non-homogeneous spaces. By means of probabilistic and dyadic techniques, together with non-homogeneous analysis, we show that g λ , μ ∗ is bounded from L p 1 ( μ ) × ⋯ × L p κ ( μ ) to L p ( μ ) under certain weak type assumptions. The multilinear non-convolution type kernels s t only need to satisfy some weaker conditions than the standard conditions of multilinear Calderón–Zygmund type kernels and the measures μ are only assumed to be upper doubling measures (non-doubling). The above results are new even under Lebesgue measures. This was done by considering first a sufficient condition for the strong type boundedness of g λ , μ ∗ based on an endpoint assumption, and then directly deduce the strong bound on a big piece from the weak type assumptions.