Boundedness of Littlewood-Paley Operators and Marcinkiewicz Integral on εa,p

Abstract:

      In this paper, we shall show that if ∈ε^a,p, then either g(f)(x)(s(f)(x),g_λ^*(f)(x),μ(f)(x))<∞ almost everywhere, or g(f)(x)(s(f)(x),g_λ^*(f)(x),μ(f)(x))<∞ almost everywhere. Furthermore, if g(f)(x)(s(f)(x),g_λ^*(f)(x),μ(f)(x))<∞ almost everywhere,then g(f)(s(f),g_λ^*(f),μ(f))∈ε^a,p and there is a constant C independent of f and x, so that ‖g(f)‖_a,p,(‖s(f)‖_a,p,‖g_λ^*(f)‖_a,p,‖μ(f)‖_a,p)≤C‖f‖_a,p.