Boundedness of higher-order Marcinkiewicz-Type integrals

Abstract

Let A be a function with derivatives of order m and DγA∈Λ˙β(0<β<1,|γ|=m). The authors in the paper proved that if Ω∈Ls(Sn−1) (s≥n/(n−β)) is homogeneous of degree zero and satisfies a vanishing condition, then both the higher-order Marcinkiewicz-type integral μΩA and its variation μ˜ΩA are bounded from Lp(ℝn) to Lq(ℝn) and from L1(ℝn) to Ln/(n−β),∞(ℝn), where 1<p<n/β and 1/q=1/p−β/n. Furthermore, if Ω satisfies some kind of Ls-Dini condition, then both μΩA and μ˜ΩA are bounded on Hardy spaces, and μΩA is also bounded from Lp(ℝn) to certain Triebel-Lizorkin space.