Boundedness of higher‐order Marcinkiewicz‐Typeintegrals

Let A be a function with derivatives of order m and . 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 and its variation 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 and are bounded on Hardy spaces, and is also bounded from Lp(ℝn) to certain Triebel‐Lizorkin space.