Variation and λ-jump inequalities on Hp spaces

Abstract

Let \phi \in S with \int \phi (x) dx = 1, and define \phi t(x) = 1 tn \phi \Bigl( x t \Bigr) , and denote the function family \{ \phi t\ast f(x)\} t>0 by \Phi \ast f(x). Let \scrJ be a subset of \BbbR (or more generally an ordered index set), and suppose that there exists a constant C1 such that \sum t\in \scrJ | \^\phi t(x)| 2 < C1 for all x \in \BbbR n. Then i) There exists a constant C2 > 0 such that \| V2(\Phi \ast f)\| Lp \leq C2\| f\| Hp, n n + 1 < p \leq 1 for all f \in Hp(\BbbR n), n n + 1 < p \leq 1. ii) The \lambda -jump operator N\lambda (\Phi \ast f) satisfies \| \lambda [N\lambda (\Phi \ast f)]1/2\| Lp \leq C3\| f\| Hp, n n + 1 < p \leq 1, uniformly in \lambda > 0 for some constant C3 > 0.