Variation and λ-Jump Inequalities on Hp Spaces

Abstract

Let \(\phi \in \mathcal{S}\) with \(\int \phi (x){\kern 1pt} dx = 1\), and define \({{\phi }_{t}}(x) = \frac{1}{{{{t}^{n}}}}\phi \left( {\frac{x}{t}} \right),\) and denote the function family \({{\{ {{\phi }_{t}} * f(x)\} }_{{t > 0}}}\) by \(\Phi * f(x)\). Let \(\mathcal{J}\) be a subset of \(\mathbb{R}\) (or more generally an ordered index set), and suppose that there exists a constant \({{C}_{1}}\) such that \(\sum\limits_{t \in \mathcal{J}} {\kern 1pt} {\kern 1pt} {\text{|}}{{\hat {\phi }}_{t}}(x){{{\text{|}}}^{2}} < {{C}_{1}}\) for all \(x \in {{\mathbb{R}}^{n}}\). Then

 (i) There exists a constant \({{C}_{2}} > 0\) such that \({\text{||}}{{\mathcal{V}}_{2}}(\Phi * f){\text{|}}{{{\text{|}}}_{{{{L}^{p}}}}} \leqslant {{C}_{2}}{\text{||}}f{\text{|}}{{{\text{|}}}_{{{{H}^{p}}}}},\quad \frac{n}{{n + 1}} < p \leqslant 1\) for all \(f \in {{H}^{p}}({{\mathbb{R}}^{n}})\), \(\frac{n}{{n + 1}} < p \leqslant 1\).

 (ii) The λ-jump operator \({{N}_{\lambda }}(\Phi * f)\) satisfies \({\text{||}}\lambda {{[{{N}_{\lambda }}(\Phi * f)]}^{{1/2}}}{\text{|}}{{{\text{|}}}_{{{{L}^{p}}}}} \leqslant {{C}_{3}}{\text{||}}f{\text{|}}{{{\text{|}}}_{{{{H}^{p}}}}},\quad \frac{n}{{n + 1}} < p \leqslant 1,\) uniformly in \(\lambda > 0\) for some constant \({{C}_{3}} > 0\).