Hp 空间上的变分与λ-跳跃不等式

IF 0.5 Q3 MATHEMATICS Russian Mathematics Pub Date : 2024-08-06 DOI:10.3103/s1066369x24700233
S. Demir
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引用次数: 0

摘要

AbstractLet \(\phi \in \mathcal{S}\) with \(\int \phi (x){\kern 1pt} dx = 1\)、并定义 \({{\phi }_{t}}(x) = \frac{1}{{{{t}^{n}}}}\phi \left( {\frac{x}{t}} \right),\)and denote the function family\({{\ {{\phi }_{t}})* f(x)\}}_{{t > 0}}}) by \(\Phi * f(x)\).让 \(\mathcal{J}\) 是 \(\mathbb{R}}\) 的一个子集(或者更笼统地说,是一个有序索引集),并假设存在一个常数 \({{C}_{1}}\) 使得 \(\sum\limits_{t \in \mathcal{J}}){text{|}}{hat {\phi }}_{t}}(x){{{text{|}}}^{2}} < {{C}_{1}}}\)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) λ-jump 算子 \({{N}_{\lambda }}(\Phi * f)\) satisfies\({\text{||}}\lambda {{[{{N}_{\lambda }}(\Phi * f)]}^{1/2}}}}{{text{|}}}{{{\text{|}}}_{{{{L}^{p}}}}}\leqslant {{C}_{3}}{text{||}}f{text{|}}{{{\text{|}}}{_{{{{H}}^{p}}}}},\quad \frac{n}{{n + 1}} < p \leqslant 1,\)在某个常数 \({{C}_{3}} > 0\) 下均匀地在\(\lambda > 0\) 中。
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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 \({{C}_{3}} > 0\).

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来源期刊
Russian Mathematics
Russian Mathematics MATHEMATICS-
CiteScore
0.90
自引率
25.00%
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期刊介绍: Russian Mathematics  is a peer reviewed periodical that encompasses the most significant research in both pure and applied mathematics.
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