拉昆序列平均数差的变异算子将 $$H_{w}^{1}(\mathbb{R})$$ 映射到 $$L_{w}^{1}(\mathbb{R})$$

AbstractLet \(f\) be a locally integrable function defined on \(\mathbb{R}\), and \(({{n}_{k}})\) be a lacunary sequence.定义（{{A}_{n}}f(x) = \frac{1}{n}\int_0^n f(x - t)dt、\)并让\({{math\cal{V}}_{\rho }}f(x) = {{\left( {\sum\limits_{k = 1}^^\infty {{{left| {{A}_{{{{n}_{k}}}}}f(x) - {{A}_{{{{n}_{k - 1}}}}}}f(x)} \right|}}^{\rho }}}\right)}^{{1/\rho }}}.\)Suppose that \(w \in {{A}_{p}}\),\(1 \leqslant p < \infty \), and\(\rho \geqslant 2\).然后，存在一个正常数（C），使得 \({{\left\| {{{\mathcal{V}}_{\rho }}f}\right\|}_{{L_{w}^{1}}}}\leqslant C{{left\| f \right\|}_{{H_{w}^{1}}}}\)for all \(f \in H_{w}^{1}(\mathbb{R})\).