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Discretization and antidiscretization of Lorentz norms with no restrictions on weights
Revista Matemática Complutense ( IF 0.8 ) Pub Date : 2021-06-22 , DOI: 10.1007/s13163-021-00399-7
Martin Křepela , Zdeněk Mihula , Hana Turčinová

We improve the discretization technique for weighted Lorentz norms by eliminating all “non-degeneracy” restrictions on the involved weights. We use the new method to provide equivalent estimates on the optimal constant C such that the inequality

$$\begin{aligned} \left( \int _0^L (f^*(t))^{q} w(t)\,\mathrm {d} t\right) ^\frac{1}{q} \le C \left( \int _0^L \left( \int _0^t u(s)\,\mathrm {d} s\right) ^{-p} \left( \int _0^t f^*(s) u(s) \,\mathrm {d} s\right) ^p v(t) \,\mathrm {d} t\right) ^\frac{1}{p} \end{aligned}$$

holds for all relevant measurable functions, where \(L\in (0,\infty ]\), \(p, q \in (0,\infty )\) and u, v, w are locally integrable weights, u being strictly positive. In the case of weights that would be otherwise excluded by the restrictions, it is shown that additional limit terms naturally appear in the characterizations of the optimal C. A weak analogue for \(p=\infty \) is also presented.



中文翻译:

对权重没有限制的洛伦兹范数的离散化和反离散化

我们通过消除对相关权重的所有“非简并性”限制来改进加权洛伦兹范数的离散化技术。我们使用新方法对最优常数C提供等效估计,使得不等式

$$\begin{aligned} \left( \int _0^L (f^*(t))^{q} w(t)\,\mathrm {d} t\right) ^\frac{1}{q } \le C \left( \int _0^L \left( \int _0^tu(s)\,\mathrm {d} s\right) ^{-p} \left( \int _0^tf^*( s) u(s) \,\mathrm {d} s\right) ^pv(t) \,\mathrm {d} t\right) ^\frac{1}{p} \end{aligned}$$

对所有相关的可测函数都成立,其中\(L\in (0,\infty ]\) , \(p, q \in (0,\infty )\)u , v , w是局部可积权重,u是严格为正。在权重将被限制排除在外的情况下,表明附加限制项自然出现在最佳C的表征中。还提供了\(p=\infty \) 的弱模拟。

更新日期:2021-06-23
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