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Discretization and antidiscretization of Lorentz norms with no restrictions on weights

Abstract

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.

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Acknowledgements

The authors would like to thank the anonymous referees for their remarks and suggestions, which have led to improvements of the final version of the article.

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Correspondence to Zdeněk Mihula.

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The first and second authors were supported by the project OPVVV CAAS CZ.02.1.01/0.0/0.0/16_019/0000778. The second and third authors were supported by the Grant P201-18-00580S of the Czech Science Foundation, by the Grant SVV-2020-260583, and by Charles University Research program No. UNCE/SCI/023.

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Křepela, M., Mihula, Z. & Turčinová, H. Discretization and antidiscretization of Lorentz norms with no restrictions on weights. Rev Mat Complut 35, 615–648 (2022). https://doi.org/10.1007/s13163-021-00399-7

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  • DOI: https://doi.org/10.1007/s13163-021-00399-7

Keywords

  • Rearrangement-invariant spaces
  • Weights
  • Discretization
  • Lorentz spaces
  • Embeddings

Mathematics Subject Classification

  • 46E30
  • 46E35