Some examples of equivalent rearrangement‐invariant quasi‐norms defined via f∗$f^*$ or f∗∗$f^{**}$

Leo R. Ya. Doktorski; Pedro Fernández‐Martínez; Teresa Signes

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Some examples of equivalent rearrangement‐invariant quasi‐norms defined via f∗$f^*$ or f∗∗$f^{**}$

机译：Some examples of equivalent rearrangement‐invariant quasi‐norms defined via f∗$f^*$ or f∗∗$f^{**}$

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Abstract We consider Lorentz–Karamata spaces, small and grand Lorentz–Karamata spaces, and the so‐called L$mathcal {L}$, R$mathcal {R}$, LL$mathcal {LL}$, LR$mathcal {LR}$, RL$mathcal {RL}$, and RR$mathcal {RR}$ spaces. The quasi‐norms for a function f in each of these spaces can be defined via the nonincreasing rearrangement f∗$f^*$ or via the maximal function f∗∗$f^{**}$. We investigate when these quasi‐norms are equivalent. Most of the proofs are based on Hardy‐type inequalities. As an application, we demonstrate how our general results can be used to establish interpolation formulae for grand and small Lorentz–Karamata spaces.

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《Mathematische Nachrichten》 |2023年第5期|1781-1802|共22页
作者
Leo R. Ya. Doktorski; Pedro Fernández‐Martínez; Teresa Signes;
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System Technologies and Image Exploitation IOSB, Department Object Recognition, Gutleuthausstr. 1;

Campus de Espinardo;

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正文语种英语
中图分类数学;
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K‐functional; real interpolation; rearrangement invariant spaces; slowly varying functions;

Some examples of equivalent rearrangement‐invariant quasi‐norms defined via f∗$f^*$ or f∗∗$f^{**}$

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