Abstract
We study Fubini-type property for Lorentz spaces \({L}^{p,r}({\mathbb{R}}^{2})\). This problem is twofold. First we assume that all linear sections of a function f in directions of coordinate axes belong to \({L}^{p,r}(\mathbb{R})\), and their one-dimensional L p, r-norms belong to \({L}^{p,r}(\mathbb{R}).\) We show that for p ≠ r it does not imply that \(f \in {L}^{p,r}({\mathbb{R}}^{2})\) (this complements one result by Cwikel). Conversely, we assume that \(f \in {L}^{p,r}({\mathbb{R}}^{2})\), and we show that then for r < p almost all linear sections of f belong to \({L}^{p,r}(\mathbb{R})\), but for p < r all linear sections may have infinite one-dimensional L p, r-norms.
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The author is grateful to the referee for his/her useful remarks.
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Dedicated to Professor Konstantin Oskolkov on the occasion of his 65th birthday.
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Kolyada, V.I. (2012). On Fubini Type Property in Lorentz Spaces. In: Bilyk, D., De Carli, L., Petukhov, A., Stokolos, A., Wick, B. (eds) Recent Advances in Harmonic Analysis and Applications. Springer Proceedings in Mathematics & Statistics, vol 25. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4565-4_16
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DOI: https://doi.org/10.1007/978-1-4614-4565-4_16
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