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A note on Lorentz–Zygmund spaces

A note on Lorentz–Zygmund spaces Let 1 ≤≤ q < p < ∞, and ℝℝ be the real line. Hörmander showed that any bounded linear translation invariant operator from L p (ℝℝ) to L q (ℝℝ) is trivial. Blozinski obtained an analogy to Hörmander in Lorentz spaces on the real line. In this paper, we generalize Blozinski's result in Lorentz–Zygmund spaces. Also, Bochkarev proved an inequality related to the Hausdorff–Young–Riesz theorem in Lorentz spaces, and the sharpness of the inequality. We improve Bochkarev's inequality in Lorentz–Zygmund spaces, and prove the sharpness of our inequality. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Georgian Mathematical Journal de Gruyter

A note on Lorentz–Zygmund spaces

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References (15)

Publisher
de Gruyter
Copyright
© de Gruyter 2011
ISSN
1072-947X
eISSN
1072-9176
DOI
10.1515/GMJ.2011.0035
Publisher site
See Article on Publisher Site

Abstract

Let 1 ≤≤ q < p < ∞, and ℝℝ be the real line. Hörmander showed that any bounded linear translation invariant operator from L p (ℝℝ) to L q (ℝℝ) is trivial. Blozinski obtained an analogy to Hörmander in Lorentz spaces on the real line. In this paper, we generalize Blozinski's result in Lorentz–Zygmund spaces. Also, Bochkarev proved an inequality related to the Hausdorff–Young–Riesz theorem in Lorentz spaces, and the sharpness of the inequality. We improve Bochkarev's inequality in Lorentz–Zygmund spaces, and prove the sharpness of our inequality.

Journal

Georgian Mathematical Journal – de Gruyter

Published: Sep 1, 2011

Keywords: Hausdorff–Young theorem; Lorentz–Zygmund space

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