A note on Lorentz–Zygmund spaces

Let 1 ≤ q < p < ∞, and ℝ be the real line. Hörmander showed that any bounded linear translation invariant operator from Lp (ℝ) to Lq (ℝ) 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.