The Fourier transform in weighted Lorentz spaces

Abstract

Necessary conditions and sufficient conditions on weights u and w are given for the Fourier transform F to be bounded as a map between the Lorentz spaces Γq(w) and Λp(u). This may be viewed as a weighted extension of a result of Jodeit and Torchinsky on operators of type (1, ∞) and (2, 2). In the case 0 < p ≤ 2 = q, the necessary and sufficient conditions are equivalent and give a simple weight condition which is equivalent to F : Γ 2 (w) → Λp(u) and also to F : Γ 2 (w) → Γp(u).

Weighted Fourier Inequalities: New Proofs and Generalizations

“…Section 3 is devoted to an exposition of Lorentz spaces and to the work of Sawyer [50] which we shall use in Section 4. We do not use the results of Flett [16] from 1973 on the classical Lorentz spaces L(p, q), nor do we use the comparably beautiful recent results of Sinnamon [52]; on the other hand, their theories do complement our approach, e. g., see the first paragraph of Section 4. We close Section 3 with a remark on Köthe spaces, which can be considered a formulation in topological vector spaces of a natural generalization of Lorentz spaces.…”

“…Because of Theorem C and Lorentz' theorem, stated in Remark 1, these weighted Lorentz spaces are in the Banach space setting. Sinnamon's work [52], referenced earlier, provides different Fourier mapping theorems by foregoing any Banach space structure.…”

Generalizations of the Riesz convergence theorem for Lorentz spaces

“…Thus by Theorem A above, we have f − f n k i → 0 as i → ∞. This contradicts (16) and the sufficient condition has been proved.…”

Symmetric conditions for a weighted Fourier transform inequality

“…Very little concerning this problem was known before the early 1980s. Notable results can be found in Muckenhoupt [8], Aguilera and Harboure [1], Jurkat and Sampson [7], Benedetto and Heinig [2], Benedetto, Heinig, and Johnson [4,5], and Sinnamon [9,10]. A survey of the subject, which includes new proofs and generalizations, can be found in Benedetto and Heinig [3].…”