Fourier transform inequalities in weighted Lebesgue spaces are proved. The inequalities are generalizations of the Plancherel theorem, they are characterized in terms of uncertainty principle relations between pairs of weights, and they are put in the context of existing weighted Fourier transform inequalities. The proofs are new and relatively elementary, and they give rise to good and explicit constants controlling the continuity of the Fourier transform operator. The smaller the constant is, the more applicable the inequality will be in establishing weighted uncertainty principle or entropy inequalities. There are two essentially different proofs, one depending on operator theory and one depending on Lorentz spaces. The results from these approaches are quantitatively compared, leading to classical questions concerning multipliers and to new questions concerning wavelets.

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Abstractcharacterization of the spaces dual to weighted Lorentz spaces are given by means of reverse Hölder inequalities (Theorems 2.1, 2.2). This principle of duality is then applied to characterize weight functions for which the identity operator, the Hardy-Littlewood maximal operator and the Hilbert transform are bounded on weighted Lorentz spaces.

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