Weighted Fourier Inequalities: New Proofs and Generalizations

Abstract

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.

Weighted Fourier inequalities: Boas’ conjecture in ℝ n

“…Relation (5) was proved in [20]; see also [12]. Thus, assuming a function to be monotone allows one to extend the range of γ as well as to reverse inequality (2) for p = q.…”

“…For n = 1, inequality (2) can be found in [5], [17], [18], [22]; for n ≥ 1, see [3], [5]. W. Beckner [2] found the sharp constant in (2) for p = q = 2 and used this to prove a logarithmic estimate for uncertainty.…”

Study on an Intelligent Inference Engine in Early-Warning System of Dam Health

“…3 can be obtained by the time-frequency analysis of trained data. This system determines the unknown weight coefficients with the forgetting factors method (Benedetto and Heinig 2003;Enns and Si 2002;Razavi and Araghinejad 2009).…”

On the Fourier transform of the symmetric decreasing rearrangements

“…[15,16,2] and the references therein). Let us mention that a weighted inequality for the Fourier transform was proved in [2] with the help of a result of Jodeit-Torchinsky [14] showing that an operator that is of type (1, ∞) and of type (2, 2) satisfies some rearrangement inequalities.…”