Inequalities for Fourier transforms

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.

show abstract

“…Although probably not best possible, in the case q = 2, the constant K q can be taken to be K q = 2, see the proof of Theorem 4.6 in [36].…”

Section: Theorem Bmentioning

confidence: 99%

Weighted Fourier Inequalities: New Proofs and Generalizations

Benedetto

Heinig

2003

Journal of Fourier Analysis and Applications

View full text Add to dashboard Cite

show abstract

“…Proof: Since T if of type (1, ∞) and (2, 2), [8,Theorem 4.6] shows that there exists a D depending only on T such that (…”

Section: Sufficient Conditionsmentioning

confidence: 99%

“…We say it is of type (1, ∞) and (2, 2). In [8], Jodeit and Torchinsky showed that a map T is of type (1, ∞) and (2, 2) if and only if there is a constant D such that…”

Section: Introductionmentioning

confidence: 99%

The Fourier transform in weighted Lorentz spaces

Sinnamon¹

2003

Publicacions Matemàtiques

View full text Add to dashboard Cite

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).

show abstract

“…The modular inequalities, like the norm inequalities, take an important role in analysis. For instance, the modular inequalities for the Fourier transform were obtained in [22]; modular versions of the Hardy-Littlewood maximal inequalities were discussed in [23] and [2]; Carro et al proved the modular inequalities for the Calderón operators and the Hardy operators in [6]. Most recently, the modular inequalities for the Laplace transform, the Hankel transform and the oscillatory integral operators were investigated in [13].…”

Section: Introductionmentioning

confidence: 99%