Pitt inequalities and restriction theorems for the Fourier transform

Carli, Laura De; Горбачев, Д. В.; Tikhonov, Sergey

doi:10.4171/rmi/955

Cited by 19 publications

(16 citation statements)

References 27 publications

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“…Thus, there is a constant C such that w p (Q) ≤ C|Q| for all cubes Q. By (7), u(E) p ′ 0 /q0 w p0 (F ) ≤ C|E|F | for all bounded sets E and F . Let F = {x : v −1 (x) > β} where β > 0 is a fixed number so that F has positive measure.…”

Section: Definitionmentioning

confidence: 95%

The Weighted Fourier Inequality, Polarity, and Reverse Hölder Inequality

Berndt

2017

J Fourier Anal Appl

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1 where E is any bounded, measurable set containing the origin, and F is any translation of its geometric polar. In fact, the set E may be translated as well, and the sets E and F may even be interchanged. Further, we discuss that this class of weights is essentially the same if E and F are replaced with a rectanguloid and its polar, or an ellipsoid and its polar. Previously in [5], we examined this inequality where E and F were cubes of reciprocal measure. The current condition is a generalization because having reciprocal measure and being polar are essentially the same for cubes.Sufficiency. We assume a weaker version of the necessary condition plus a reverse Hölder condition to prove weighted restricted weak-type estimates for the Fourier transform. This is accomplished by comparing either the measure u or v to Lebesgue measure and then applying the Hausdorff-Young inequality. The result actually implies restricted weak-type for a range of p and q; we are then able to interpolate and obtain the (strong-type) weighted Fourier inequality.

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Section: Definitionmentioning

confidence: 95%

The Weighted Fourier Inequality, Polarity, and Reverse Hölder Inequality

Berndt

2017

J Fourier Anal Appl

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“…In what follows we assume u(x) = x −β ′ q , v(x) = x γp with β 1 − γ 1 = β 2 − γ 2 and s(x) = w(x) = x δ , δ > 0 in (1.4). Piecewise power weights have been considered for the study of weighted restriction Fourier inequalities [9,16], and moreover they play a fundamental role in the study of weighted norm inequalities for the Jacobi transform [24] (see also [28]). For the sake of generality, we first give sufficient conditions for (1.32) to hold, and then we also study necessary conditions for (1.33) to hold, i.e., with non-mixed power weights.…”

Section: Weighted Norm Inequalities For Transforms With Powertype Kernelmentioning

confidence: 99%

“…Important examples of applications of the above inequalities are the study of uncertainty principle relations (cf. [4]) or restriction inequalities [21,42,16]. Inequality (1.2) and its variants have been extensively studied, see [1,3,5,6,27,33] and the references therein.…”

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confidence: 99%

Weighted norm inequalities for generalized Fourier-type transforms and applications

Debernardi

2020

Publicacions Matemàtiques

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We obtain necessary and sufficient conditions on weights for the generalized Fourier-type transforms to be bounded between weighted L p − L q spaces. As an important example, we investigate transforms with kernel of power type, as for instance the sine, Hankel or Hα transforms. The obtained necessary and sufficient conditions are given in terms of weights, but not in terms of their decreasing rearrangements, as in several previous investigations.

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“…Power weights will play an important role in our study of the Hankel transform, while piecewise power weights will be important for the analysis of the Jacobi transform. Note that piecewise power weights were considered earlier in the study of weighted Fourier inequalities, see, e.g., [9,17]. 2 , and w(x) = (2/π)x 2 .…”

Section: Integral Transforms Of Fourier Typementioning

confidence: 99%

“…Another approach to obtain (1.14) with general weights has recently been considered in [17]. It is based on restriction inequalities for the Fourier transform on the unit sphere of R n (see, e.g., [9]).…”

Section: Integral Transforms Of Fourier Typementioning

confidence: 99%

Weighted norm inequalities for integral transforms

Горбачев¹,

Liflyand²,

Tikhonov³

2018

Indiana Univ. Math. J.

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Weighted (L p , L q) inequalities are studied for a variety of integral transforms of Fourier type. In particular, weighted norm inequalities for the Fourier, Hankel, and Jacobi transforms are derived from Calderón type rearrangement estimates. The obtained results keep their novelty even in the simplest cases of the studied transforms, the cosine and sine Fourier transforms. Sharpness of the conditions on weights is discussed.

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Pitt inequalities and restriction theorems for the Fourier transform

Cited by 19 publications

References 27 publications

The Weighted Fourier Inequality, Polarity, and Reverse Hölder Inequality

The Weighted Fourier Inequality, Polarity, and Reverse Hölder Inequality

Weighted norm inequalities for generalized Fourier-type transforms and applications

Weighted norm inequalities for integral transforms

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