Weighted Fourier Inequalities in Lebesgue and Lorentz Spaces
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(2020) 26:57
Weighted Fourier Inequalities in Lebesgue and Lorentz Spaces Erlan Nursultanov1,2 · Sergey Tikhonov3,4,5 Received: 5 December 2019 / Revised: 15 May 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, we obtain sufficient conditions for the weighted Fourier-type transforms to be bounded in Lebesgue and Lorentz spaces. Two types of results are discussed. First, we review the method based on rearrangement inequalities and the corresponding Hardy’s inequalities. Second, we present Hörmander-type conditions on weights so that Fourier-type integral operators are bounded in Lebesgue and Lorentz spaces. Both restricted weak- and strong-type results are obtained. In the case of regular weights necessary and sufficient conditions are given. Keywords Fourier transforms · Weights · Lebesgue and Lorentz spaces · Integral operators · Rearrangements · Hardy inequalities · Hörmander-type conditions Mathematics Subject Classification Primary 42B10 · Secondary 46E30 · 42B35
Communicated by Hans G. Feichtinger. The research was partially supported by MTM 2017-87409-P, 2017 SGR 358, the CERCA Programme of the Generalitat de Catalunya, and the Ministry of Education and Science of the Republic of Kazakhstan (AP051 32071, AP051 32590).
B
Sergey Tikhonov [email protected] Erlan Nursultanov [email protected]
1
Lomonosov Moscow State University (Kazakh Branch), Gumilyov Eurasian National University, Munatpasova 7, 010010 Astana, Kazakhstan
2
Institute of Mathematics and Mathematical Modelling, 050010 Almaty, Kazakhstan
3
Centre de Recerca Matemàtica, Campus de Bellaterra, Edifici C, 08193 Bellaterra (Barcelona), Spain
4
ICREA, Pg. Lluís Companys 23, 08010 Barcelona, Spain
5
Universitat Autònoma de Barcelona, Barcelona, Spain 0123456789().: V,-vol
57
Page 2 of 29
Journal of Fourier Analysis and Applications
(2020) 26:57
1 Introduction We will let f (ξ ) = Rn ei xξ f (x) d x, ξ ∈ Rn , be the Fourier transform in L 1 (Rn ), and · p be the norm in L p (Rn ). Let ν and ω be nonnegative weights. A weighted Fourier inequality states that ν f q ≤ Cω f p , that is, Rn
| f (ξ )ν(ξ )|q dξ
1 q
≤C
Rn
| f (x)ω(x)| p d x
1
p
.
(1.1)
Throughout this paper, F G means that F ≤ C G; by C we denote positive constants depending only on non-essential parameters that may be different on different occasions. Moreover, F G means that F G F. The standard method to obtain sufficient conditions for (1.1) ∞ is to use the Hardy– Littlewood–Pólya inequality, which claims that Rn f g ≤ 0 f ∗ g ∗ . Here h ∗ is the decreasing rearrangements of h, i.e., where m(σ, f ) = {x ∈ Rn : | f (x)| > σ }.
h ∗ (t) = inf{σ : m(σ, h) ≤ t}, In more detail, denote
μ(x) =
1 . ω(x)
(1.2)
Then by the Hardy–Littlewood–Pólya inequality the estimate
∞
( f )∗ (t)q ν ∗ (t)q dt
1 q
≤C
0
∞
f ∗ (t) p
0
1 1 p dt μ∗ (t) p
(1.3)
implies (1.1) for any 0 < p, q < ∞, with the same constant C. In its turn, the usual way to obtain sufficient conditions
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