Majorant problems for trigonometric series

Boas, R. P.

doi:10.1007/bf02790309

Cited by 31 publications

(43 citation statements)

References 4 publications

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“…The failure of the majorization property for p / ∈ 2N was shown by Boas [8] (see also [7] for arbitrarily large constants, and also [16,22] for further comments and similar results in other groups.) Montgomery conjectured that it fails also if we restrict to majorants belonging to P, see [24, p. 144].…”

Section: Introductionmentioning

confidence: 62%

Integral concentration of idempotent trigonometric polynomials with gaps

Bonami¹,

Révész²

2009

ajm

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Abstract. We prove that for all p > 1/2 there exists a constant γ p > 0 such that, for any symmetric measurable set of positive measure E ⊂ T and for any γ < γ p , there is an idempotent trigonometrical polynomial f satisfying E |f | p > γ T |f | p . This disproves a conjecture of Anderson, Ash, Jones, Rider and Saffari, who proved the existence of γ p > 0 for p > 1 and conjectured that it does not exists for p = 1.Furthermore, we prove that one can take γ p = 1 when p > 1 is not an even integer, and that polynomials f can be chosen with arbitrarily large gaps when p = 2. This shows striking differences with the case p = 2, for which the best constant is strictly smaller than 1/2, as it has been known for twenty years, and for which having arbitrarily large gaps with such concentration of the integral is not possible, according to a classical theorem of Wiener.We find sharper results for 0 < p ≤ 1 when we restrict to open sets, or when we enlarge the class of idempotent trigonometric polynomials to all positive definite ones. (2000): Primary 42A05. Secondary 42A16, 42A61, 42A55, 42A82, 42B05. Mathematics Subject Classification

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Section: Introductionmentioning

confidence: 62%

Integral concentration of idempotent trigonometric polynomials with gaps

Bonami¹,

Révész²

2009

ajm

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“…This fact is sometimes referred to as the failure of the upper Hardy-Littlewood majorant property for L p -norms when p ∈ 2N ∪ {∞}. Indeed, the existence of such functions f can be inferred from known counterexamples on the circle group T given by trigonometric polynomials P ∈ L 2 (T); see [3,31,32]. If we take f (x) = λ 1/2p e −πλx 2 P (x) and choose 0 < λ ≪ 1 sufficiently small, we can produce the desired examples f ∈ S(R d ) for d = 1.…”

Section: Indeed a Corresponding Version Of Theorem 1 Remains Truementioning

confidence: 99%

A Sharp Rearrangement Principle in Fourier Space and Symmetry Results for PDEs with Arbitrary Order

Lenzmann

Sok

2020

International Mathematics Research Notices

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We prove sharp inequalities for the symmetric-decreasing rearrangement in Fourier space of functions in R d . Our main result can be applied to a general class of (pseudo-)differential operators in R d of arbitrary order with radial Fourier multipliers. For example, we can take any positive power of the Laplacian (−∆) s with s > 0 and, in particular, any polyharmonic operator (−∆) m with integer m 1. As applications, we prove radial symmetry and real-valuedness (up to trivial symmetries) of optimizers for: i) Gagliardo-Nirenberg inequalities with derivatives of arbitrary order, ii) ground states for bi-and polyharmonic NLS, and iii) Adams-Moser-Trudinger type inequalities for H d/2 (R d ) in any dimension d 1. As a technical key result, we solve a phase retrieval problem for the Fourier transform in R d . To achieve this, we classify the case of equality in the corresponding Hardy-Littlewood majorant problem for the Fourier transform in R d . Applications: Radial Symmetry of OptimizersIn this section, we discuss some applications of Theorem 1 to show radial symmetry of optimizers. More precisely, we will consider the following examples.• Gagliardo-Nirenberg type inequalities with derivatives of arbitrary order.• Ground states for generalized NLS type (e. g. fourth-order NLS).

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“…It was thus quite surprising that Peller [3] announced that (3) fails for some infinite matrices whenever p is not an even integer. In correspondence, Peller described his counterexample which relies on his beautiful but elaborate theory of ^ Hankel operators (4) and on a paper of Boas (2). It follows from Peller's example that (3) must fail for some finite N but it is not clear for which N. Our purpose here is to give explicit N and to avoid the complications of Peller's ^-Hankel theory.…”

Section: Pointwise Domination Of Matrices and Comparison Of ^ Normsmentioning

confidence: 99%

Pointwise domination of matrices and comparison of ℐ_pnorms

Simon¹

1981

Pacific J. Math.

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Majorant problems for trigonometric series

Cited by 31 publications

References 4 publications

Integral concentration of idempotent trigonometric polynomials with gaps

Integral concentration of idempotent trigonometric polynomials with gaps

A Sharp Rearrangement Principle in Fourier Space and Symmetry Results for PDEs with Arbitrary Order

Pointwise domination of matrices and comparison of ℐ_pnorms

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Majorant problems for trigonometric series

Cited by 31 publications

References 4 publications

Integral concentration of idempotent trigonometric polynomials with gaps

Integral concentration of idempotent trigonometric polynomials with gaps

A Sharp Rearrangement Principle in Fourier Space and Symmetry Results for PDEs with Arbitrary Order

Pointwise domination of matrices and comparison of ℐpnorms

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Pointwise domination of matrices and comparison of ℐ_pnorms