On the Failure of Multilinear Multiplier Theorem with Endpoint Smoothness Conditions

Park, Bae Jun

doi:10.1007/s11118-020-09877-x

Cited by 8 publications

(4 citation statements)

References 6 publications

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“…We remark that the condition (1.9) is sharp in the sense that if the condition does not hold, then there exists σ such that the estimate (1.10) fails for the corresponding operators T σ . Refer to [21] for more details.…”

Section: Introductionmentioning

confidence: 99%

On the boundedness of multilinear Fourier multipliers on Hardy spaces

Lee¹,

Park²

2022

Preprint

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In this paper, we study multilinear Fourier multiplier operators on Hardy spaces. In particular, we prove that the multilinear Fourier multiplier operator of Hörmander type is bounded from H p 1 × • • • × H pm to H p for 0 < p1, . . . , pm ≤ 1 with 1/p1 + • • • 1/pm = 1/p, under suitable cancellation conditions. As a result, we extend the trilinear estimates in [17] to general multilinear ones and improve the boundedness result in [18] in limiting situations.

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Section: Introductionmentioning

confidence: 99%

On the boundedness of multilinear Fourier multipliers on Hardy spaces

Lee¹,

Park²

2022

Preprint

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“…Here, the space L 2 (s 1 ,...,sm) ((R n ) m ) indicates the product type Sobolev space on (R n ) m , in which the norm is defined by replacing the term (1+4π 2 | ξ| 2 ) s in (1.6) by m j=1 1+4π 2 |ξ j | 2 s j . It is known in [27] that the condition (1.8) is sharp in the sense that if the condition does not hold, then there exists σ such that the corresponding operator T σ does not satisfy (1.10). We also refer the reader to [6,11] for weighted estimates for multilinear Fourier multipliers.…”

Section: Introductionmentioning

confidence: 99%

Trilinear Fourier multipliers on Hardy spaces

Lee¹,

Park²

2021

Preprint

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In this paper, we obtain the H p 1 × H p 2 × H p 3 → H p boundedness for trilinear Fourier multiplier operators, which is a trilinear analogue of the multiplier theorem of Calderón and Torchinsky [4]. Our result improves the trilinear estimate in [22] by additionally assuming an appropriate vanishing moment condition, which is natural in the boundedness into the Hardy space H p for 0 < p ≤ 1.

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“…Necessary condition. In order to prove the direction (1.8) ⇒ (1.7) in Theorem 1.1, two different multipliers will be constructed based on an idea contained in [25]. However, the methods in [25] essentially rely on Plancherel's theorem to obtain the upper bound of…”

Section: Introductionmentioning

confidence: 99%

Characterization of multilinear multipliers in terms of Sobolev space regularity

Grafakos¹,

Park²

2020

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We provide necessary and sufficient conditions for multilinear multiplier operators with symbols in L r -based product-type Sobolev spaces uniformly over all annuli to be bounded from products of Hardy spaces to a Lebesgue space. We consider the case 1 < r ≤ 2 and we characterize boundedness in terms of inequalities relating the Lebesgue indices (or Hardy indices), the dimension, and the regularity and integrability indices of the Sobolev space. The case r > 2 cannot be handled by known techniques and remains open. Our result not only extends but also establishes the sharpness of previous results of Miyachi, Nguyen, Tomita, and the first author [13,14,15,23], who only considered the case r = 2.

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On the Failure of Multilinear Multiplier Theorem with Endpoint Smoothness Conditions

Cited by 8 publications

References 6 publications

On the boundedness of multilinear Fourier multipliers on Hardy spaces

On the boundedness of multilinear Fourier multipliers on Hardy spaces

Trilinear Fourier multipliers on Hardy spaces

Characterization of multilinear multipliers in terms of Sobolev space regularity

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