On a hybrid of bilinear Hilbert transform and paraproduct

Dong, Deshuai; Li, Xiao Chun

doi:10.1007/s10114-017-6415-9

Cited by 3 publications

(8 citation statements)

References 10 publications

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“…For the definition of the correlation degree, we refer the reader to P. 2 in [12]. There are many other related works; see, for example, [11,13,14,15,18,32].…”

Section: Background and Motivationmentioning

confidence: 99%

Bilinear Hilbert Transforms and (Sub)Bilinear Maximal Functions along Convex Curves

2020

Preprint

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“…For the definition of the correlation degree, we refer the reader to P. 2 in [12]. There are many other related works; see, for example, [11,13,14,15,18,32].…”

Section: Background and Motivationmentioning

confidence: 99%

Bilinear Hilbert Transforms and (Sub)Bilinear Maximal Functions along Convex Curves

2020

Preprint

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“…In this section, we prove Proposition 3.3. The proofs of some variants of this proposition already appear in [2] and [9]. For the convenience of the reader, we include the details here.…”

Section: Size Estimatesmentioning

confidence: 99%

“…Then the tiles associated with these functions have structures similar to that of the tri-tiles as in the study of BHT. Therefor, the proof of Theorem 1.1 given in [2] still applies to I, and we omit the details. This finishes the proof of Theorem 1.3.…”

Section: Telescopingmentioning

confidence: 99%

“…Remarks. (1) Strictly speaking, this theorem is proved in [2] only in the case α = β = 1, but this restriction is inessential. The proof given in [2] works for any homogeneous-scale case.…”

mentioning

confidence: 99%

“…The definition of tri-linear correspondence of Π Γ (f 1 , f 2 ) was given in [2], where the author and Li introduced the following class of operators T α,β that can be viewed a hybrid of BHT and paraproduct:…”

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confidence: 99%

See 2 more Smart Citations

Quasi Pieces of the Bilinear Hilbert Transform Incorporated into a Paraproduct

Dong

2018

J Geom Anal

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Quasi-linear analysis of dispersion relation preservation for nonlinear schemes

Pan

Qin

et al. 2022

Adv. Aerodyn.

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In numerical simulations of complex flows with discontinuities, it is necessary to use nonlinear schemes. The spectrum of the scheme used has a significant impact on the resolution and stability of the computation. Based on the approximate dispersion relation method, we combine the corresponding spectral property with the dispersion relation preservation proposed by De and Eswaran (J Comput Phys 218:398–416, 2006) and propose a quasi-linear dispersion relation preservation (QL-GRP) analysis method, through which the group velocity of the nonlinear scheme can be determined. In particular, we derive the group velocity property when a high-order Runge–Kutta scheme is used and compare the performance of different time schemes with QL-GRP. The rationality of the QL-GRP method is verified by a numerical simulation and the discrete Fourier transform method. To further evaluate the performance of a nonlinear scheme in finding the group velocity, new hyperbolic equations are designed. The validity of QL-GRP and the group velocity preservation of several schemes are investigated using two examples of the equation for one-dimensional wave propagation and the new hyperbolic equations. The results show that the QL-GRP method integrated with high-order time schemes can determine the group velocity for nonlinear schemes and evaluate their performance reasonably and efficiently.

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On a hybrid of bilinear Hilbert transform and paraproduct

Cited by 3 publications

References 10 publications

Bilinear Hilbert Transforms and (Sub)Bilinear Maximal Functions along Convex Curves

Bilinear Hilbert Transforms and (Sub)Bilinear Maximal Functions along Convex Curves

Quasi Pieces of the Bilinear Hilbert Transform Incorporated into a Paraproduct

Quasi-linear analysis of dispersion relation preservation for nonlinear schemes

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