Relations between bilinear multipliers on ℝ n , % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC % vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz % ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqipC0xg9qqqrpepC0xbb % L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe % pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam % aaeaqbaaGcbaWefv3ySLgznfgDOjdarCqr1ngBPrginfgDObcv39ga % iyaacqWFtcpvaaa!4660! $$ \mathbb{T} $$ n and ℤ n

Bose, Debashish; Madan, Shobha; Mohanty, Parasar; Shrivastava, Saurabh

doi:10.1007/s12044-009-0038-8

Cited by 4 publications

(4 citation statements)

References 13 publications

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“…In the year of 1985, Rubio de Francia [18] proved above inequality (4) for the case of arbitrary sequence of disjoint intervals in R. In both the previous results p ≥ 2 is an optimal condition.…”

Section: Introductionmentioning

confidence: 85%

“…For this we will use the ideas developed in [11], adapted to our setting. We would like to refer the reader to [4], [2], [3], [8] for some work related to transference techniques in the bilinear settings.…”

Section: Relation Between B P (R) and L P (R) Multipliersmentioning

confidence: 99%

“…He proved the boundedness of a version of non-smooth bilinear LittlewoodPaley operator. In particular, he obtained the bilinear analogue of Carleson's Littlewood-Paley inequality (4) …”

Section: Introductionmentioning

confidence: 99%

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Bilinear Littlewood-Paley for circle and transference

Mohanty¹,

Shrivastava²

2011

Publicacions Matemàtiques

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Section: Introductionmentioning

confidence: 85%

Section: Relation Between B P (R) and L P (R) Multipliersmentioning

confidence: 99%

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Bilinear Littlewood-Paley for circle and transference

Mohanty¹,

Shrivastava²

2011

Publicacions Matemàtiques

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“…We will use a result proved in [1], where it is shown that the periodic extension of a compactly supported bilinear symbol is again a bilinear symbol for a bounded operator. …”

Section: Erratamentioning

confidence: 99%

Jodeit's extensions for bilinear multipliers (Bull. London Math Soc. 40 (2008) 937-944)

Madan

Mohanty²

2013

Bulletin of the London Mathematical Society

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We are writing this note to correct the above-mentioned paper and provide errata for some of the calculations in the paper. Unfortunately, these shortcomings were overlooked by the authors. Errata(1) Page 941, line 12 should be replaced by Observe that in the above proof, essentially we have proved the boundedness of the following extension, given by the symbolwhere1 2 ]ξ η}. First, the set P n should be written asFurther, we do not know if this assertion is true. Indeed, if this were so, and Theorem 3.5 held as stated, these would then give us another proof of the boundedness of the bilinear Hilbert transform for p 3 1. To be precise, consider φ n = δ 0 (n) where δ 0 is Dirac mass at 0. Thenwhich is the case for p 3 1 (it is the Fourier transform of the constant function 1). Hence, by dilation one can get that χ {(ξ,η):ξ>η} belongs to M p3 p1,p2 (R). Next, Theorems 3.3 and 3.5 can be restated with a suitable, but crucial modification in the constants. The constants now depend on the L 1 norms of the kernels. In order to prove this, we use Lacey's and Thiele's result [2]. We restate these theorems and give the proof in detail.

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A class of bilinear multipliers given by Littlewood–Paley decomposition

Shrivastava

2016

Journal of Mathematical Analysis and Applications

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In this paper we study a particular class of bilinear multipliers which are given by Littlewood-Paley decompositions. In the first part of the paper, we show that if φ(ξ − η) is a bilinear multiplier for (p, q, r), 1 ≤ p, q ≤ ∞ satisfying the Hölder's condition 1 p + 1 q = 1 r and have support inside [0, 1), then its periodization φ (ξ) = j∈Z φ(ξ − j) is also a bilinear multiplier for the same triplet (p, q, r). Further, we show that for a given triplet (p, q, r) of exponents outside the local L 2 -range, there exists sequence {φ j } j∈Z of uniformly bounded bilinear multipliers so that the function σ(ξ) = j∈Z φ j (ξ) is not a bilinear multiplier for the triplet (p, q, r). In the second part, we describe several results for bilinear multipliers of the type m (ξ, η) which are similar to the first part in nature. In particular, we point out that the results described by P. Honzik (2014) [13] can be generalized to a more general setting.

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Abstract: In this paper we prove the bilinear analogue of de Leeuw's result for periodic bilinear multipliers and some Jodeit type extension results for bilinear multipliers.

Cited by 4 publications

References 13 publications

Bilinear Littlewood-Paley for circle and transference

Bilinear Littlewood-Paley for circle and transference

Jodeit's extensions for bilinear multipliers (Bull. London Math Soc. 40 (2008) 937-944)

A class of bilinear multipliers given by Littlewood–Paley decomposition

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