A note on bi-linear multipliers
HTML articles powered by AMS MathViewer
- by Saurabh Shrivastava PDF
- Proc. Amer. Math. Soc. 143 (2015), 3055-3061 Request permission
Abstract:
In this paper we prove that if $\chi _{_E}(\xi -\eta )$ – the indicator function of a measurable set $E\subseteq \mathbb {R}^d$ – is a bi-linear multiplier symbol for exponents $p,q,r$ satisfying the Hölder’s condition $\frac {1}{p}+\frac {1}{q}=\frac {1}{r}$ and exactly one of $p,q,$ or $r’=\frac {r}{r-1}$ is less than $2,$ then $E$ is equivalent to an open subset of $\mathbb {R}^d.$References
- Oscar Blasco, Bilinear multipliers and transference, Int. J. Math. Math. Sci. 4 (2005), 545–554. MR 2172393, DOI 10.1155/IJMMS.2005.545
- Karel de Leeuw, On $L_{p}$ multipliers, Ann. of Math. (2) 81 (1965), 364–379. MR 174937, DOI 10.2307/1970621
- Loukas Grafakos and Rodolfo H. Torres, Multilinear Calderón-Zygmund theory, Adv. Math. 165 (2002), no. 1, 124–164. MR 1880324, DOI 10.1006/aima.2001.2028
- Michael Lacey and Christoph Thiele, $L^p$ estimates on the bilinear Hilbert transform for $2<p<\infty$, Ann. of Math. (2) 146 (1997), no. 3, 693–724. MR 1491450, DOI 10.2307/2952458
- Michael Lacey and Christoph Thiele, On Calderón’s conjecture, Ann. of Math. (2) 149 (1999), no. 2, 475–496. MR 1689336, DOI 10.2307/120971
- V. Lebedev and A. Olevskiĭ, Idempotents of Fourier multiplier algebra, Geom. Funct. Anal. 4 (1994), no. 5, 539–544. MR 1296567, DOI 10.1007/BF01896407
- Parasar Mohanty and Saurabh Shrivastava, A note on the bilinear Littlewood-Paley square function, Proc. Amer. Math. Soc. 138 (2010), no. 6, 2095–2098. MR 2596047, DOI 10.1090/S0002-9939-10-10233-0
- Parasar Mohanty and Saurabh Shrivastava, Bilinear Littlewood-Paley for circle and transference, Publ. Mat. 55 (2011), no. 2, 501–519. MR 2839453, DOI 10.5565/PUBLMAT_{5}5211_{1}1
Additional Information
- Saurabh Shrivastava
- Affiliation: Department of Mathematics, Indian Institute of Science Education and Research Bhopal, Indore By-pass road Bhauri, Bhopal-462066, India
- MR Author ID: 894393
- Email: saurabhk@iiserb.ac.in
- Received by editor(s): March 18, 2014
- Published electronically: March 18, 2015
- Communicated by: Alexander Iosevich
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 3055-3061
- MSC (2010): Primary 42A45, 42B15; Secondary 42B25
- DOI: https://doi.org/10.1090/S0002-9939-2015-12679-2
- MathSciNet review: 3336630