Illinois Journal of Mathematics

Transference of bilinear multiplier operators on Lorentz spaces

Oscar Blasco and Francisco Villarroya

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Abstract

Let $m(\xi,\eta)$ be a bounded continuous function in $\mathbb{R}\times\mathbb{R}$, let $0< p_i,q_i<\infty$ for $i=1,2$, and let $0<p_3,q_3\le\infty$, be such that $1/p_1+1/p_2=1/p_3$. It is shown that

\[ C_m (f,g)(x)=\int_{\mathbb{R}} \int_{\mathbb{R}} \hat f(\xi) \hat g(\eta) m(\xi,\eta) e^{2\pi i x(\xi +\eta )}d\xi d\eta \]

is a bounded bilinear operator from $L^{p_1,q_1}(\mathbb{R})\times L^{p_2,q_2}(\mathbb{R})$ into $L^{p_3,q_3}(\mathbb{R})$ if and only if

\[ P_{D_{\varepsilon^{-1}}m} (f,g)(\theta)=\sum_{k\in \mathbb{Z}} \sum_{k'\in \mathbb{Z}} \hat f(k) \hat g(k') m(\varepsilon k, \varepsilon k') e^{2\pi i \theta(k +k' )} \]

are bounded bilinear operators from $L^{p_1,q_1}(\T)\times L^{p_2,q_2}(\T)$ into $L^{p_3,q_3}(\T)$ with norm bounded by a uniform constant for all $\epsilon >0$.

Article information

Source
Illinois J. Math., Volume 47, Number 4 (2003), 1327-1343.

Dates
First available in Project Euclid: 13 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1258138107

Digital Object Identifier
doi:10.1215/ijm/1258138107

Mathematical Reviews number (MathSciNet)
MR2037006

Zentralblatt MATH identifier
1056.42010

Subjects
Primary: 42A45: Multipliers

Citation

Blasco, Oscar; Villarroya, Francisco. Transference of bilinear multiplier operators on Lorentz spaces. Illinois J. Math. 47 (2003), no. 4, 1327--1343. doi:10.1215/ijm/1258138107. https://projecteuclid.org/euclid.ijm/1258138107


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