×

Notes on the spaces of bilinear multipliers. (English) Zbl 1209.42007

A bounded operator from \(L^{P_1}(\mathbb{R}) \times L^{P_2}(\mathbb{R}) \rightarrow L^{P_3}(\mathbb{R})\) of the form \[ \displaystyle B (f,g) (x) = \int\limits_{\mathbb{R}^n}\int\limits_{\mathbb{R}^n}\hat{f}(\xi)\hat{g}(\eta) m(\xi, \eta) e^{2 \pi \iota (\xi + \eta, x)} d \xi d \eta, \] where \(m\) is a locally integrable function, is called a bilinear multiplier on \(\mathbb{R}\) of type \((p_1, p_2, p_3)\). In this well written paper, the author studies basic properties of bilinear multipliers and the space of bilinear multipliers of type \((p_1, p_2, p_3)\). About half the paper is devoted to the special class of multipliers of the form \(m(\xi, \eta)= M(\xi - \eta)\) for a real-valued function \(M\). This class includes the important examples of the bilinear fractional integral and the bilinear Hilbert transform.

MSC:

42B15 Multipliers for harmonic analysis in several variables
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
PDFBibTeX XMLCite
Full Text: arXiv