Bilinear Hilbert transform on measure spaces
In 2001, D.~Fan and S.~Sato showed the boundedness (\(L^{p_1}(\mathbb T)\times L^{p_2}(\mathbb T)\to L^{p_3}(\mathbb T)\), \(p_1,p_2>1, 1/p_1+1/p_2=1/p_3<3/2\)) of the bilinear Hilbert transform on the torus \(H_{\mathbb T}(f,g)(x)=\int_{\mathbb T}f(x-y)g(x+y)\,dy\), by transfering the Lacey-Thiele's result on \(\mathbb R\). The authors give such boundedness results (on the same range but \(p_3\geq1\)) of the discrete Hilbert transform \(H_{\mathbb Z}(a,b)(m)=\sum_{n\neq 0}a_{m-n}b_{n+m}/n\), of the ergodic Hilbert transform \(H_T(f,g)(x)=\sum_{n\neq 0}T^nf(x)T^{-n}g(x)/n\), where \(T\) is an ergodic transformation acting on \(L^{p_i}(\Omega)\) for a certain \(\sigma\)-finite measure space \(\Omega\), and in particular \(H_D(f,g)(x)=\sum_{n\neq 0}f(x-n)g(x+n)/n\) on \(\mathbb R\). They also give a new proof of the result of Fan and Sato. To prove these, they give a transferance theorem improving the one developed by \textit{R. R. Coifman} and \textit{G. Weiss} [``Transferance methods in analysis'', Expository lectures from the CBMS Regional Conference held at the University of Nebraska, May 31--June 4, 1976. Regional Conference Series in Mathematics. No.31. Providence, R.I.: American Mathematical Society (AMS) (1977; Zbl 0377.43001)] and some discretization techniques.
scientific article; zbMATH DE number 5013631