Transference of certain maximal Hilbert transforms on the torus (Q310915)

Let \({H}\) be the periodic bilinear Hilbert transform on the torus, i.\,e. NEWLINE\[NEWLINE {H}(f,g)(x)= \text{p.v.} \int_{ | t | < 1/2} f(x-t)g(x+t) \cot(\pi t)\, dt. NEWLINE\]NEWLINE By using a transference method, \textit{D. Fan} and \textit{S. Sato} [J. Aust. Math. Soc. 70, No. 1, 37--55 (2001; Zbl 0984.42006)] proved the \(L^p \times L^q \to L^r\) boundedness of \(H\) for \(r>2/3\).NEWLINENEWLINEThe authors consider the maximal operator NEWLINE\[NEWLINE {H}^{*}(f,g)(x)= \sup_{\varepsilon >0} \left| \text{p.v.} \int_{ \varepsilon < | t | < 1/2} f(x-t)g(x+t) \cot(\pi t)\, dt \right| NEWLINE\]NEWLINE and prove that NEWLINE\[NEWLINE \| {H}^{*}(f,g) \|_{L^r({\mathbb T})} \leq C \| f \|_{L^p({\mathbb T})} \| g \|_{L^q({\mathbb T})}, NEWLINE\]NEWLINE where \(1 < p,q \leq \infty, 1/r = 1/p + 1/q \) and \(r > 2/3\).NEWLINENEWLINEThey also study the bilinear Riesz transforms on the \(n\)-torus \({\mathbb T}^n\).