Almost orthogonal operators on the bitorus (Q971914)

The result of \textit{C. Fefferman} [Ann. Math. (2) 98, 551--571 (1973); erratum ibid. 146, 239 (1997; Zbl 0268.42009)] presents a new proof of a theorem of Carleson and Hunt: The Fourier series of an \(L^p\) function on \([0, 2\pi]\) converges almost everywhere. In the article under review, the author first studies the operators \(S_p f (x, y)\) acting as a kind of Fourier coefficients on one variable and as a kind of truncated Hilbert transforms with a phase \(N (x, y)\) on the other variable. Then, for the sum of such operators \(S_p f (x, y)\), under the basic assumption \(N (x, y)\) mainly a function of \(y\) and the additional assumption \(N (x, y)\) non-decreasing in \(x\) for every \(y\) fixed, the author proves an \(L^2\) -estimate, which is an extension of an argument of almost orthogonality in Fefferman's result [loc. cit.] to two dimensions.