The Carleson-Hunt theorem states that: (a) If \(1< p\leq 2\), then there is a constant \(C_ p> 0\) such that \(\|{\mathcal S}^*_ 1 f\|_ p\leq C_ p\| f\|_ p\) for all \(f\in L^ p({\mathfrak R})\); and (b) if \(f\in L^ p({\mathfrak R})\), \(1< p\leq 2\), then \(\lim_{R\to \infty} S_ R f(x)= f(x)\) for a.a. \(x\in {\mathfrak R}\). Here \(S_ R\) denotes the partial sums of the inverse Fourier transform of the Fourier transform \(\widehat f\) of \(f\), \(S_ R f(x)= \int^ R_{-R} \widehat f(y) e^{ixy} dy\) (\(R> 0\) and all \(x\in {\mathfrak R}\)), and \({\mathcal S}^*_ 1\) is the maximal function \({\mathcal S}^*_ 1 f(x)= \sup_{R> 0} | S_ R f(x)|\). This paper considers extensions of the Carleson-Hunt theorem to higher dimensional Euclidean spaces and to some symmetric spaces.
For the entire collection see [Zbl 0782.00061].