On convergence of some integral transforms. (English) Zbl 0813.42007
Doust, Ian (ed.) et al., Proceedings of the miniconference on probability and analysis, held at the University of New South Wales, Sydney, Australia, July 24-26, 1991. Canberra: Centre for Mathematics and Its Applications, Australian National University. Proc. Cent. Math. Appl. Aust. Natl. Univ. 29, 145-162 (1991).
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].
For the entire collection see [Zbl 0782.00061].
Reviewer: B.P.Duggal (Al Khod / Oman)
MSC:
42B10 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
42B25 | Maximal functions, Littlewood-Paley theory |
22E30 | Analysis on real and complex Lie groups |