Amri, Besma; Rachdi, Lakhdar T. The Littlewood-Paley \(g\)-function associated with the Riemann-Liouville operator. (English) Zbl 1302.43003 Ann. Univ. Paedagog. Crac. 128, Stud. Math. 12, 31-58 (2013). The authors define Gauss and Poisson kernels associated with the Riemann-Liouville operator. They prove that the Gauss kernel has an approximation of the identity. Then, they define Gauss and Poisson semigroups associated with the Riemann-Liouville operator such that their operators are bounded from \(L^{p}\left( dv_{\alpha }\right) \) into itself. Also using these semigroups, they define maximal functions and check their boundedness. This paper has a interesting result that is important for the Littlewood-Paley theory. This result is the boundedness of the Littlewood-Paley \(g\)-function associated with the Riemann-Liouville operator acting on \(L^{p}\left( dv_{\alpha }\right) \), \(p\epsilon (1,2]\) (see Theorem 4.7). Reviewer: Öznur Kulak (Görele) Cited in 6 Documents MSC: 43A32 Other transforms and operators of Fourier type 42B25 Maximal functions, Littlewood-Paley theory Keywords:Gauss semigroup; Poisson semigroup; Riemann-Liouville operator; maximal function; Littlewood-Paley g-function PDFBibTeX XMLCite \textit{B. Amri} and \textit{L. T. Rachdi}, Ann. Univ. Paedagog. Crac., Stud. Math. 128(12), 31--58 (2013; Zbl 1302.43003)