×

The Littlewood-Paley \(g\)-function associated with the Riemann-Liouville operator. (English) Zbl 1302.43003

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).

MSC:

43A32 Other transforms and operators of Fourier type
42B25 Maximal functions, Littlewood-Paley theory
PDFBibTeX XMLCite