Extremal Functions and Calderon’s Formulas for the Riemann-Liouville Two-wavelet Transform

The Riemann-Liouville operator has been extensively investigated and his witnessed a remarkable development in numerous fields of harmonic analysis. Knowing the fact of the study of the time-frequency analysis are both theoritically interesting and pratically useful, we investigated several problems for this subject on the setting of the Riemann-Liouville wavelet transform. Firstly, we introduce the notion of Riemann-Liouville two-wavelet and we present generalized version of Parseval’s, Plancherel’s, inversion and Calderon’s reproducing formulas. Next, using the theory of reproducing kernels, we give best estimates and an integral representation of the extremal functions related to the Riemann-Liouville wavelet transform on weighted Sobolev spaces.