Continuous wavelet transform on semisimple Lie groups and inversion of the Abel transform and its dual

In this work we define and study wavelets and continuous wavelet transform on semisimple Lie groups $G$ of real rank $\ell$. We prove for this transform Plancherel and inversion formulas. Next using the Abel transform $\mathcal{A}$ on $G$ and its dual $\mathcal{A}^\ast$, we give relations between the continuous wavelet transform on $G$ and the classical continuous wavelet transform on $\mathbb{R}^\ell$, and we deduce the formulas which give the inverse operators of the operators $\mathcal{A}$ and $\mathcal{A}^\ast$.