Wavelet-type Transform and Bessel Potentials Associated with the Generalized Translation

Abstract

Wavelet–type transform associated with singular Laplace–Bessel differential operator $\Delta _\nu = \sum\limits_{k = 1}^n {\frac{{\partial ^2 }} {{\partial x_k^2 }}} + \frac{{2\nu}} {{x_n }}\frac{\partial } {{\partial x_n }}$ is introduced and the relevant Calderon–type reproducing formula is established. Representations of the generalized Bessel potentials $(E - \Delta _\nu )^{ - \alpha /2} f,\quad (Re \alpha > 0)$ and their inverses via the wavelet–type transform are obtained.