Wavelet and Weyl transforms associated with the spherical mean operator (Q1764251)

The spherical mean operator \(R\) is defined on \(S^*(\mathbb{R}\times \mathbb{R}^n)\) by \[ Rf(r,x)= \int_{S^n} f(r\eta, x+ r\xi)\, d\sigma(\eta,\xi), \] where, \(\forall(r, x)\in [0+\infty]\times \mathbb{R}^n\), \(S^*(\mathbb{R}\times\mathbb{R}^n)\) is the space of infinitely differentiable functions on \(\mathbb{R}\times\mathbb{R}^n\), even with respect to the first variable, rapidly decreasing together with all their derivatives, \(d\sigma\) is the normalized surface measure on the unit sphere \(S^n\). The translation operator, the convolution, and the Fourier transform are considered in association with this spherical mean operator. These are well-known as generalized shift operator, the corresponding convolution, and the Fourier-Bessel transform on the weighted \(L_p\) spaces. In this paper the admissible wavelets associated with the spherical mean operator are defined. The Weyl transform is established as a bounded operator on weighted \(L_p\) spaces.