Uncertainty principle for the spherical mean operator (Q2928361)

In harmonic analysis associated to the spherical mean operator \(\mathcal R\), the classical Fourier transform on \({\mathbb R}\times{\mathbb R}^{n}\), even with respect to the first variable, with the kernel \(\cos(rs)e^{-i\langle y,x\rangle}\) is changed to the transform \({\mathcal F}\) with the kernel \(\phi_{(s,y)}(r,x)={\mathcal R}(\cos(s\cdot)e^{-i\langle y,\cdot\rangle})(r,x)\). In this paper Miyachi's classical uncertainty principle on \({\mathbb R}^n\) is rewritten to the associated Fourier transform \({\mathcal F}\).