Some Results for the Windowed Fourier Transform Related to the Spherical Mean Operator

Abstract

We define and study the windowed Fourier transform associated with the spherical mean operator. We prove the boundedness and compactness of localization operators of this windowed. Next, we establish new uncertainty principles for the Fourier and the windowed Fourier transforms associated with the spherical mean operator. More precisely, we give a Shapiro-type uncertainty inequality for the Fourier transform that is, for s > 0 and {αk}k be an orthonormal sequence in L2(dνn+ 1) $$ \sum\limits_{k=1}^{M}\left( \||(r,x)|^{s}\alpha_{k}\|^{2}_{2,\nu_{n+1}}+\||(\mu,\lambda)|^{s}\mathcal{\tilde{F}}(\alpha_{k})\|^{2}_{2,\nu_{n+1}}\right) \geqslant R_{n,s}M^{1+\frac{s}{2n+1}}. $$ Finally, we prove an analogous inequality for the windowed Fourier transform.