SHAPIRO TYPE INEQUALITIES FOR THE WEINSTEIN AND THE WEINSTEIN-GABOR TRANSFORMS

Abstract

The aim of this paper is to prove new uncertainty principles for the Weinstein and the Weinstein-Gabor transforms associated with the Weinstein operator dened on the half  space $\mathbb{R}^d_{+}$ by $\Delta_W =\sum_{i=1}^{d } \frac{\partial}{\partial x_i^2}+ \frac{2\alpha+1}{x_{d}}\frac{\partial}{\partial x_{d-1}};\ \ \ \ \ d\ge2,\ \alpha>-1/2.$ More precisely, we give a Shapiro-type uncertainty inequality for the Weinstein transform that is, for $s>0$ and $\{\phi_n\}_n$ be an orthonormal sequence in $L^2_\alpha(\mathbb{R}^d_{+})$, $\sum_{n=1}^N(\Vert \vert x\vert^s \phi_n\Vert_{{L_\alpha^2(\mathbb{R}^d_{+})}}^{2}+ \Vert \vert\xi\vert^s \mathcal{F}_W(\phi_n)\Vert_{{L_\alpha^2(\mathbb{R}^d_{+})}}^{2 })\geq KN^{1+\frac{s}{2\alpha+d+1}},$ where $K$ is a constant which depends only on $d$; $s$ and $\alpha$. Next, we establish an analogous inequality for the Weinstein-Gabor transform

Keywords

• Weinstein operator; Shapiro uncertainty inequality,time frequency concentration.

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