标题： 与平均相关的傅里叶不确定性原理的极值化问题 显示英文标题

标题： Extremizers of a Fourier uncertainty principle related to averaging

摘要： We study the uncertainty principle $$\lVert\widehat{\mu}(\xi) |\xi|^\beta\rVert_\infty^{\alpha} \left(\int |x|^\alpha d \mu\right)^{\beta} \geq C(\alpha,\beta,d){\lVert{\mu}\rVert_{TV}^{\alpha+\beta}}$$ for finite non-negative measures on $\mathbb{R}^d $. We prove that $C(\alpha,\beta,d)>0$ for all $\alpha,\beta>0$ and that extremizers exist. Moreover, we obtain an abstract characterization of the extremizers, which allows us to describe their asymptotic behavior and, for certain parameter values, to determine them explicitly. 显示英文摘要

摘要： We study the uncertainty principle $$\lVert\widehat{\mu}(\xi) |\xi|^\beta\rVert_\infty^{\alpha} \left(\int |x|^\alpha d \mu\right)^{\beta} \geq C(\alpha,\beta,d){\lVert{\mu}\rVert_{TV}^{\alpha+\beta}}$$ for finite non-negative measures on $\mathbb{R}^d $. We prove that $C(\alpha,\beta,d)>0$ for all $\alpha,\beta>0$ and that extremizers exist. Moreover, we obtain an abstract characterization of the extremizers, which allows us to describe their asymptotic behavior and, for certain parameter values, to determine them explicitly.