Uncertainty inequalities associated to some homogeneous functions.
For a pair \(E,\widehat{E}\) of subsets of \(\mathbb{R}^d\) the author defines the \textit{annihilating constant} \(A(E,\widehat{E})\) as the smallest constant \(A\) such that \(\int_{\widehat{E}}| \widehat{f\chi_{E}}| ^2\leq A\| f\| _{L^2}^2\). For \(k=(k_1,\dots,k_d)\in \mathbb{Z}^d\) and \(\gamma=(\gamma_1,\dots,\gamma_d)\in (1,\infty)^d\), one defines \(Q(k,\gamma)=\{x\in\mathbb{R}^d: | x_i| \in [\gamma_i^{k_i},\gamma_i^{k_i+1})\}\). The main result states that there is a constant \(K(\gamma,d)\) depending only on \(\gamma\) and the dimension such that, for all measurable pairs \((E,\widehat{E})\), one has \[ A(E,\widehat{E})\leq K(\gamma,d) \sum_{m,n\in \mathbb{Z}} \Biggl( \sup_{\sum k_i= m;\,\;\sum p_j= n} A(E,\widehat{E},k,p)\Biggr)(1+| n| +| m| )^{2(d-1)} \] where \(A(E,\widehat{E},k,p) =A(E\cap Q(k,\gamma),\widehat{E}\cap Q(p,\gamma))\). It is also shown that the inequality \[ \| f\| _{L^2_\alpha}\| \hat{f}\| _{L^2_\alpha}\geq K(\alpha) \| f\| _{L^2}^2 \] follows from this, where \(\| f\| _{L^2_\alpha}^2=\int \prod_{i=1}^d | x_i| ^\alpha_i | f(x)| ^2\, dx\).
scientific article; zbMATH DE number 2182146