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Sharp local uncertainty inequalities. (English) Zbl 0636.46029

Summary: For each measurable \(E\subseteq {\mathbb{R}}^ d\) and \(\alpha >d/2\), the local uncertainty principle inequality \[ \int_{E}| F(\omega)|^ 2 d\Omega < \text{ const }\cdot m(E)\| f\|_ 2^{2-d/\alpha}\| | t|^{\quad \alpha}f\|_ 2^{d/\alpha} \] is established for all \(f\in L^ 2({\mathbb{R}}^ d)\) and the best constant found, where F denotes the Fouriertransform of f. The constant only depends on d and \(\alpha\).

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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