A (𝑝,𝑞) version of Bourgain’s theorem

Benedetto, John; Powell, Alexander

doi:10.1090/S0002-9947-05-03737-2

A $(p,q)$ version of Bourgain’s theorem
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by John J. Benedetto and Alexander M. Powell PDF

Trans. Amer. Math. Soc. 358 (2006), 2489-2505 Request permission

Abstract:

Let $1<p,q<\infty$ satisfy $\frac {1}{p} + \frac {1}{q} =1.$ We construct an orthonormal basis $\{ b_n \}$ for $L^2 (\mathbb {R})$ such that $\Delta _p ( b_n )$ and $\Delta _q (\widehat {b_n})$ are both uniformly bounded in $n$. Here $\Delta _{\lambda } (f) \equiv \textrm {inf}_{a \in \mathbb {R}} \left ( \int |x - a|^{\lambda } |f(x)|^2 dx \right )^{\frac {1}{2}}$. This generalizes a theorem of Bourgain and is closely related to recent results on the Balian-Low theorem.

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