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Refined Hardy inequalities. (English) Zbl 1121.43006
As a generalization of the classical Hardy inequality on \(\mathbb R^N\), the authors obtain the following two inequalities: For \(0<s<N/2\) there exists a constant \(C\) such that \[ \int_{\mathbb R^N}\frac{|u|^2(x)}{|x|^{2s}}dx \leq C\,\|u\|^2_{\Dot{H}^s}, \] where \(\Dot{H}^s\) is the Sobolev space of order \(s\), and for \(2\leq q<2N/(N-2s)<p\leq\infty\), let \(\alpha=pq(1/q-1/2+s/N)/(p-q)\). Then there exists a constant \(C\) such that \[ \Big(\int_{\mathbb R^N}\frac{|u|^2(x)}{|x|^{2s}}dx\Big)^{1/2} \leq C\,\|u\|^\alpha_{\Dot{B}^{s-N(1/2-1/p)}_{p,2}} \|u\|^{1-\alpha}_{\Dot{B}^{s-N(1/2-1/q)}_{q,2}}, \] where \(\Dot{B}^s_{p,q}\) is the Besov space. Moreover, they show that these inequalities also hold on the Heisenberg group. In order to prove the first one they use the Littlewood-Paley theory and prove that \(|x|^{-2s}\in\Dot{B}^{N-s}_{1,\infty}\) and \(u^2\in\Dot{B}^{2s-N/2}_{2,1}\). For the second one, they use the paraproduct algorithm introduced by J.-M. Bony. Furthermore, they deduce that the second one is invariant under oscillations, that is, both sides are of the same order of magnitude when \(u\) are highly oscillatory functions.

MSC:
43A80 Analysis on other specific Lie groups
42B30 \(H^p\)-spaces
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