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Discretization and anti-discretization of rearrangement-invariant norms. (English) Zbl 1066.46023
A new method of discretization and anti-discretization of weighted inequalities is developed which is applied to norms in classical Lorentz spaces and to spaces endowed with the so-called Hilbert norm. As an application of these results, some integral conditions characterizing embeddings \(\Gamma^{p}(v)\to \Lambda^{q}(w)\) and an integral characterization of the associate space to \(\Gamma^{p}(v)\) are given, where \(p,q\in (0,\infty),\) \(v,w\) are weights on \([0,\infty)\) and \[ \| f\| _{\Lambda^{p}(v)}=\left(\int_{0}^{\infty}f^{\ast}(t)^{p}v(t) \,dt\right)^{1/p},\qquad \| f\| _{\Gamma^{p}(v)}= \left(\int_{0}^{\infty} f^{\ast\ast}(t)^{p}v(t) \,dt\right)^{1/p}. \]

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
26D10 Inequalities involving derivatives and differential and integral operators
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