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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 $\| f\| _{\Lambda^{p}(v)}=\left(\int_{0}^{\infty}f^{\ast}(t)^{p}v(t) \,dt\right)^{1/p},\qquad \| f\| _{\Gamma^{p}(v)}= \left(\int_{0}^{\infty} f^{\ast\ast}(t)^{p}v(t) \,dt\right)^{1/p}.$

##### MSC:
 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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