On decompositions in generalised Lorentz-Zygmund spaces

Neves, J. S.

Bollettino della Unione matematica italiana, Série 8, 4B (2001) no. 1, pp. 239-267.

Résumé

Il lavoro presenta diverse caratterizzazioni degli spazi Lorentz-Zygmund generalizzati (GLZ) $L_{p, q; \mathbf{\alpha}}(R)$, con $p, q \in (0, +\infty]$, $m \in \mathbb{N}$, $\mathbf{\alpha}\in \mathbb{R}^{m}$ e $(R, \mu)$ spazio misurato con misura $\mu(R)$ finita. Dato uno spazio misurato $(R, \mu)$ e $\mathbf{\alpha} \in \mathcal{R}^{m}_{-}$ , otteniamo representazioni equivalenti per la (quasi-) norma dello spazio GLZ $L_{\infty, \infty; \mathbf{\alpha}} (R)$. Inoltre, se $(R, \mu)$ è uno spazio misurato con misura finita e $\mathbf{\alpha} \in \mathcal{R}^{m}_{+}$, viene presentata in termini di decomposizioni una norma equivalente per lo spazio $L_{1, 1; \mathbf{\alpha}}(R)$. Si dimostra che le norme equivalenti considerate per $L_{\infty, \infty; \mathbf{\alpha}}(R)$, con $(R, \mu)$ uno spazio a misura finita, e la norma di decomposizione in $L_{1, 1; \mathbf{\alpha}}(R)$ possono essere utilizzate per ottenere semplici dimostrazioni di alcuni risultati di estrapolazione concernenti questi spazi.

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