Categories of results in variable Lebesgue spaces theory

Fiorenza, Alberto

Rendiconto della Accademia delle scienze fisiche e matematiche, Série 4, Tome 86 (2019) no. 1, pp. 79-102.

Abstract (VO)

Variable (exponent) Lebesgue spaces represent a relevant research area within the theory of Banach function spaces. Much attention is devoted to look for sufficient conditions on the variable exponent to establish the assertions within the theory. In this Note we try to show the beauty of the research in this field, mainly quoting some known results organized into “categories", each of them characterized by a common typology of conditions on the variable exponent. New results involve the failure of rearrangement-invariant property, the rearrangement of the exponent, and a generalization of a formula known for constant exponents. Riassunto – Gli spazi di Lebesgue con esponente variabile rappresentano un set

Riassunto

Gli spazi di Lebesgue con esponente variabile rappresentano un settore di rilievo nell’ambito della teoria degli spazi funzionali di Banach. Di notevole interesse è la ricerca di condizioni, da imporre alla funzione esponente, sufficienti ad assicurare il verificarsi di determinate affermazioni. In questa Nota ci proponiamo di mostrare il fascino della ricerca in questo settore, segnalando essenzialmente alcuni noti risultati organizzati in “categorie", ognuna delle quali caratterizzata da una comune tipologia di condizioni sulla funzione esponente. I risultati originali sono relativi alla non invarianza per riordinamento, al riordinamento dell’esponente e ad una generalizzazione di una formula nota per esponenti costanti.

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