Czechoslovak Mathematical Journal, Vol. 68, No. 4, pp. 1011-1020, 2018


Uniform convexity and associate spaces

Petteri Harjulehto, Peter Hästö

Received February 8, 2017.   Published online April 13, 2018.

Abstract:  We prove that the associate space of a generalized Orlicz space $L^{\phi(\cdot)}$ is given by the conjugate modular $\phi^*$ even without the assumption that simple functions belong to the space. Second, we show that every weakly doubling $\Phi$-function is equivalent to a doubling $\Phi$-function. As a consequence, we conclude that $L^{\phi(\cdot)}$ is uniformly convex if $\phi$ and $\phi^*$ are weakly doubling.
Keywords:  generalized Orlicz space; Musielak-Orlicz space; nonstandard growth; variable exponent; double phase; uniform convexity; associate space
Classification MSC:  46E30, 46A25


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Affiliations:   Petteri Harjulehto, Department of Mathematics and Statistics, FI-20014 University of Turku, Finland, e-mail: petteri.harjulehto@utu.fi; Peter Hästö, Department of Mathematics and Statistics, FI-20014 University of Turku, Finland, and Department of Mathematical Sciences, FI-90014 University of Oulu, Finland, e-mail: peter.hasto@oulu.fi


 
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