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Uniform convexity and associate spaces
Czechoslovak Mathematical Journal, Tome 68 (2018) no. 4, pp. 1011-1020
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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.
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.
DOI : 10.21136/CMJ.2018.0054-17
Classification : 46A25, 46E30
Keywords: generalized Orlicz space; Musielak-Orlicz space; nonstandard growth; variable exponent; double phase; uniform convexity; associate space 
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Harjulehto, Petteri; Hästö, Peter. Uniform convexity and associate spaces. Czechoslovak Mathematical Journal, Tome 68 (2018) no. 4, pp. 1011-1020. doi: 10.21136/CMJ.2018.0054-17

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