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Keywords:
generalized Orlicz space; Musielak-Orlicz space; nonstandard growth; variable exponent; double phase; uniform convexity; associate space
generalized Orlicz space; Musielak-Orlicz space; nonstandard growth; variable exponent; double phase; uniform convexity; associate space
Summary:
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.
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