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Characterization of generalized Orlicz spaces by:5 (Source: Crossref)

    The norm in classical Sobolev spaces can be expressed as a difference quotient. This expression can be used to generalize the space to the fractional smoothness case. Because the difference quotient is based on shifting the function, it cannot be used in generalized Orlicz spaces. In its place, we introduce a smoothed difference quotient and show that it can be used to characterize the generalized Orlicz–Sobolev space. Our results are new even in Orlicz spaces and variable exponent spaces.

    During the reviewing process of this manuscript, the authors became aware that Sibei Yang, Dachun Yang, and Wen Yuan had completed a related manuscript [35]. Their manuscript is dated of 28 January 2018, over one year after our work was posted on arXiv (see arXiv:1612.04566).

    Their work is based on the abstract assumption that the maximal operator is bounded, whereas we assume (2.3) (which implies the maximal inequality [25]). On the other hand, our results are more general in the sense that we cover arbitrary open sets Ω, instead of the entire Euclidean space n, and we assume only local integrability, whereas they assume global integrability.

    AMSC: 46E35, 46E30


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