Given the Lebesgue space with variable exponent $L^{s(\cdot )}(\mathrm{\Omega })$ whose norm is denoted by $||\cdot |{|}_{s(\cdot )}$, we show the following equivalence: $\underset{|E|\to 0}{\lim }||{\chi }_E|{|}_{s(\cdot )}=0$ if and only if $\underset{p\to +\infty }{\lim }\frac{1}{p}{\left[\underset{\mathrm{\Omega }}{⨍}s{(x)}^{p}dx\right]}^{\frac{1}{p}}=0$, where ${\chi }_E$ is the characteristic function of the measurable set E and $|E|$ its Lebesgue measure. We apply such results to characterize compactness of some inclusions.

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对可变指数空间 Lp(⋅) 的紧致性结果的评论

给定具有可变指数的 Lebesgue 空间 ${升}^{秒(\cdot )}(\mathrm{\Omega })$ 其范数表示为 $||\cdot |{|}_{秒(\cdot )}$，我们证明以下等价： $\underset{|乙|\to 0}{\mathrm{林}}||{\chi }_{乙}|{|}_{秒(\cdot )}=0$ 当且仅当 $\underset{p\to +\infty }{\mathrm{林}}\frac{1}{p}{\left[\underset{\mathrm{\Omega }}{⨍}秒{(X)}^{p}dX\right]}^{\frac{1}{p}}=0$， 在哪里 ${\chi }_{乙}$是可测集E的特征函数和$|乙|$其勒贝格测度。我们应用这些结果来表征一些夹杂物的致密性。