Growth Envelopes of Some Variable and Mixed Function Spaces

We study unboundedness properties of functions belonging to Lebesgue and Lorentz spaces with variable and mixed norms using growth envelopes. Our results extend the ones for the corresponding classical spaces in a natural way. In the case of spaces with mixed norms, it turns out that the unboundedness in the worst direction, i.e., in the direction where $$p_{i}$$ p i is the smallest, is crucial. More precisely, the growth envelope is given by $${\mathfrak {E}}_{{\mathsf {G}}}(L_{\overrightarrow{p}}(\varOmega )) = (t^{-1/\min \{p_{1}, \ldots , p_{d} \}},\min \{p_{1}, \ldots , p_{d} \})$$ E G ( L p → ( Ω ) ) = ( t - 1 / min { p 1 , … , p d } , min { p 1 , … , p d } ) for mixed Lebesgue and $${\mathfrak {E}}_{{\mathsf {G}}}(L_{\overrightarrow{p},q}(\varOmega )) = (t^{-1/\min \{p_{1}, \ldots , p_{d} \}},q)$$ E G ( L p → , q ( Ω ) ) = ( t - 1 / min { p 1 , … , p d } , q ) for mixed Lorentz spaces, respectively. For the variable Lebesgue spaces, we obtain $${\mathfrak {E}}_{{\mathsf {G}}}(L_{p(\cdot )}(\varOmega )) = (t^{-1/p_{-}},p_{-})$$ E G ( L p ( · ) ( Ω ) ) = ( t - 1 / p - , p - ) , where $$p_{-}$$ p - is the essential infimum of $$p(\cdot )$$ p ( · ) , subject to some further assumptions. Similarly, for the variable Lorentz space, it holds$${\mathfrak {E}}_{{\mathsf {G}}}(L_{p(\cdot ),q}(\varOmega )) = (t^{-1/p_{-}},q)$$ E G ( L p ( · ) , q ( Ω ) ) = ( t - 1 / p - , q ) . The growth envelope is used for Hardy-type inequalities and limiting embeddings. In particular, as a by-product, we determine the smallest classical Lebesgue (Lorentz) space which contains a fixed mixed or variable Lebesgue (Lorentz) space, respectively.