一类变函数空间和混合函数空间的增长包络
Growth envelopes of some variable and mixed function spaces
摘要：
利用增长包络研究了具有变范数和混合范数的Lebesgue空间和Lorentz空间的函数的无界性。我们的结果以一种自然的方式推广了相应经典空间的结果。在具有混合范数的空间的情况下，在最坏方向上的无界性，即，在$p_{i}$最小的方向上是至关重要的。更准确地说，增长包络分别由混合勒贝格空间的$E_G(L_{\vec{p}}(Ω)) = (t^{-1/\min\{p_{1}, \ldots, p_{d} \}},\min\{p_{1}, \ldots, p_{d} \})$和混合洛伦兹空间的$E_G(L_{\vec{p},q}(Ω)) = (t^{-1/\min\{p_{1}, \ldots, p_{d} \}},q)$给出。对于可变Lebesgue空间，我们得到了$E_G(L_{p(\cdot)}(Ω)) = (t^{-1/p_{-}},p_{-})$，其中$p_{-}$是$p(\cdot)$的本质下确界，并作了进一步的假设.类似地，对于可变洛伦兹空间，它保持$E_G(L_{p(\cdot),q}(Ω)) = (t^{-1/p_{-}},q)$。增长包络用于Hardy型不等式和极限嵌入。特别是，作为一个副产品，我们确定最小的经典勒贝格（洛伦兹）空间，其中包含一个固定的混合或可变勒贝格（洛伦兹）空间，分别。
英文摘要：
We study unboundedness properties of functions belonging 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 $p_{i}$ is the smallest, is crucial. More precisely, the growth envelope is given by $E_G(L_{\vec{p}}(Ω)) = (t^{-1/\min\{p_{1}, \ldots, p_{d} \}},\min\{p_{1}, \ldots, p_{d} \})$ for mixed Lebesgue and $E_G(L_{\vec{p},q}(Ω)) = (t^{-1/\min\{p_{1}, \ldots, p_{d} \}},q)$ for mixed Lorentz spaces, respectively. For the variable Lebesgue spaces we obtain $E_G(L_{p(\cdot)}(Ω)) = (t^{-1/p_{-}},p_{-})$, where $p_{-}$ is the essential infimum of $p(\cdot)$, subject to some further assumptions. Similarly, for the variable Lorentz space it holds $E_G(L_{p(\cdot),q}(Ω)) = (t^{-1/p_{-}},q)$. 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.
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