论文标题
某些变量和混合功能空间的生长信封
Growth envelopes of some variable and mixed function spaces
论文作者
论文摘要
我们研究了属于Lebesgue和Lorentz空间具有可变和混合规范功能的功能的无界性能。我们的结果以自然方式扩展了相应的经典空间的结果。对于具有混合规范的空间，事实证明，最糟糕的方向的无限性，即在$ p_ {i} $的方向上，最小的是至关重要的。更准确地说，生长封装由$ e_g（l _ {\ vec {p}}（ω））=（t^{ - 1/\ min \ {p_ {p_ {1}，\ ldots，p_ {d}}}}}}}，\ min \ min \ {p_ {p_ {p_ {p_ {p_ {p_ {1}}混合Lebesgue和$ e_g（l _ {\ vec {p}，q}（ω））=（t^{ - 1/\ min \ {p_ {1}，\ ldots，p_ {d}}}}，q）分别用于混合Lorentz空间。对于可变lebesgue空间，我们获得$ e_g（l_ {p（\ cdot）}（ω）））=（t^{ - 1/p _ { - }}，p _ { - }）$，其中$ p _ { - } $是$ p（\ cdot）$的基本信息。同样，对于可变lorentz空间，它保留$ e_g（l_ {p（\ cdot），q}（ω））=（t^{ - 1/p _ { - }}，q）$。生长包膜用于强壮的型不平等和限制嵌入。特别是，作为副产品，我们确定了分别包含固定混合或可变的Lebesgue（Lorentz）空间的最小古典Lebesgue（Lorentz）空间。
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.
