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 $E_G(L_{ \vec{ p}}(\Omega)) = (t^{ -1/\min\{ p_{ 1}, \ldots, p_{ d} \}},\min\{ p_{ 1}, \ldots, p_{ d} \})$ for mixed Lebesgue and $E_G(L_{ \vec{ p},q}(\Omega)) = (t^{ -1/\min\{ p_{ 1}, \ldots, p_{ d} \}},q)$ for mixed Lorentz spaces, respectively. For the variable Lebesgue spaces we obtain $E_G(L_{ p(\cdot)}(\Omega)) = (t^{ -1/p_{ -}},p_{ -})$, where $p_{ -}$ is the essential infimum of $p(\cdot)$, subject to some further assumptions. Similarly, for the variable Lorentz space it holds $E_G(L_{ p(\cdot),q}(\Omega)) = (t^{ -1/p_{ -}},q)$. 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.
Categories: math.FA