Abstract
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} 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} }) for mixed Lebesgue and {mathfrak {E}}_{{mathsf {G}}}(L_{overrightarrow{p},q}(varOmega )) = (t^{-1/min {p_{1}, ldots , 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_{-}), where p_{-} is the essential infimum of p(cdot ), 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). 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.
Highlights
Using Sobolev embeddings, the integrability properties of a real function can be deduced from those of its derivatives
We study unboundedness properties of functions belonging to Lebesgue and Lorentz spaces with variable and mixed norms using growth envelopes
We use the notation χA for the characteristic function of a set A. This approach can be seen as a generalization of the classical Lorentz space L p,q
Summary
Using Sobolev embeddings, the integrability properties of a real function can be deduced from those of its derivatives. The growth envelope function EGX is monotonically decreasing, see Haroske [20, Prop. The pair EG(X ) := EGX , uGX is called the growth envelope of the function space X. We use the notation χA for the characteristic function of a set A This approach can be seen as a generalization of the classical Lorentz space L p,q. It will turn out, that if the measure of Ω is finite, for the growth envelopes, we have. Replacing the constant exponent p in the classical L p-norm by an exponent function p(·), the variable Lebesgue space L p(·) is obtained. The space L p(·) consists of the functions f , whose quasi-norm
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