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.

Spaces of Sobolev type with positive smoothness on ℝn, embeddings and growth envelopes

We characterize Triebel-Lizorkin spaces with positive smoothness onℝn, obtained by different approaches. First we present three settingsFp,qs(ℝn),Fp,qs(ℝn),ℑp,qs(ℝn)associated to definitions by differences, Fourier-analytical methods and subatomic decompositions. We study their connections and diversity, as well as embeddings between these spaces and into Lorentz spaces. Secondly, relying on previous results obtained for Besov spaces𝔅p,qs(ℝn), we determine their growth envelopes𝔈G(Fp,qs(ℝn))for0≺p≺∞,0≺q≤∞,s≻0, and finally discuss some applications.

Sobolev Spaces with Muckenhoupt Weights, Singularities and Inequalities

We use the recently introduced concept of growth envelopes to characterize weighted spaces of type , where 𝑤 belongs to some Muckenhoupt 𝐴𝑝 class, and discuss some applications.