Growth envelopes of anisotropic function spaces (Q2427613)

Let \(\alpha = (\alpha_1, \dots, \alpha_n)\), \(0<\alpha_1 \leq \alpha_2 \leq \dots \leq \alpha_n\), \(\sum^n_{j=1} \alpha_j =n\) be an anisotropy in \(\mathbb R^n\) and let \[ B^{s,\alpha}_{pq} (\mathbb R^n), \quad F^{s, \alpha}_{pq} (\mathbb R^n), \quad s \in\mathbb R, \quad 0<p,q\leq \infty, \quad(p<\infty\text{ for the \(F\)-spaces}) \] be the related anisotropic spaces. With \(A\in\{B,F\}\), the authors prove in the sub-critical case \[ n\left(\frac{1}{p}-1\right)_+<s<\frac{n}{p},\quad s - \frac{n}{p}=-\frac{n}{r},\quad 1<r<\infty, \] sharp embeddings of the type \[ \left(\int^{\varepsilon}_0(t^{1/r}f^*(t))^v\,\frac{dt}{t}\right)^{1/v}\leq c\,\| f|A^{s,\alpha}_{pq}(\mathbb R^n)\|\qquad\text{(Theorem 4.3)}. \] They are independent of the anisotropy \(\alpha\). In the critical case \(s = n/p\), one has corresponding results with \((\log t)^\lambda\) in place of \(t^{1/r}\) (Theorem 5.1). All this is formulated in terms of growth envelopes. It extends well-known corresponding assertions from the isotropic case \(\alpha_1 = \dots = \alpha_n =1\) to the anisotropic case. As a consequence, one gets sharp Hardy inequalities.