A new class of function spaces connecting Triebel--Lizorkin spaces and \(Q\) spaces (Q960548)

Let \(\dot{F}^s_{pq}(\mathbb R^n)\) with \(s \in \mathbb R\), \(0<p,q\leq\infty\) be the homogeneous Triebel--Lizorkin spaces in \(\mathbb R^n\). Let \(0< \alpha <1\), \(0<p\leq \infty\), \( 1\leq q \leq \infty\). Then the \(Q\)-spaces \(Q^{\alpha,q}_p (\mathbb R^n)\) are defined via \[ \|f|Q^{\alpha,q}_p(\mathbb R^n)\|=\sup|I|^{\frac{1}{p}-\frac{1}{q}}\Big(\int_{I\times I}\frac{|f(x)-f(y)|^q}{|x-y|^{n+q\alpha}}\,dx\,dy\Big)^{1/q}, \] where the supremum is taken over all (dyadic) cubes \(I\) in \(\mathbb R^n\). The authors introduce spaces \(\dot{F}^{s,\tau}_{pq}(\mathbb R^n)\) which cover both types of spaces and study their properties such as embeddings. In addition, further classes of spaces are introduced based on Carleson measures and Hausdorff capacities (tent spaces). This gives the possibility to study dual spaces, preduals, and their relations to the spaces \(\dot{F}^{s,\tau}_{pq}(\mathbb R^n)\).