$A^p_\alpha$ classes in the Dirichlet range: inner-outer factorization, Carleson measures and weak products

We study properties of $A^p_\alpha$ spaces in the Dirichlet range, recently defined by Brevig, Kulikov, Seip and Zlotnikov as the set of all holomorphic functions on the unit disc $\mathbb{D}$ such that $$\int_{\mathbb{D}} |f(z)|^{p-2} |f'(z)|^2 (1 - |z|^2)^{\alpha} \, dA(z) < \infty,$$ when $0<\alpha < 1$ and $p > 0$. We answer in the negative two questions posed by Brevig et al. by showing that, if $p\ne2$ and $p > \frac{1}{2}$, $A^p_\alpha$ is not a vector space and that the norm is in general not increasing in $p$. This is achieved by means of an equivalent description for $A^p_\alpha$ which is given in terms of the Poisson integral of the boundary function of its inhabitants. Such norm also leads to a description of $A^p_\alpha$ functions in the Dirichlet range given in terms of their inner and outer factors. As a corollary, we show that $A^1_\alpha$ is contained in the weak product of a Dirichlet-type space.