Tight fitting conjecture for Hardy and weighted Bergman spaces on the unit ball

Let $\mathbb B_n$ be the unit ball in $\mathbb C^n$. For $0<p,q<\infty$ and $\alpha,\beta>-1$, let $A^p_\alpha$ and $A^q_\beta$ denote weighted Bergman spaces, and let $H^p$ denote the Hardy space. An embedding is a tight fitting when it is proper, contractive, and non-compact.

Tight fitting conjecture. (a) The embedding $A^p_\alpha\subset A^q_\beta$ is a tight fitting if and only if

$$p<q\quad\text{and}\quad (n+1+\alpha)q=(n+1+\beta)p.$$

Moreover, equality of norms holds if and only if

$$f(z)=\frac{c}{(1-\langle z,a\rangle)^{2(n+1+\alpha)/p}},$$

where $c\in\mathbb C$ and $a\in\mathbb B_n$.

(b) The embedding $H^p\subset A^q_\alpha$ is a tight fitting if and only if

$$p<q\quad\text{and}\quad nq=(n+1+\alpha)p.$$

Moreover, equality of norms holds if and only if

$$f(z)=\frac{c}{(1-\langle z,a\rangle)^{2n/p}},$$

where $c\in\mathbb C$ and $a\in\mathbb B_n$.

This conjecture aims to characterize all proper, contractive, non-compact embeddings between Hardy and weighted Bergman spaces on the unit ball. Earlier results established several non-compact contractive embeddings, while Kulikov settled the previous conjectures for the disc; the corresponding complete characterization in the stated ball setting is presented here as a conjecture.