Fix $n \in \mathbb{N}$ and consider the Hardy space $H^1 := H^1(\mathbb{D}^n)$, consisting of holomorphic functions $f$ on the unit polydisk $\mathbb{D}^n=\mathbb{D}\times\dots\times\mathbb{D}$ such that

$$\sup_{0\leq r<1}\int_{\mathbb{T}^n}|f(rw)|\frac{dw}{(2\pi)^n}<\infty,$$

and let $H^2 := H^2(\mathbb{D}^n)$ be the collection of all holomorphic functions $f$ on $\mathbb{D}^n$ with square-summable power-series coefficients (equivalently, they satisfy the above inequality with $|f(rw)|^2$ instead).

It is clear that the map

$$\mu_n : H^2 \times H^2 \to H^1$$ $$\mu_n(f,g) = fg$$

is well-defined, and it is `known' to be onto if and only if $n = 1$. However, I had never seen an example of an $H^1$ function that **CANNOT** be written as a product of two $H^2$ functions.

According to Theorem 4.2.2 from Rudin's *`Function theory in polydiscs'*, we can construct an explicit example for $n=4$ (hence for all $n > 4$ as well) but he leaves it as an open problem to generalize this to the $n=2$ case (Problem 4.2.3 (a)). I am unable to see how his method generalizes immediately to this case, and I was also not able to find any good reference for this online.

Some classes of functions in $H^1$ that **CAN** be factored as such can be found in Theorem 4.8.4 and Exercise 5.4.9 (b), so it is clear that the example that I am looking for must be outside this class but I do not know where to start. Does anyone have an example at hand?