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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?

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    $\begingroup$ I do no think that at the time the book was written an answer to this problem was known. Rudin says that the problems "raise questions which I have not been able to answer". For sure it is still an open problem if H^2 of he bidisc is equal to the weak product of H^1 with itself. I wouldn't be surprised if also this problem is still open. $\endgroup$
    – an_ordinary_mathematician
    Commented Feb 12 at 22:43
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    $\begingroup$ Thanks for the reply. I have always seen that this is treated as a fact, so I thought it must be written down somewhere and I used Rudin's book because I don't know of any other reference for Hardy spaces in SCV. Also, weak factorization of $H^1$ is already shown to exist: See Corollary 1.3 and the following remark in this paper by Ferguson and Lacey. $\endgroup$
    – EG2023
    Commented Feb 12 at 23:17
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    $\begingroup$ Unfortunatelly this paper is known to contain a gap in the proof whhich was discovered recently. See muse.jhu.edu/article/785253/pdf. About the "simple" factorization do you have a reference where this is treated like a fact ? $\endgroup$
    – an_ordinary_mathematician
    Commented Feb 13 at 6:17
  • $\begingroup$ Oh I see. I wasn't aware of the gap in the paper. Good to know! Actually, the reason why I asked this question is because I never saw any reference to this, but heard it in passing several times. In Rudin's book, he writes ``$\mu_n$ is undoubtedly not onto if $n>1$" but then only shows it for $n > 3$. $\endgroup$
    – EG2023
    Commented Feb 13 at 13:32

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I asked the polydisc experts for a reference and in fact it is known. It was proved by by J.P. Rosay in this paper. It is in french but it shouldn't be difficult to understand.

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  • $\begingroup$ Perfect! Exactly what I was looking for. Thanks! :) $\endgroup$
    – EG2023
    Commented May 21 at 15:04

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