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It seems so, otherwise it seems incompatible with the condition of Theorem 6.2.4. By the way in my answer you need one last step with the one dimensional Hilbert transform in order to pass from the Fourier multiplier $|\xi_i|^3$ to the Fourier multiplier $\xi_i^3$. Do you need to fill in the details ?
I think it exists. You should probabaly modify the classical example of P. Du Bois Reymond of of a continuous periodic function which has Fourier series which diverges at some point.
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 ?
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.
