So, Folks, here's the deal:
After looking at this question, posted a little earlier on this site, and getting quite inspired by the beauty of this kind of result, I have got quite interested on this subject.
Maybe because of the lack of experience, maybe because of my own stupidness, I could not properly solve the problem. But, instead of getting me mad, this had just the inverse reaction: I really want to get deeper on it, but my original working area is Harmonic Analysis, with not a big emphasis on Complex Analysis.
And you come in exactly now: how can I work on something that has to do with both Harmonic and Fourier Analysis and this kind of more Potential-Theoretic stuff, like analytic capacity?
Recently, the connections between Harmonic and Complex Analysis have been made visible; for those who are unfamiliar with these, I'd recommend the (excelent) text by Emanuel Carneiro, available here, where some relations between extremal problems and complex-analytic theorems with the Fourier Transform are showed, and the book by John Garnett, Bounded Analytic Functions, where some applications of classical Harmonic Analysis on recent theory of Bounded Analytic Function are also proved.
If this kind of intersection is nonempty, can somebody recommend me any books for me to start studying, or, as the theory is not that old, at least some good references for papers in this area?
Any help will be (hugely) appreciated.