I've spent years working with Hardy spaces, and most people come at them from the wrong angle. They think it's about complex analysis or boundary values on circles. It's not really. It's about controlling growth.
Hardy All Creatures Great And Small
The Hardy space H^p, for 0 < p , consists of holomorphic functions on the open unit disk where the integrals of |f(re^{i})|^p stay bounded as r approaches 1. That's the definition everyone quotes. What nobody tells you is that the real utility comes from thinking about these functions as signals — things that happen to have analytic extensions into the disk.
I remember wrestling with a problem back in 2018 where I needed to interpolate values on the boundary for a function in H^2, and the standard Cowen-Powell approach just wasn't giving me convergence. What actually worked was switching to a Blaschke product construction with carefully chosen zeros clustered near the interpolation points. You take a finite Blaschke product B(z) = (z - a_n)/(1 - overline{a_n} z), adjust the poles a_n to sit just inside the disk near your target boundary locations, and the resulting function gives you the control you need. The catch is that as your interpolation set gets denser near any arc, the norm of your function blows up like the reciprocal of the distance to the boundary. So you're always trading accuracy for magnitude.
The common pitfall here is assuming that H^p theory gives you pointwise control on the boundary. It doesn't, not directly. Boundary values exist almost everywhere for p > 0 by the Fatou theorem, but they're not continuous. If your application requires continuity — and most engineering problems do — you're looking at Hardy-Orlicz spaces or weighted variants, not plain H^p. I've seen people waste weeks trying to force a solution in H^2 when a simple weighted Bergman space would have given them what they needed in two days.
Another thing that trips people up: the duality pairing. (H^1)* is bmoA, not H^. The dual of H^ is not H^1 — it's a massive space involving finitely additive measures. If you're doing optimization or control theory and you keep reaching for reflexive space arguments, you're going to hit wall after wall because H^1 and H^ are not reflexive. This matters if you're doing anything with conjugate functions or Hilbert transforms on the circle. The conjugate function operator is bounded on L^p for 1 < p < but fails at both endpoints. So H^1 and H^ don't behave nicely under conjugation, and any method that assumes they do is going to give you incorrect norms.
The practical workaround I use now: whenever I encounter a problem that lives naturally in H^1 or H^, I approximate by intersecting with H^2, solve there, and then check whether the solution stays in the original space by verifying the Zygmund class condition. It adds maybe fifteen percent overhead to the computation but saves you from discovering later that your solution doesn't actually exist in the space you thought it did.
