I've been reading up on Hardy spaces and (sub)harmonic functions over the open unit disk $\mathbb{D}\subset\mathbb{C}$, and I've found myself working with atypical objects in mostly-typical situations. Whenever I look up Hardy spaces (in texts, or on the internet), they're virtually always referred to as “spaces of holomorphic/analytic functions”. However, the space-defining norm works for non-holomorphic functions, as well. The prototypical case I'm presently dealing with is as follows: let $f:\mathbb{D}\rightarrow\mathbb{C}$ be holomorphic, and let: $$g\left(z\right)\overset{\textrm{def}}{=}\left(1-\left|z\right|\right)^{1/2}f\left(z\right)$$ Then, the $H^{2}\left(\mathbb{D}\right)$-norm of $g\left(z\right)$ is given by: $$\left\Vert g\right\Vert _{H^{2}\left(\mathbb{D}\right)}=\sqrt{\sup_{r\in\left(0,1\right)}\int_{0}^{1}\left|g\left(re^{2\pi it}\right)\right|^{2}dt}=\sqrt{\sup_{r\in\left(0,1\right)}\left(1-r\right)\int_{0}^{1}\left|f\left(re^{2\pi it}\right)\right|^{2}dt}$$ Letting $f$ admit the power series expansion: $$f\left(z\right)=\sum_{n=0}^{\infty}a_{n}z^{n}$$ the Parseval-Gutzmer Theorem then allows us to write: $$\left\Vert g\right\Vert _{H^{2}\left(\mathbb{D}\right)}=\sqrt{\sup_{r\in\left(0,1\right)}\left(1-r\right)\sum_{n=0}^{\infty}\left|a_{n}\right|^{2}r^{2n}}$$ If, for example, the $a_{n}$s satisfy: $$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}\left|a_{n}\right|^{2}=\alpha\in\left(0,\infty\right)$$ then: $$\lim_{r\uparrow1}\left(1-r\right)\sum_{n=0}^{\infty}\left|a_{n}\right|^{2}r^{2n}=\alpha$$ and hence, necessarily the supremum over $r\in\left(0,1\right)$ is finite, which then guarantees the same for $\left\Vert g\right\Vert _{H^{2}\left(\mathbb{D}\right)}$. Despite all this, however, $g\left(z\right)$ is most definitely not holomorphic on $\mathbb{D}$, even when $f$'s coefficients satisfy the above asymptotic conditions.

In light of this, I then ask: to what extent, if any, does the 'classical' Hardy space theory apply to these functions? More generally, if we create an “extended” version of $H^{2}\left(\mathbb{D}\right)$ by considering the space of all, say, continuous functions $g:\mathbb{D}\rightarrow\mathbb{C}$ with finite $H^{2}\left(\mathbb{D}\right)$-norm, do we end up with a Banach space, and—if so—does it have any noteworthy properties? References to any books or papers on this sort of thing would be much appreciated.

Additionally, supposing that $f$ satisfies the conditions in the preivous pragraphs, of particular interest to me is the behavior of: $$\int_{0}^{1}\left|g\left(re^{2\pi it}\right)\right|^{2}dt=\left(1-r\right)\int_{0}^{1}\left|f\left(re^{2\pi it}\right)\right|^{2}dt$$ as a real-valued function of $r\in\left[0,1\right)$. I know that, for any subharmonic function $v:\mathbb{D}\rightarrow\left[-\infty,\infty\right]$, the quantity: $$\int_{0}^{1}\left|v\left(re^{2\pi it}\right)\right|^{2}dt$$ is strictly increasing as a function of $r$, and from this, one can express notions of harmonic majorants, and the like. To that end, it is easy to show that $\left|g\left(z\right)\right|^{2}$ is subharmonic on the annulus $A=\left\{ z\in\mathbb{D}:\frac{1}{2}\leq\left|z\right|<1\right\}$. Consequently, the mean value inequality property: $$\left|g\left(z\right)\right|^{2}\leq\int_{0}^{1}\left|g\left(z+re^{2\pi it}\right)\right|^{2}dt$$ holds for all z in the interior of $A$ whenever $r$ is sufficiently small. What happens, though, if we integrate $\left|g\left(z\right)\right|^{2}$ along a circle centered at $0$ of radius $r\in\left(\frac{1}{2},1\right)$?