The Hardy spaces $H^p$ were originally defined in terms of complex functions on the unit disk. Namely, an element of $H^p$ is a holomorphic function $f$ on the unit disk such that $\sup_{r} \int_{0}^{2 \pi} |f(re^{i \theta})|^p d \theta$ is finite. This quantity (in analog with the $L^p$ norms) is used to define the norm on $H^p$, so this is a Banach space for $p \geq 1$ (and a Hilbert space for $p=2$). There is a very rich and interesting theory of these complex Hardy spaces. For instance, radial limits exist almost everywhere (though this is also true for the broader case of $f$ in the Nevanlinna class), and have vanishing Fourier coefficients at negative indices. The function $f$ can be reconstructed from the boundary data via a Poisson integral. More interestingly, the corona theorem is a statement about the spectrum of the Banach algebra $H^\infty$; it states that the ideals $M_z := {f: f(z)=0}$ for $z$ in the unit disk are dense. All this is based upon the complex-variable theory, which came first.
However, this original definition via complex-variable theory was "wrong" in the sense in that it had to be modified to allow for the real-variable theory in higher dimensions. The "real-variable" definition of an element of $H^p$ is defined in terms of distributions with "maximal functions" (defined with respect to a normalized Schwarz function) in $L^p$. Much of the modern theory of Hardy spaces (e.g. duality of $H^1$ and $BMO$, stability under singular integrals) was developed, I think, in this more general setting (which, according to Stein, took hold in the 1960s).