What's the role of $H^{p}(\mathbb{R}^{n})$ in modern (harmonic) analysis? The classical theory of $H^p$,due to it's heavy dependence on the complex function theory(such as Blaschke products), seemed to have an insurmountable obstacle barrying its extension to several variables.However, C.Fefferman and E.Stein's remarkable paper'$H^{p}$ spaces of several variables' showed that $H^{p}$ classes can be characterized without any recourse to analytic functions, conjugacy of harmonic functions, etc. Thus $H^{p}$ classes have an intrinsic real variable meaning of their own. Another surprising result they got was that the predual of BMO(functions of bounded mean oscillation) was exactly $H^1$.
Well,what I'm particularly interested is its applications in mordern analysis.For instance,From Ferfferman's work,I know that $H^1$ is sometimes a  proper subsitute for $L^1$,this can be seen from the CZOs(Calderon-Zygmund operators),which are bounded from $H^1$ to itself,but not on $L^{1}$. This is useful when evaluating some singular integral operators through complex interpolation.
Sometimes it's also very convenient to prove a bounded function to be $L^p$ multipliers through $H^1$,for example
$m(\xi)=\psi(\xi)e^{i|\xi|^{a}}|\xi|^{-b}$($b>0$,$a>0$,$a\neq 1$),where $\psi \in C^{\infty}$ is 0 nere 0,and 1 when $|\xi|$ large.Then m is a $L^{p}$ multiplier iff $n|\frac{1}{2}-\frac{1}{p}|\leq \frac{b}{a}$
My question is what's the role of $H^{p}$ in modern (harmonic) analysis,and how people get useful results by choosing $H^p$.
I would appreciate any good examples, as well as some historical outlines on the topics development 
 A: The real Hardy spaces are equivalent to $L^p(\mathbb{R}^n)$ space when $p>1$, and they are much easier to to use than $L^p(\mathbb{R}^n)$ when $p\leq 1$. Since $H^p(\mathbb{R}^n)$ has a maximal function and singular integral generalisation, $H^p(\mathbb{R}^n)$ gives us an extension of maximal function/singular integral results to $p\leq 1$, when they were originally designed for $L^p(\mathbb{R}^n),~p>1$. For example, suppose we have a distribution $K$ with bounded Fourier transform satisfying bounds on the derivative of $K$ equivalent to the smoothness condition known as the Hörmander condition (away from the origin, ie, in $\mathbb{R}^n\setminus 0$). Then we can ascertain that the operator $T:H^p(\mathbb{R}^n)\to H^p(\mathbb{R}^n)$ defined by the convolution
\begin{equation*}
Tf=f\ast K,~~f\in H^p(\mathbb{R}^n)
\end{equation*}
is bounded for $p\leq 1$. 
Check out the following references:


*

*"Modern Fourier Analysis" by Loukas Grafakos

*"Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscilllatory Integrals" by Elias Stein

*"Singular Integrals and Differentiability Properties of Functions" by Elias Stein


They cover what you are looking for in detail.
