2 added 1 characters in body; deleted 12 characters in body; added 19 characters in body; added 1 characters in body

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 $m(\xi)=\psi(\xi)e^{i|\xi|^{a}}|\xi|^{-b}$($b>0$,$0 m(\xi)=\psi(\xi)e^{i|\xi|^{a}}|\xi|^{-b}$($b>0$,$a>0$,$a\neq 1$),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 reslts results by choosing $H^p$.
I would appreciate any good examples, as well as some historical outlines on the topics development

1

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 $m(\xi)=\psi(\xi)e^{i|\xi|^{a}}|\xi|^{-b}$($b>0$,$0 My question is what's the role of$H^{p}$in modern (harmonic) analysis,and how people get useful reslts by choosing$H^p\$.
I would appreciate any good examples, as well as some historical outlines on the topics development