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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

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The div-curl lemma (in the formulation of Coifman, Lions, Meyer, and Semmes), which roughly speaking showed that the inner product of a divergence-free $L^2$ vector field and a curl-free $L^2$ vector field was in $H^1$ and not just in $L^1$, was useful in the theory of harmonic maps, in particular used in the original argument of Helein on the regularity of 2D harmonic maps. See e.g. the ICM lecture of Lions. – Terry Tao Jul 16 at 18:11

1 Answer 1

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.

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