Hardy Spaces, BMOA, and the Real-Variable $H^1$¶

Job: develop the real-variable Hardy spaces $H^p$ (atomic decomposition, maximal/Poisson), the $H^1$–BMO duality (Fefferman–Stein), the singular integrals on $H^1$ via cancellation, the Carleson measure / $H^1$-BMOA, and the $H^p(\mathbb{D})$ boundary / CR — with the $H^1$–BMO duality + atomic CZ as the worked example.

Core¶

The Hardy spaces $H^p$ and the real-variable¶

The Hardy space $H^p(\mathbb{R}^n)$ for $0<p\le\infty$: the real-variable $H^p$, defined via the maximal function $f^+=\sup_{t>0}|P_t*f|$ (the Poisson maximal), $f\in H^p$ iff $f^+\in L^p$. For $p>1$, $H^p=L^p$; for $p\le1$, $H^p\subset L^p$ is a proper subspace (the cancellation / the mean-zero).

$H^p$ / atomic / maximal	Description
$H^p(\mathbb{R}^n)$	the real-variable Hardy ($f^+=\sup_t
Atomic decomposition	$f=\sum\lambda_j a_j$ (atoms $a_j$: $\|a_j\|_\infty\le
Maximal / Poisson	$f^+=\sup_t

The BMOA and the $H^1$–BMO duality¶

The BMO (bounded mean oscillation, recalled from functional analysis / harmonic): $\|f\|_{\mathrm{BMO}}=\sup_Q\frac1{|Q|}\int_Q|f-f_Q|<\infty$ (the John–Nirenberg). The BMOA (the BMO of the analytic / the boundary values of $H^\infty(\mathbb{D})$, the dyadic BMOA).

Theorem (Fefferman–Stein). $(H^1)^*=\mathrm{BMO}$ — the $H^1$–BMO duality, the real-variable pairing $\langle f,g\rangle=\int fg$ for $f\in H^1, g\in\mathrm{BMO}$.

BMO / BMOA / $H^1$-BMO	Description
BMO	$\|f\|_{\mathrm{BMO}}=\sup_Q\frac1{
BMOA	the BMO of the analytic / the boundary $H^\infty(\mathbb{D})$ (the dyadic BMOA)
$H^1$–BMO duality	$(H^1)^*=\mathrm{BMO}$ (the Fefferman–Stein, the $H^1$ the Hardy / the BMO the dual)

The singular integrals on $H^1$ and the cancellation¶

Singular on $H^1$ / cancellation / wavelet	Description
$T:H^1\to L^1$	the CZ the $H^1$ the bounded (the $T(a_j)$ the atom the $L^1$, the $\sum
Cancellation	the atom $a_j$ the mean-zero $\int a_j=0$ (the cancellation the key, the CZ the bounded via the cancellation)
Wavelet (Meyer)	the wavelet the unconditional basis for $H^1$ / BMO (the Meyer, the functional analysis / function spaces)

The Carleson measure and the $H^1$ / the BMOA¶

Carleson / $H^1$-BMOA / interpolation	Description
Carleson measure	$\mu$ on $\mathbb{R}^{n+1}_+$ the Carleson ($\sup_Q\mu(Q\times(0,r))/
$H^1$ / BMOA and Carleson	the $H^1$ the Carleson measure / the BMOA the interpolation (the Carleson / the interpolation)
Interpolation	the $H^1$ / BMO / $L^p$ the interpolation (the Carleson the measure)

The Hardy spaces and the boundary / the complex¶

$H^p(\mathbb{D})$ / real-variable / CR	Description
$H^p(\mathbb{D})$	the holomorphic $H^p$ on the disk (the boundary values, the F. & M. Riesz)
Real-variable vs complex	the $H^p(\mathbb{R}^n)$ the real-variable / the $H^p(\mathbb{D})$ the complex (the boundary / the SCV / CR)
SCV connection	the Hardy $H^p$ the boundary the CR, the SCV

See it¶

$H^1$–BMO duality (Fefferman–Stein) and the atomic decomposition / the CZ on $H^1$. The $H^1(\mathbb{R}^n)$ (the real-variable Hardy, the $f\in H^1$ the maximal / the atomic). The atomic: the $f=\sum\lambda_j a_j$ the atoms (the $a_j$ the mean-zero / the moment, the $\|a_j\|_\infty\le|Q_j|^{-1}$, the $\int a_j=0$ the cancellation), the $\|f\|_{H^1}\sim\sum|\lambda_j|$. The $H^1$–BMO duality: the $(H^1)^*=\mathrm{BMO}$ (the Fefferman–Stein, the $H^1$ the Hardy / the BMO the dual, the pairing $\langle f,g\rangle=\int fg$ for $f\in H^1, g\in\mathrm{BMO}$). The CZ on $H^1$: the $T:H^1\to L^1$ (the $T(a_j)$ the atom the $L^1$, the $\sum|\lambda_j|$ the $H^1$ the norm, the CZ the bounded). The key point: the $H^1$ the atomic / the $H^1$–BMO duality (Fefferman–Stein) / the CZ the bounded — the real-variable Hardy the harmonic analysis the dual pair.

Check¶

Show that the atomic decomposition $f=\sum\lambda_j a_j$ with $\sum|\lambda_j|<\infty$ gives $f\in H^1$, and explain the cancellation.

The atoms $a_j$: $\|a_j\|_\infty\le|Q_j|^{-1}$, $\mathrm{supp}\,a_j\subset Q_j$, $\int a_j=0$ (the mean-zero). The $H^1$ norm: $\|f\|_{H^1}\sim\inf\sum|\lambda_j|$ (the infimum over the atomic decompositions). The cancellation: the $\int a_j=0$ the key (the mean-zero the atom the $H^1$, the CZ the $T(a_j)$ the $L^1$ via the cancellation / the mean-zero, the $T(a_j)$ the $L^1$ the decay from the kernel). Without the cancellation (the $\int a_j\ne0$), the $a_j$ the $L^1$ but not the $H^1$ (the $H^1$ the strict, the $L^1$ the larger). So the atomic the $H^1$ (the $\sum|\lambda_j|$), the cancellation the key.

Explain the Fefferman–Stein duality $(H^1)^*=\mathrm{BMO}$, and contrast with $(L^1)^*=L^\infty$.

The $(L^1)^*=L^\infty$ (the classical duality, the $L^1$ the dual $L^\infty$). The $H^1\subset L^1$ (the proper, the cancellation), the $(H^1)^*=\mathrm{BMO}$ (the Fefferman–Stein, the BMO $\supset L^\infty$ the larger, the $H^1$ the smaller the dual the larger). The pairing: $\langle f,g\rangle=\int fg$ for $f\in H^1, g\in\mathrm{BMO}$ (the $fg\in L^1$ the $H^1$-BMO, the duality). The BMO the $L^\infty$ + the oscillation (the John–Nirenberg), the $H^1$ the $L^1$ + the cancellation (the atomic). So the $(H^1)^*=\mathrm{BMO}$ (the Fefferman–Stein), the $(L^1)^*=L^\infty$ (the classical) — the $H^1$-BMO the real-variable dual pair.

Pitfalls¶

The $H^1$ the real-variable ($p\le1$ the non-trivial, the $H^p\subset L^p$ the proper) — the $p>1$ the $H^p=L^p$, the $p\le1$ the $H^p$ the strict (the cancellation).
The BMO not $L^\infty$ (the BMO the $L^\infty$ + the oscillation, the John–Nirenberg the exponential) — the BMO the larger, the $L^\infty\subset\mathrm{BMO}$.
The BMOA the analytic (the BMO of the analytic / the boundary $H^\infty(\mathbb{D})$, the dyadic BMOA) — the BMOA the analytic, the BMO the real-variable.
The wavelet the unconditional (the Meyer: the wavelet the unconditional basis for $H^1$ / BMO) — the wavelet the unconditional, the Haar the conditional.
The Carleson measure the $H^1$-BMOA the interpolation (the Carleson the measure, the $H^1$ / BMO / $L^p$ the interpolation).

Refs¶

Fefferman–Stein, "$H^p$ spaces of several variables" (Acta 1972); the $H^1$–BMO duality
Coifman–Rochberg–Weiss, "Factorization theorems for Hardy spaces" (1976); the atomic decomposition
John–Nirenberg, "On functions of bounded mean oscillation" (1961); Meyer, Wavelets and Operators (the unconditional basis)
Carleson, "Interpolation by bounded analytic functions" (1962); F. & M. Riesz, the boundary values

Next¶

FA index · Foundations · Restriction & Strichartz · Oscillatory & Carleson · Abstract Harmonic & Pontryagin · Functional analysis / harmonic · Measure theory / harmonic

BMO / BMOA / \(H^1\)-BMO	Description
BMO	$\|f\|_{\mathrm{BMO}}=\sup_Q\frac1{
BMOA	the BMO of the analytic / the boundary \(H^\infty(\mathbb{D})\) (the dyadic BMOA)
\(H^1\)–BMO duality	\((H^1)^*=\mathrm{BMO}\) (the Fefferman–Stein, the \(H^1\) the Hardy / the BMO the dual)

Singular on \(H^1\) / cancellation / wavelet	Description
\(T:H^1\to L^1\)	the CZ the \(H^1\) the bounded (the \(T(a_j)\) the atom the \(L^1\), the $\sum
Cancellation	the atom \(a_j\) the mean-zero \(\int a_j=0\) (the cancellation the key, the CZ the bounded via the cancellation)
Wavelet (Meyer)	the wavelet the unconditional basis for \(H^1\) / BMO (the Meyer, the functional analysis / function spaces)

Carleson / \(H^1\)-BMOA / interpolation	Description
Carleson measure	\(\mu\) on \(\mathbb{R}^{n+1}_+\) the Carleson ($\sup_Q\mu(Q\times(0,r))/
\(H^1\) / BMOA and Carleson	the \(H^1\) the Carleson measure / the BMOA the interpolation (the Carleson / the interpolation)
Interpolation	the \(H^1\) / BMO / \(L^p\) the interpolation (the Carleson the measure)

\(H^p(\mathbb{D})\) / real-variable / CR	Description
\(H^p(\mathbb{D})\)	the holomorphic \(H^p\) on the disk (the boundary values, the F. & M. Riesz)
Real-variable vs complex	the \(H^p(\mathbb{R}^n)\) the real-variable / the \(H^p(\mathbb{D})\) the complex (the boundary / the SCV / CR)
SCV connection	the Hardy \(H^p\) the boundary the CR, the SCV

Hardy Spaces, BMOA, and the Real-Variable \(H^1\)¶

Core¶

The Hardy spaces \(H^p\) and the real-variable¶

The BMOA and the \(H^1\)–BMO duality¶

The singular integrals on \(H^1\) and the cancellation¶

The Carleson measure and the \(H^1\) / the BMOA¶