Real-Variable Hardy Spaces H^p and the Atomic Decomposition

Intuition Beginner

The space of absolutely-integrable functions — the ones whose total accumulated size is finite — is the natural home for a great deal of analysis. But it has a defect that shows up the moment you apply the most useful operations in the subject, the ones that recover a signal's twin or its derivatives from averaged data. Those operations take a perfectly fine integrable input and produce an output whose total size is infinite, by a hair: the output decays just slowly enough at infinity to escape being integrable. The integrable functions are not closed under the operations you most want to use.

Real Hardy space is the repair. It is a slightly smaller class of inputs, carved out by one extra demand: the function must have a kind of built-in balance. A bounded function automatically lives in the integrable world; a Hardy-space function must additionally average out to zero in a precise sense, and at the finest level it is glued together from tiny standardized pieces, each one a bump that integrates to zero. Because every piece is balanced, the dangerous slow tails that the useful operations produce get cancelled before they can accumulate to infinity.

The headline payoff: the operations that wreck the integrable class send Hardy space neatly back into the integrable class — and send Hardy space back into itself. The class of balanced inputs is exactly the right place to stand so that singular operations stay tame. And there is a dual side of the same coin: the functions that pair nicely against these balanced inputs are precisely the ones with bounded mean oscillation, the slowly-growing functions met in the previous unit.

The one-sentence takeaway: real Hardy space is the corrected version of the integrable functions — built from standardized balanced bumps — on which the singular operations of analysis stay bounded, exactly where the plain integrable class fails.

Visual Beginner

Picture a single building block: a bump supported in one small box, with equal area above and below the zero line, so its total signed area is zero. Stacking many such balanced bumps, scaled and shifted, with the sizes of the scalings adding up in a controlled way, reconstructs any Hardy-space function. The balance of each piece is the whole point; an unbalanced bump (all positive, like a plain spike) would not qualify.

The lower panel carries the punchline. A Hardy-space function is not measured by its raw height or its total size; it is measured by the cheapest way to write it as a balanced sum of these standardized boxed bumps, adding up the coefficient sizes. A plain integrable function can fail to have any such balanced decomposition, which is exactly why it can be wrecked by singular operations. A Hardy-space function, by construction, always does — and the operations act atom by atom, each balanced piece staying tame.

Worked example Beginner

We build the simplest possible balanced building block on the line and check the two defining properties by hand.

Step 1. Take the box to be the interval from $0$ to $2$, of width $2$. Define a bump $a$ that equals $+1$ on the left half, from $0$ to $1$, equals $-1$ on the right half, from $1$ to $2$, and equals $0$ everywhere outside the box.

Step 2. Check the balance (the zero-average property). The signed area is the value times the width on each half: on the left it is $(+1)\times 1=1$, on the right it is $(-1)\times 1=-1$. Adding them, $1+(-1)=0$. The bump integrates to zero, so it is balanced.

Step 3. Check the size cap. A standardized building block on a box of width $w$ is required to be no taller than one divided by $w$. Here $w=2$, so the cap is one-half. But our bump has height $1$, which is bigger than one-half. So $a$ itself is twice too tall; the genuine standardized block is $a$ divided by $2$, namely the bump equal to $+0.5$ on the left half and $-0.5$ on the right half. That one is balanced and obeys the size cap.

Step 4. Read off the coefficient. To write the original $a$ as a coefficient times a standardized block, we use $a=2\times (\text{standardized block})$. The coefficient is $2$. The cost we record for this one-piece decomposition is the size of the coefficient, namely $2$.

What this tells us: every Hardy-space function is assembled from balanced boxed bumps like this standardized block, and the price of the assembly is the sum of the coefficient sizes. Balance plus a size cap are the only two demands on each piece. The plain absolute value used to measure integrable functions is replaced by this assembly cost, and the balance is what tames the singular operations.

Check your understanding Beginner

Formal definition Intermediate+

Throughout, $0<p\le 1$ unless stated otherwise; cubes $Q$ have sides parallel to the axes, $|Q|$ is Lebesgue measure, and $f_Q={\text{fint}}_{Q}f$. The Fourier transform is that of 02.10.04. Fix a Schwartz function $\Phi$ with $\int \Phi =1$ and write ${\Phi }_t(x)=t^{-n}\Phi (x/t)$.

Definition (grand maximal function). For $N\in \mathbb{N}$ let ${\mathcal{A}}_N=\{\phi \in \mathcal{S}:{\sup }_{|\alpha |,|\beta |\le N}\Vert x^{\beta }{\partial }^{\alpha }\phi {\Vert }_{\infty }\le 1\}$. The grand maximal function of a tempered distribution $f$ is $${\mathcal{M}}_{N}f(x)=\underset{\phi \in {\mathcal{A}}_N}{\sup }\underset{t>0}{\sup }|({\phi }_t\ast f)(x)|,{\phi }_t(y)=t^{-n}\phi (y/t).$$ The non-tangential maximal function attached to a single $\Phi$ is ${\mathcal{M}}_{\Phi }^{\ast }f(x)={\sup }_{|y-x|<t}|({\Phi }_t\ast f)(y)|$, and the *radial* (vertical) maximal function is ${\mathcal{M}}_{\Phi }f(x)={\sup }_{t>0}|({\Phi }_t\ast f)(x)|$.

Definition (real Hardy space, maximal characterization). For $0<p\le 1$ and $N=N(n,p)$ large enough, the real Hardy space is $$H^p({\mathbb{R}}^n)=\{f\in {\mathcal{S}}^{\prime }:{\mathcal{M}}_{N}f\in L^p({\mathbb{R}}^n)\},\Vert f{\Vert }_{H^p}=\Vert {\mathcal{M}}_{N}f{\Vert }_{L^p}.$$ This is a (quasi-)Banach space; for $p<1$, $\Vert \cdot {\Vert }_{H^p}^p$ is subadditive but $\Vert \cdot {\Vert }_{H^p}$ is only a quasinorm. The Fefferman-Stein theorem (below) makes ${\mathcal{M}}_N$, ${\mathcal{M}}_{\Phi }^{\ast }$, and ${\mathcal{M}}_{\Phi }$ define the same space with comparable (quasi)norms, so the choices of $\Phi$ and $N$ are immaterial.

Definition ($(p,q)$-atom). Let $0<p\le 1\le q\le \infty$ with $p<q$, and set the moment order $N_p=\lfloor n(1/p-1)\rfloor$. A function $a$ is a $(p,q)$-atom if there is a cube $Q$ with:

1. (Support.) $\mathrm{supp}a\subseteq Q$.
2. (Size.) $\Vert a{\Vert }_{L^q}\le |Q{|}^{1/q-1/p}$.
3. (Moments / cancellation.) $\int a(x)x^{\alpha }dx=0$ for every multi-index $\alpha$ with $|\alpha |\le N_p$.

For $p=1$ the moment order is $N_1=0$, so a $(1,q)$-atom is a single-mean-zero bump with $\Vert a{\Vert }_{L^q}\le |Q{|}^{1/q-1}$; the case $q=\infty$ gives the height cap $\Vert a{\Vert }_{\infty }\le |Q{|}^{-1}$.

Definition (atomic Hardy space). The atomic Hardy space $H_{\mathrm{at},q}^p$ consists of all $f={\sum }_j{\lambda }_j{a}_j$ (convergence in ${\mathcal{S}}^{\prime }$) with each $a_j$ a $(p,q)$-atom and ${\sum }_j|{\lambda }_j{|}^p<\infty$, normed by $$\Vert f{\Vert }_{H_{\mathrm{at},q}^p}=\inf (\sum _j|{\lambda }_j{|}^p{)}^{1/p},$$ the infimum over all atomic representations.

Counterexamples to common slips Intermediate+

• $H^p$ is not a space of functions for $p<1$ in the naïve sense; its elements are distributions whose maximal function is in $L^p$. For $p<1$ even $H^p$-functions need not be locally integrable, and the cancellation forces them to be genuinely oscillatory; treating $H^p$ as a subspace of $L^p$ is wrong — $L^p⊈H^p$ and $H^p⊈L^p$ for $p<1$.

• More vanishing moments are needed as $p$ decreases — a single mean-zero condition does not suffice below $p=n/(n+1)$. The moment order $N_p=\lfloor n(1/p-1)\rfloor$ grows without bound as $p\to 0$. Using only $\int a=0$ for small $p$ breaks the atom-to-maximal estimate; the higher moments are essential.

• The size normalization is tied to $L^q$, not $L^{\infty }$. Stating the size bound as $\Vert a{\Vert }_{\infty }\le |Q{|}^{-1/p}$ only describes $(p,\infty )$-atoms; the general $(p,q)$-atom uses $\Vert a{\Vert }_{L^q}\le |Q{|}^{1/q-1/p}$. All choices of $q\in (p,\infty ]$ give the same atomic space — a theorem, not a definition.

• $L^1$ is not $H^1$. Every $H^1$ function is in $L^1$ with $\Vert f{\Vert }_{L^1}\le C\Vert f{\Vert }_{H^1}$, but the inclusion is strict: a single non-balanced bump (mean nonzero) is in $L^1\setminus H^1$. The defining difference is the global cancellation $\int f=0$ together with its maximal-function refinement.

Key theorem with proof Intermediate+

Theorem (atomic decomposition of $H^p$; Coifman 1974, Latter 1978). Fix $0<p\le 1$ and $p<q\le \infty$, $q\ge 1$. A tempered distribution $f$ lies in $H^p({\mathbb{R}}^n)$ if and only if it admits a decomposition $f={\sum }_j{\lambda }_j{a}_j$ into $(p,q)$-atoms with ${\sum }_j|{\lambda }_j{|}^p<\infty$, and $$\Vert f{\Vert }_{H^p}\approx \inf (\sum _j|{\lambda }_j{|}^p{)}^{1/p},$$ with constants depending only on $n,p,q$. In particular all the atomic spaces $H_{\mathrm{at},q}^p$ ($p<q\le \infty$) coincide with $H^p$ and with each other.

Proof. We prove the harder direction, the construction of atoms from the maximal-function definition (the "$\Rightarrow$"), in the model case $p=1$, $q=\infty$; the easy direction follows the Bridge. Assume $\Vert f{\Vert }_{H^1}=\Vert {\mathcal{M}}_{\Phi }f{\Vert }_{L^1}<\infty$ with $\mathcal{M}={\mathcal{M}}_{\Phi }$ the radial maximal function; one may take $f\in L^1$ with $\int f=0$ for the construction (the general distributional case is reached by approximation).

Step 1 (Calderón-Zygmund selection on the maximal function). For $k\in \mathbb{Z}$ set the open level sets ${\Omega }_k=\{x:\mathcal{M}f(x)>2^k\}$. Each ${\Omega }_k$ has finite measure, $|{\Omega }_k|\le 2^{-k}\Vert \mathcal{M}f{\Vert }_{L^1}$, and ${\Omega }_{k+1}\subseteq {\Omega }_k$. Apply a Whitney decomposition 02.19.03 to each ${\Omega }_k$, obtaining cubes $\{Q_{k,j}{\}}_j$ with bounded overlap of their dilates $\{3Q_{k,j}\}$, ${\bigcup }_j{Q}_{k,j}={\Omega }_k$, and side length comparable to the distance to ${\Omega }_k^c$.

Step 2 (the bad/atomic split). Choose a partition of unity $\{{\eta }_{k,j}\}$ subordinate to $\{3Q_{k,j}\}$ with ${\sum }_j{\eta }_{k,j}={\chi }_{{\Omega }_k}$. Define the good part at level $k$ by freezing $f$ off ${\Omega }_k$ and replacing it on each Whitney cube by the $\eta$-weighted local mean, $$g_k=f{\chi }_{{\Omega }_k^c}+\sum _j{c}_{k,j}{\eta }_{k,j},c_{k,j}=\frac{\int f{\eta }_{k,j}}{\int {\eta }_{k,j}},$$ so that $b_{k,j}=(f-c_{k,j}){\eta }_{k,j}$ has $\int b_{k,j}=0$. The telescoping difference $g_{k+1}-g_k$ is supported in ${\Omega }_k$, has integral zero on each cube, and — using that $|f|\le \mathcal{M}f\le 2^{k+1}$ off ${\Omega }_{k+1}$ and the bounded-overlap geometry — satisfies the pointwise bound $\Vert g_{k+1}-g_k{\Vert }_{\infty }\le C{2}^k$.

Step 3 (atoms emerge). Write the difference as a sum over cubes of mean-zero bumps, $$g_{k+1}-g_k=\sum _j{A}_{k,j},A_{k,j}\text{ supported in }3Q_{k,j},\int A_{k,j}=0,\Vert A_{k,j}{\Vert }_{\infty }\le C{2}^k.$$ Set ${\lambda }_{k,j}=C{2}^k|3Q_{k,j}|$ and $a_{k,j}={\lambda }_{k,j}^{-1}{A}_{k,j}$. Then $a_{k,j}$ is supported in the cube $3Q_{k,j}$, has mean zero, and obeys $\Vert a_{k,j}{\Vert }_{\infty }\le |3Q_{k,j}{|}^{-1}$ — a $(1,\infty )$-atom.

Step 4 (summability of coefficients). Since $g_k\to f$ in $L^1$ as $k\to +\infty$ and $g_k\to 0$ as $k\to -\infty$ (because $\int f=0$), telescoping gives $f={\sum }_{k,j}{\lambda }_{k,j}{a}_{k,j}$ in $L^1$, hence in ${\mathcal{S}}^{\prime }$. The coefficient sum is controlled by the maximal function: using ${\sum }_j|3Q_{k,j}|\le C|{\Omega }_k|$ (bounded overlap) and the level-set identity, $$\sum _{k,j}|{\lambda }_{k,j}|=C\sum _k{2}^k\sum _j|3Q_{k,j}|\le C^{\prime }\sum _k{2}^k|{\Omega }_k|\le C^{\prime \prime }{\int }_0^{\infty }|\{\mathcal{M}f>\lambda \}|d\lambda =C^{\prime \prime }\Vert \mathcal{M}f{\Vert }_{L^1}=C^{\prime \prime }\Vert f{\Vert }_{H^1}.$$ Thus ${\sum }_{k,j}|{\lambda }_{k,j}|\le C\Vert f{\Vert }_{H^1}$, which is the $p=1$ coefficient bound. $\square$

Bridge. This decomposition builds toward every boundedness theorem for $H^p$, and the same atoms appear again in the Fefferman duality $(H^1{)}^{\ast }=\mathrm{BMO}$ and in the $H^1\to L^1$ continuity of Calderón-Zygmund operators. The foundational reason the construction works is that the Calderón-Zygmund stopping-time selection of 02.19.03, run on the level sets of the maximal function rather than on $|f|$ itself, manufactures the cancellation by hand: each telescoped difference $g_{k+1}-g_k$ is forced to have mean zero on every Whitney cube, so the atoms are balanced by fiat and their coefficient sum is exactly the layer-cake integral of $\mathcal{M}f$. This is exactly the dual mechanism to the John-Nirenberg recursion of 02.20.01: there the stopping time upgraded mean oscillation to exponential integrability, here it disassembles a maximal-function bound into balanced unit pieces, and putting these together the pairing $⟨a,\phi ⟩=\int a(\phi -{\phi }_Q)$ converges because the atom's cancellation meets the $\mathrm{BMO}$ function's John-Nirenberg integrability. The easy direction — that any atomic sum has $\Vert f{\Vert }_{H^p}\le C(\sum |{\lambda }_j{|}^p{)}^{1/p}$ — is dual to this and generalises the single-atom estimate $\Vert \mathcal{M}a{\Vert }_{L^p}\le C$, the central insight being that one normalized atom has $H^p$-quasinorm bounded by a constant, after which subadditivity of $\Vert \cdot {\Vert }_{H^p}^p$ assembles the sum.

Exercises Intermediate+

Advanced results Master

Theorem 1 (maximal-function characterizations coincide; Fefferman-Stein 1972 Acta Math. 129, 137; Burkholder-Gundy-Silverstein 1971). For $0<p\le 1$ and $N=N(n,p)$ sufficiently large, the grand, non-tangential, and radial maximal functions all define the same space, with $$\Vert {\mathcal{M}}_{N}f{\Vert }_{L^p}\approx \Vert {\mathcal{M}}_{\Phi }^{\ast }f{\Vert }_{L^p}\approx \Vert {\mathcal{M}}_{\Phi }f{\Vert }_{L^p},$$ constants depending only on $n,p,\Phi$. The radial maximal function alone suffices: a tempered distribution with ${\sup }_{t>0}|{\Phi }_t\ast f|\in L^p$ already has all the stronger maximal functions in $L^p$. This is the theorem that makes the maximal definition of $H^p$ robust ^{[Fefferman-Stein 1972]}.

Theorem 2 (atomic decomposition; Coifman 1974 Studia Math. 51, 269; Latter 1978 Studia Math. 62, 93). For $0<p\le 1$ and any $p<q\le \infty$, $q\ge 1$, the atomic space $H_{\mathrm{at},q}^p$ equals $H^p$ with comparable (quasi)norms, and the moment order $N_p=\lfloor n(1/p-1)\rfloor$ is sharp: $(p,q)$-atoms with fewer vanishing moments do not generate $H^p$. The independence from $q$ means one may always reduce to the most convenient atom size, $L^{\infty }$ or $L^2$ ^{[Latter 1978]}.

Theorem 3 (Calderón-Zygmund operators on the endpoints). A Calderón-Zygmund operator $T$ bounded on $L^2$ with Hörmander kernel maps $H^1\to L^1$ boundedly, and under the additional cancellation $T^{\ast }1=0$ (i.e. $T$ maps atoms to atoms in a suitable sense) maps $H^p\to H^p$ for $\frac{n}{n+1}<p\le 1$. The pair $(L^{\infty }\to \mathrm{BMO},H^1\to L^1)$ is the correct endpoint replacement for the failed $L^{\infty }\to L^{\infty }$ and $L^1\to L^1$ bounds; interpolating $H^1\to L^1$ with $L^2\to L^2$ recovers $L^p$ for $1<p<2$ ^{[Stein 1993]}.

Theorem 4 (Fefferman duality $(H^1)^ = \mathrm{BMO}$; Fefferman 1971; Fefferman-Stein 1972 Acta Math. 129, 137).* The dual of $H^1({\mathbb{R}}^n)$ is $\mathrm{BMO}({\mathbb{R}}^n)$: every $\phi \in \mathrm{BMO}$ defines a bounded functional $f\mapsto \int f\phi$ on $H^1$ (interpreted via atoms) with $\Vert \phi {\Vert }_{(H^1{)}^{\ast }}\approx \Vert \phi {\Vert }_{\ast }$, and every bounded functional on $H^1$ arises this way. The pairing converges because the atom's cancellation $\int a=0$ meets the John-Nirenberg integrability of $\mathrm{BMO}$ 02.20.01, so $\int a\phi =\int a(\phi -{\phi }_Q)$ is absolutely convergent ^{[Fefferman-Stein 1972]}.

Theorem 5 (interpolation: the $H^p$ scale). The complex and real interpolation spaces between $H^{p_0}$ and $H^{p_1}$ ($0<p_0<p_1\le \infty$, with $H^{\infty }$ read as $\mathrm{BMO}$) are again Hardy spaces: $[H^{p_0},H^{p_1}{]}_{\theta }=H^p$ with $1/p=(1-\theta )/p_0+\theta /p_1$. In particular interpolation between $H^1$ and $\mathrm{BMO}=(H^1{)}^{\ast }$ reproduces the full $L^p$ scale $1<p<\infty$, placing $H^1$ and $\mathrm{BMO}$ as the genuine endpoints below $L^1$ and above $L^{\infty }$ ^{[Fefferman-Stein 1972]}.

Theorem 6 (Coifman-Weiss spaces of homogeneous type; Coifman-Weiss 1977 Bull. AMS 83, 569). The atomic definition of $H^p$ requires only a quasimetric and a doubling measure: on any space of homogeneous type the atomic Hardy space $H_{\mathrm{at}}^p$ is well defined for $p$ near $1$, the $H^1$-$\mathrm{BMO}$ duality persists, and Calderón-Zygmund operators remain bounded. This abstraction reveals that the Euclidean Fourier-analytic apparatus is inessential: cancellation against a doubling geometry is the true content ^{[Coifman-Weiss 1977]}.

Synthesis. The atomic decomposition is the foundational reason that real Hardy space behaves well under singular operations: it disassembles every $H^p$ element into balanced unit pieces, and this is exactly the structural device — Calderón-Zygmund stopping-time selection of 02.19.03 run on the maximal function's level sets — that converts a single maximal-function bound into a sum of mean-zero atoms with controlled coefficients. The central insight is a chain of dualities centered on cancellation. Putting these together, the four characterizations (grand maximal, non-tangential, atomic, Riesz-transform) are the same object seen from four directions, and the equivalence is what lets each boundedness proof choose its most convenient face: atoms for $H^1\to L^1$ (Theorem 3), the maximal function for interpolation (Theorem 5). This is dual to the BMO theory of 02.20.01: the John-Nirenberg exponential integrability there is exactly what makes the atomic pairing $\int a(\phi -{\phi }_Q)$ converge here, so $(H^1{)}^{\ast }=\mathrm{BMO}$ is the precise statement that atoms and oscillation are paired endpoints. The bridge unifying the picture is that $H^1$ and $\mathrm{BMO}$ bracket the Lebesgue scale: interpolating between them generalises Plancherel into the continuum $1<p<\infty$, and the whole edifice ports — by Coifman-Weiss (Theorem 6) — to any doubling geometry, revealing that the Euclidean Fourier transform was never the point; the central insight is that cancellation against a doubling measure is.

Full proof set Master

Proposition 1 ($H^p=L^p$ for $p>1$). For $1<p\le \infty$ and the radial maximal function, $\Vert f{\Vert }_{H^p}\approx \Vert f{\Vert }_{L^p}$.

Proof. For $p>1$: ${\mathcal{M}}_{\Phi }f(x)={\sup }_t|({\Phi }_t\ast f)(x)|\le \Vert \Phi {\Vert }_{L^1}Mf(x)$ is not quite right because $\Phi$ need not be radial-decreasing; instead dominate $|\Phi |\le C{\sum }_k{2}^{-k(n+1)}{\chi }_{B(0,2^k)}/|B(0,2^k)|$ by a radial-decreasing majorant, giving ${\mathcal{M}}_{\Phi }f\le CMf$ pointwise. Since $M$ is bounded on $L^p$ for $p>1$ 02.19.03, $\Vert f{\Vert }_{H^p}=\Vert {\mathcal{M}}_{\Phi }f{\Vert }_p\le C\Vert Mf{\Vert }_p\le C\Vert f{\Vert }_p$. Conversely, ${\Phi }_t\ast f\to f$ in $L^p$ and a.e. as $t\to 0$ (approximate identity), so $|f(x)|\le {\mathcal{M}}_{\Phi }f(x)$ at Lebesgue points, whence $\Vert f{\Vert }_p\le \Vert {\mathcal{M}}_{\Phi }f{\Vert }_p=\Vert f{\Vert }_{H^p}$. $\square$

Proposition 2 (a single atom has bounded $H^p$-quasinorm). For $0<p\le 1$ and a $(p,\infty )$-atom $a$ with the required $N_p$ vanishing moments, $\Vert a{\Vert }_{H^p}\le C_{n,p}$.

Proof. Let $a$ be supported in $Q$, $\Vert a{\Vert }_{\infty }\le |Q{|}^{-1/p}$, with vanishing moments up to order $N_p$. Split ${\mathbb{R}}^n=Q^{\ast }\cup (Q^{\ast }{)}^c$, $Q^{\ast }=2\sqrt{n}Q$. On $Q^{\ast }$: $\mathcal{M}a\le CMa$, and by Hölder with exponent $>1/p$ and the $L^r$-boundedness of $M$, $${\int }_{Q^{\ast }}(\mathcal{M}a{)}^p\le |Q^{\ast }{|}^{1-p/r}(\int (Ma{)}^r{)}^{p/r}\le C|Q{|}^{1-p/r}\Vert a{\Vert }_r^p\le C|Q{|}^{1-p/r}(|Q{|}^{1/r}|Q{|}^{-1/p}{)}^p=C.$$ Off $Q^{\ast }$: a Taylor expansion of ${\Phi }_t$ to order $N_p$, killed against the vanishing moments, yields $\mathcal{M}a(x)\le C|Q{|}^{(N_p+1)/n}|x-x_Q{|}^{-n-N_p-1}|Q{|}^{-1/p+1}$ after the size normalization; since $N_p+1>n(1/p-1)$, the exponent makes $(\mathcal{M}a{)}^p$ integrable off $Q^{\ast }$ with ${\int }_{(Q^{\ast }{)}^c}(\mathcal{M}a{)}^p\le C_{n,p}$. Adding, $\Vert a{\Vert }_{H^p}^p\le C_{n,p}$. The vanishing-moment order $N_p$ is exactly the smallest making the far tail $p$-integrable. $\square$

Proposition 3 (atomic sums converge in $H^p$). If ${\sum }_j|{\lambda }_j{|}^p<\infty$ and each $a_j$ is a $(p,\infty )$-atom, then $f={\sum }_j{\lambda }_j{a}_j$ converges in ${\mathcal{S}}^{\prime }$ and $\Vert f{\Vert }_{H^p}^p\le C_{n,p}{\sum }_j|{\lambda }_j{|}^p$.

Proof. By $p$-subadditivity of the maximal functional (the maximal operator is sublinear, and $t\mapsto t^p$ is subadditive for $p\le 1$), $\mathcal{M}f\le ({\sum }_j|{\lambda }_j{|}^p(\mathcal{M}{a}_j{)}^p{)}^{1/p}$ pointwise in the sense that $(\mathcal{M}f{)}^p\le {\sum }_j|{\lambda }_j{|}^p(\mathcal{M}{a}_j{)}^p$. Integrating and using Proposition 2, $$\Vert f{\Vert }_{H^p}^p=\int (\mathcal{M}f{)}^p\le \sum _j|{\lambda }_j{|}^p\int (\mathcal{M}{a}_j{)}^p=\sum _j|{\lambda }_j{|}^p\Vert a_j{\Vert }_{H^p}^p\le C_{n,p}\sum _j|{\lambda }_j{|}^p.$$ Convergence in ${\mathcal{S}}^{\prime }$ follows because partial sums are Cauchy in $H^p$ and $H^p\hookrightarrow {\mathcal{S}}^{\prime }$ continuously. This is the easy direction; Theorem 2 supplies the converse. $\square$

Proposition 4 ($H^1⊊L^1$). $H^1({\mathbb{R}}^n)\subseteq L^1({\mathbb{R}}^n)$ with $\Vert f{\Vert }_{L^1}\le \Vert f{\Vert }_{H^1}$, and the inclusion is strict.

Proof. For $f\in H^1$, $|f(x)|\le {\mathcal{M}}_{\Phi }f(x)$ a.e. (Lebesgue points of the approximate identity), so $\Vert f{\Vert }_{L^1}\le \Vert {\mathcal{M}}_{\Phi }f{\Vert }_{L^1}=\Vert f{\Vert }_{H^1}$. Strictness: take $f={\chi }_{[0,1]}$ on $\mathbb{R}$, a nonnegative bump with $\int f=1\ne 0$. Then for large $|x|$, ${\Phi }_t\ast f(x)\sim {\Phi }_t(x)\int f\sim |x{|}^{-1}$ at the optimal scale $t\sim |x|$, so ${\mathcal{M}}_{\Phi }f(x)\gtrsim |x{|}^{-1}$, which is not integrable on $\mathbb{R}$. Hence $f\in L^1\setminus H^1$. The obstruction is precisely the nonvanishing mean: $\int f\ne 0$ forces a non-integrable maximal tail. $\square$

Proposition 5 (necessity of $\int f=0$ for $H^1$). If $f\in H^1({\mathbb{R}}^n)$ then $\int f=0$.

Proof. Since $f\in L^1$ (Proposition 4), $\hat{f}$ is continuous and $\hat{f}(0)=\int f$. For $f\in H^1$, ${\mathcal{M}}_{\Phi }f\in L^1$; choosing $\Phi$ with $\hat{\Phi }(0)=1$, the bound $|{\Phi }_t\ast f(x)|\le {\mathcal{M}}_{\Phi }f(x)$ with ${\Phi }_t\ast f\to f$ shows $\int {\mathcal{M}}_{\Phi }f<\infty$ forces the mean of $f$ to vanish: were $\int f=c\ne 0$, the tail estimate of Proposition 4 gives ${\mathcal{M}}_{\Phi }f(x)\gtrsim |c||x{|}^{-n}\notin L^1$, contradiction. Hence $\hat{f}(0)=\int f=0$. $\square$

Proposition 6 (BMO functionals are well-defined on $H^1$). For $\phi \in \mathrm{BMO}$ the functional ${\ell }_{\phi }(f)=\int f\phi$ is well-defined and bounded on $H^1$ with $\Vert {\ell }_{\phi }\Vert \le C\Vert \phi {\Vert }_{\ast }$.

Proof. This is Exercise 8: on finite $(1,2)$-atomic sums $f=\sum {\lambda }_j{a}_j$, $|{\ell }_{\phi }(f)|\le \sum |{\lambda }_j||\int a_j\phi |\le C\Vert \phi {\Vert }_{\ast }\sum |{\lambda }_j|$ by the pairing bound (Exercise 6: $|\int a\phi |\le \Vert a{\Vert }_2|Q{|}^{1/2}({\text{fint}}_Q|\phi -{\phi }_Q{|}^2{)}^{1/2}\le C\Vert \phi {\Vert }_{\ast }$ using the atom size and John-Nirenberg $L^2$-oscillation 02.20.01). Infimizing over representations and using $\Vert f{\Vert }_{H_{\mathrm{at}}^1}\approx \Vert f{\Vert }_{H^1}$ (Theorem 2) gives $|{\ell }_{\phi }(f)|\le C\Vert \phi {\Vert }_{\ast }\Vert f{\Vert }_{H^1}$; density of finite atomic sums extends ${\ell }_{\phi }$ to all of $H^1$. Representation-independence follows from the uniform bound. $\square$

Connections Master

• BMO and the John-Nirenberg inequality 02.20.01. The dual partner and direct prerequisite. The Fefferman duality $(H^1{)}^{\ast }=\mathrm{BMO}$ pairs the mean-zero atoms of $H^1$ against $\mathrm{BMO}$ functions, and the absolute convergence of the pairing $\int a(\phi -{\phi }_Q)$ is supplied exactly by the John-Nirenberg $L^2$- and exponential-oscillation integrability proved there; the BMO unit's Theorem 4 and this unit's Theorem 4 are the two faces of one duality.

• Calderón-Zygmund singular integral operators 02.19.03. The direct prerequisite and the engine of the atomic decomposition. The Whitney/stopping-time selection that builds atoms from the maximal function is the Calderón-Zygmund decomposition run on level sets, and the same Hörmander kernel estimate that gives weak-$(1,1)$ there gives $H^1\to L^1$ here — the endpoint that replaces the failed $L^1\to L^1$ on which singular integrals are genuinely bounded.

• $L^p$ spaces, Hölder, Minkowski, Riesz-Fischer completeness 02.07.06. The direct prerequisite carrying the layer-cake calculus behind the coefficient-sum bound, the Cauchy-Schwarz and Hölder estimates in the single-atom and pairing proofs, and the interpolation scaffolding that places $H^1$ and $\mathrm{BMO}$ as the endpoints bracketing the $L^p$ scale via Theorem 5.

• Fourier transform on ${\mathbb{R}}^n$ and the Plancherel theorem 02.10.04. The direct prerequisite supplying the Riesz-transform symbols $-i{\xi }_j/|\xi |$, the harmonic-conjugate (Stein-Weiss) system behind the Riesz characterization, and the mean-vanishing consequence $\hat{f}(0)=\int f=0$ that distinguishes $H^1$ from $L^1$.

• Littlewood-Paley theory and square functions 02.20.03. The analytic sibling. The square-function characterization of $H^p$ — $\Vert f{\Vert }_{H^p}\approx \Vert Sf{\Vert }_{L^p}$ for the Littlewood-Paley square function $S$ — is the third major description of Hardy space, dual to the Carleson-measure characterization of $\mathrm{BMO}$ and equivalent to both the maximal and atomic definitions developed here.

Historical & philosophical context Master

The Hardy spaces $H^p$ originate in complex analysis: G. H. Hardy and others studied, for the unit disc, the analytic functions whose boundary values have controlled $L^p$ means, and the conjugate-function theorems of M. Riesz and Kolmogorov gave the first $L^p$ and weak-$(1,1)$ bounds. The real-variable theory — freeing $H^p$ from analyticity and conjugate harmonic functions — was inaugurated by Elias Stein and Guido Weiss, who replaced the single analytic function by a system of conjugate harmonic functions (the Riesz-transform system) and defined $H^p$ for $0<p\le 1$ in higher dimensions through it.

The maximal-function characterization, which removed even the harmonic structure, was the achievement of Charles Fefferman and Elias Stein in their 1972 Acta Mathematica paper $H^p$ spaces of several variables ^{[Fefferman-Stein 1972]}, building on the disc-model maximal theorem of Donald Burkholder, Richard Gundy, and Martin Silverstein ^{[Burkholder-Gundy-Silverstein 1971]}; the same paper established the duality $(H^1{)}^{\ast }=\mathrm{BMO}$. The atomic decomposition — reducing the entire space to balanced standardized pieces — was found by Ronald Coifman in dimension one ^{[Coifman 1974]} and extended to ${\mathbb{R}}^n$ by Robert Latter ^{[Latter 1978]}. Ronald Coifman and Guido Weiss then showed that atoms require only a doubling quasimetric, defining $H^p$ on spaces of homogeneous type ^{[Coifman-Weiss 1977]} and severing the theory's last tie to the Fourier transform.

Bibliography Master

@article{FeffermanStein1972,
  author  = {Fefferman, Charles and Stein, Elias M.},
  title   = {$H^p$ spaces of several variables},
  journal = {Acta Mathematica},
  volume  = {129},
  year    = {1972},
  pages   = {137--193}
}

@article{BurkholderGundySilverstein1971,
  author  = {Burkholder, Donald L. and Gundy, Richard F. and Silverstein, Martin L.},
  title   = {A maximal function characterization of the class $H^p$},
  journal = {Transactions of the American Mathematical Society},
  volume  = {157},
  year    = {1971},
  pages   = {137--153}
}

@article{Coifman1974,
  author  = {Coifman, Ronald R.},
  title   = {A real variable characterization of $H^p$},
  journal = {Studia Mathematica},
  volume  = {51},
  year    = {1974},
  pages   = {269--274}
}

@article{Latter1978,
  author  = {Latter, Robert H.},
  title   = {A characterization of $H^p(\mathbb{R}^n)$ in terms of atoms},
  journal = {Studia Mathematica},
  volume  = {62},
  year    = {1978},
  pages   = {93--101}
}

@article{CoifmanWeiss1977,
  author  = {Coifman, Ronald R. and Weiss, Guido},
  title   = {Extensions of Hardy spaces and their use in analysis},
  journal = {Bulletin of the American Mathematical Society},
  volume  = {83},
  year    = {1977},
  pages   = {569--645}
}

@article{SteinWeiss1960,
  author  = {Stein, Elias M. and Weiss, Guido},
  title   = {On the theory of harmonic functions of several variables I},
  journal = {Acta Mathematica},
  volume  = {103},
  year    = {1960},
  pages   = {25--62}
}

@book{Stein1993,
  author    = {Stein, Elias M.},
  title     = {Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals},
  publisher = {Princeton University Press},
  year      = {1993}
}

@book{Grafakos2014Modern,
  author    = {Grafakos, Loukas},
  title     = {Modern Fourier Analysis},
  edition   = {3},
  publisher = {Springer},
  year      = {2014}
}