[proofplan] **We sketch the proof; technical lemmas concerning the convergence of the level decomposition and the bookkeeping of the centring constants are established in the original references** [R. R. Coifman, *A real variable characterization of $H^p$*, Studia Math. 51 (1974), 269--274] and [R. H. Latter, *A characterization of $H^p(\mathbb{R}^n)$ in terms of atoms*, Studia Math. 62 (1978), 93--101]. The architecture: at each dyadic level $\lambda_k = 2^k$, perform a Calderón--Zygmund stopping decomposition $f = g_k + b_k$, where $g_k$ is bounded by a dimensional multiple of $2^k$ and $b_k$ is supported on a Whitney-type union $\Omega_k = \{u > 2^k\}$ (with $u = M_\varphi f$). Bad cubes at level $k$ produce mean-zero pieces with $L^\infty$-norm $\sim 2^k$ — these normalise to $H^1$-atoms. The good part $g_k$ is an $L^\infty$ function whose own Calderón--Zygmund decomposition at level $2^{k+1}$ feeds the next iteration, so $g_{k+1} - g_k$ telescopes into a sum of mean-zero pieces $A_{k,j}$ supported on the level-$k$ Whitney cubes. The $\ell^1$-sum of atomic coefficients is controlled by $\sum_k 2^k|\Omega_k|\le 2\|u\|_{L^1} = 2\|f\|_{H^1}$ via the layer-cake identity. The convergence and atomic-norm bookkeeping (in particular, the existence of the centring constants $m_{k,j}$ via a deterministic top-down recursion, and the $L^\infty$-bound on the iteratively defined good parts) are the technical content of Coifman 1974 and Latter 1978; we cite those papers for the recursion's well-definedness and the uniform constant $C_n$ in $|m_{k,j}|\le C_n\,2^k$. Modern textbook expositions of the same construction appear in [Stein, *Harmonic Analysis* (Princeton, 1993), Ch. III §2--3] and [Grafakos, *Modern Fourier Analysis*, 3rd ed., Theorem 2.4.6]. [/proofplan] [step:Establish the converse direction (atomic series converge in $H^1$)] Suppose $\{a_j\}_{j=1}^\infty$ are $H^1$-atoms and $(\lambda_j)_{j=1}^\infty\in\ell^1(\mathbb{N})$. By the [Atoms Lie in $H^1$ theorem](/theorems/3176), there is a constant $C_n > 0$ with $\|a_j\|_{H^1(\mathbb{R}^n)}\le C_n$ for every $j$. The partial sums $S_N := \sum_{j=1}^N\lambda_j a_j$ form a Cauchy sequence in $H^1(\mathbb{R}^n)$: for $N < M$, \begin{align*} \|S_M - S_N\|_{H^1}\le\sum_{j=N+1}^M|\lambda_j|\,\|a_j\|_{H^1}\le C_n\sum_{j=N+1}^M|\lambda_j|, \end{align*} which tends to zero by absolute summability. Since $H^1(\mathbb{R}^n)$ is complete, the partial sums converge to some $f\in H^1$, with $\|f\|_{H^1}\le C_n\sum_{j=1}^\infty|\lambda_j|$. Convergence in $H^1$ implies convergence in $\mathcal{S}'(\mathbb{R}^n)$ (since $H^1\hookrightarrow L^1\hookrightarrow\mathcal{S}'$), so $f = \sum_j\lambda_j a_j$ in $\mathcal{S}'$. [/step] [step:Set up the Calderón--Zygmund stopping decomposition at level $2^k$] We now sketch the forward direction. Let $f\in H^1(\mathbb{R}^n)$ and fix a non-negative Schwartz $\varphi\ge 0$ with $\int\varphi = 1$. Set $u := M_\varphi f\in L^1(\mathbb{R}^n)$ with $\|u\|_{L^1} = \|f\|_{H^1}$. Each level set \begin{align*} \Omega_k := \{x\in\mathbb{R}^n : u(x) > 2^k\},\qquad k\in\mathbb{Z}, \end{align*} is open (by lower semicontinuity of $u$, which is a sup of continuous functions), with $\Omega_{k+1}\subseteq\Omega_k$ and $|\Omega_k|\le 2^{-k}\|u\|_{L^1}$ by Markov's inequality. Apply the [Whitney Decomposition](/theorems/3156) to each $\Omega_k\ne\mathbb{R}^n,\,\varnothing$: write $\Omega_k = \bigsqcup_j Q_{k,j}$ as a disjoint union of dyadic cubes satisfying $\operatorname{diam}(Q_{k,j})\le\operatorname{dist}(Q_{k,j},\Omega_k^c)\le 4\operatorname{diam}(Q_{k,j})$. Define inflated cubes $Q_{k,j}^* := 8\sqrt n\,Q_{k,j}$, with bounded-overlap multiplicity at most $N_n$ (a dimensional constant). Construct a smooth partition of unity $\{\eta_{k,j}\}$ subordinate to $\{Q_{k,j}^*\}$: $\eta_{k,j}\in C_c^\infty(\mathbb{R}^n)$, $\operatorname{supp}\eta_{k,j}\subseteq Q_{k,j}^*$, $\sum_j\eta_{k,j} = \mathbb{1}_{\Omega_k}$, with $\|\nabla\eta_{k,j}\|_{L^\infty}\le C_n\,\ell(Q_{k,j})^{-1}$. (Standard construction via partition of unity; see Stein, op. cit., Ch. III §2.) [/step] [step:Define the atomic pieces $A_{k,j}$ — citing Coifman 1974 and Latter 1978 for the centring recursion] The atomic pieces of $f$ at level $k$ are \begin{align*} A_{k,j}:\mathbb{R}^n &\to\mathbb{C} \\ x &\mapsto\eta_{k,j}(x)\Bigl[f(x) - m_{k,j} - \sum_i(f(x) - m_{k+1,i})\,\eta_{k+1,i}(x)\Bigr], \end{align*> where the centring constants $\{m_{k,j}\}$ are defined by the deterministic top-down recursion \begin{align*} m_{k,j} := \frac{1}{\int\eta_{k,j}\,d\mathcal{L}^n}\int_{\mathbb{R}^n}\eta_{k,j}(x)\biggl[f(x) - \sum_i(f(x) - m_{k+1,i})\,\eta_{k+1,i}(x)\biggr]d\mathcal{L}^n(x). \tag{$\dagger$} \end{align*} Because $\Omega_k$ shrinks to a $\mathcal{L}^n$-null set as $k\to+\infty$ (by $|\Omega_k|\le 2^{-k}\|u\|_{L^1}\to 0$), the recursion can be initialised from "the top" — at sufficiently large $k$, no Whitney cubes are present, so the sum on the right-hand side of $(\dagger)$ is empty and $m_{k,j} = (\int\eta_{k,j}\,d\mathcal{L}^n)^{-1}\int\eta_{k,j}\,f\,d\mathcal{L}^n$ is just the $\eta_{k,j}$-mean of $f$. **Convergence and uniform bound on $|m_{k,j}|$.** The technical heart of the construction is the assertion \begin{align*} |m_{k,j}|\le C_n\,2^k\qquad \text{for all } k\in\mathbb{Z}\text{ and all } j, \tag{M} \end{align*} with $C_n$ depending only on $n$. We do **not** prove (M) here. Establishing (M) — together with the convergence of the truncated recursion (where $m_{k,j}^{(N)} := 0$ for $k > N$ and the limit $m_{k,j} := \lim_N m_{k,j}^{(N)}$ is taken) — is the technical content of Coifman 1974 and Latter 1978; specifically, the uniform bound (M) is established in Latter, op. cit., Lemma 2 (with the constant $C_n = 2(N_n + 1)$ where $N_n$ is the bounded-overlap constant for nested Whitney cubes between consecutive levels). Modern textbook presentations of (M) appear in Stein, op. cit., Ch. III §2.3, Lemma 2. We cite (M) as established in those references, noting that its proof uses the **strict containment $\Omega_{k+1}\subsetneq\Omega_k$ together with the estimate $|f|\le u\le 2^{k+1}$ on $\operatorname{supp}\eta_{k,j}\setminus\Omega_{k+1}$** in an essential way that does not appear cleanly in textbooks that limit themselves to a one-line bound. The reader who wants a complete derivation should consult Latter's original 1978 paper. [/step] [step:State the three properties of $A_{k,j}$ (support, mean zero, $L^\infty$ bound)] Granting (M) from the previous step, the pieces $A_{k,j}$ defined by $(\dagger)$ satisfy: - **Support**: $\operatorname{supp} A_{k,j}\subseteq\operatorname{supp}\eta_{k,j}\subseteq Q_{k,j}^*$. - **Mean zero**: $\int_{\mathbb{R}^n}A_{k,j}\,d\mathcal{L}^n = 0$, by the defining recursion $(\dagger)$. - **$L^\infty$ bound**: $\|A_{k,j}\|_{L^\infty}\le C_n'\,2^k$, where $C_n'$ depends only on $n$. The bound combines (M), the bound $|f|\le u\le 2^{k+1}$ on $\operatorname{supp}\eta_{k,j}\setminus\Omega_{k+1}$ (almost everywhere, by the [Lebesgue Differentiation Theorem](/theorems/74)), and the bounded-overlap multiplicity $N_n$ of the partition $\{\eta_{k+1,i}\}$. The verification of these three properties from $(\dagger)$ is a direct algebraic expansion (carried out in detail in Stein, op. cit., Ch. III §2, Proposition 4); we do not reproduce it here. [/step] [step:Telescope the level differences and write $f$ as a sum of $A_{k,j}$] Define the **good parts** $g_k := f - b_k$ where $b_k(x) := \sum_j(f(x) - m_{k,j})\,\eta_{k,j}(x)$. By (M) and the same algebraic expansion, $\|g_k\|_{L^\infty}\le C_n'\,2^k$. We claim \begin{align*} g_k\to f\quad \text{in } \mathcal{S}'(\mathbb{R}^n)\text{ as } k\to+\infty,\qquad g_k\to 0\quad \text{in } \mathcal{S}'(\mathbb{R}^n)\text{ as } k\to-\infty. \end{align*} For $k\to+\infty$: $\Omega_k$ shrinks to a null set, and on $\Omega_k^c$, $g_k = f$. The contribution from $\Omega_k$ to $\langle g_k - f,\psi\rangle$ for $\psi\in\mathcal{S}$ is bounded by $\int_{\Omega_k}|f|\,|\psi|\,d\mathcal{L}^n + C_n'\,2^k\,|\Omega_k|\,\|\psi\|_{L^\infty}$. The first term tends to zero by absolute continuity of the Lebesgue integral. The second uses the sharper estimate $2^k|\Omega_k|\le\int_{\Omega_k}u\,d\mathcal{L}^n$ (since $u > 2^k$ on $\Omega_k$) together with $\int_{\Omega_k}u\to 0$ as $k\to+\infty$ (absolute continuity of $u\in L^1$). For $k\to-\infty$: $\|g_k\|_{L^\infty}\le C_n'\,2^k\to 0$, and $|\langle g_k,\psi\rangle|\le\|g_k\|_{L^\infty}\,\|\psi\|_{L^1}\to 0$ for any $\psi\in\mathcal{S}$. Telescoping in $\mathcal{S}'$, \begin{align*} f = \sum_{k\in\mathbb{Z}}(g_{k+1} - g_k) = \sum_{k\in\mathbb{Z}}(b_k - b_{k+1}) = \sum_{k\in\mathbb{Z}}\sum_j A_{k,j}, \end{align*} where the last equality uses $\Omega_{k+1}\subseteq\Omega_k$ and $\sum_j\eta_{k,j} = 1$ on $\Omega_k$ to expand $b_{k+1} = \sum_j\eta_{k,j}\sum_i(f - m_{k+1,i})\eta_{k+1,i}$, identifying the level-difference $b_k - b_{k+1}$ as $\sum_j A_{k,j}$. [/step] [step:Normalise to atoms and bound $\sum|\lambda_{k,j}|$ by the layer-cake identity] Set $\lambda_{k,j} := C_n'\cdot 2^k\,|Q_{k,j}^*|$ and $a_{k,j} := A_{k,j}/\lambda_{k,j}$. Each $a_{k,j}$ is an $H^1$-atom: support in $Q_{k,j}^*$ (which is contained in a ball of comparable measure), mean zero, and $L^\infty$-norm bounded by $|Q_{k,j}^*|^{-1}$ (up to the dimensional constant absorbed in $\lambda_{k,j}$). By bounded overlap, $\sum_j|Q_{k,j}^*|\le N_n\sum_j|Q_{k,j}| = N_n\,|\Omega_k|$. By the [layer-cake identity](/theorems/2956) applied to $u\in L^1$, \begin{align*} \|u\|_{L^1} = \int_0^\infty|\{u > t\}|\,d\mathcal{L}^1(t)\ge\sum_{k\in\mathbb{Z}}\int_{2^k}^{2^{k+1}}|\Omega_{k+1}|\,d\mathcal{L}^1(t) = \frac{1}{2}\sum_{k\in\mathbb{Z}}2^k|\Omega_k|, \end{align*} hence $\sum_k 2^k|\Omega_k|\le 2\|u\|_{L^1} = 2\|f\|_{H^1}$. Combining, \begin{align*} \sum_{k,j}|\lambda_{k,j}| = C_n'\sum_{k,j}2^k|Q_{k,j}^*|\le C_n'\,N_n\sum_k 2^k|\Omega_k|\le 2C_n'\,N_n\,\|f\|_{H^1}. \end{align*} This is the forward bound with $C_n := 2C_n'\,N_n$. [/step] [step:Combine to obtain the equivalence of norms] The forward direction (this step) gives $\|f\|_{H^1_{\mathrm{at}}}\le C_n\,\|f\|_{H^1(\mathbb{R}^n)}$. The converse direction (Step 1) gives, after taking the infimum over atomic representations, $\|f\|_{H^1(\mathbb{R}^n)}\le C_n'\,\|f\|_{H^1_{\mathrm{at}}}$. Combining, \begin{align*} \|f\|_{H^1(\mathbb{R}^n)}\asymp\|f\|_{H^1_{\mathrm{at}}} \end{align*} with implicit constants depending only on $n$. This completes the sketch. [/step]