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Atomic characterisation of real $H^{p}$ for $0 < p \leq 1$

Statement

Assume Countable Choice. Let $n \geq 1$ , $0 < p \leq 1$ , fix the admissible kernel $φ$ defining $H^{p}$ , and set $s = ⌊ n (1 / p - 1) ⌋$ . Fix an integer $K \geq n / p$ and the associated reproducing pair from Calderon reproducing pair and the telescoping identity in $S^{'}$ . Fix an admissible grand-maximal order $N \geq \max {N_{0} (n, p, φ), n + s + 1}$ as in the two cited lemmas. For $f \in S^{'} (R^{n})$ the following are equivalent:

$f \in H^{p} (R^{n})$ in the sense of The real Hardy space $H^{p}$ defined by a radial maximal function;
there exist a sequence $(λ_{j}) \in ℓ^{p}$ and a sequence $(a_{j})$ of $(p, \infty, s)$ -atoms ( $H^{p}$ atoms with a prescribed moment order) with $f = \sum_{j} λ_{j} a_{j}$ converging in $S^{'} (R^{n})$ .

In that case $∥ f ∥_{H^{p}} ≍_{n, p, N, K, φ} \inf (\sum_{j} ∣ λ_{j} ∣^{p})^{1 / p},$ the infimum being taken over all atomic representations of $f$ , and every such series converges also in the $H^{p}$ quasi-norm.

Facts & Assumptions

Given: Countable Choice, $n \geq 1$ , $0 < p \leq 1$ , the fixed kernel $φ$ and reproducing order $K \geq n / p$ , $s = ⌊ n (1 / p - 1) ⌋$ , an admissible order $N \geq \max {N_{0} (n, p, φ), n + s + 1}$ as in the two cited lemmas, and $f \in S^{'}$ .

[F1]

Level decomposition: if $f \in H^{p}$ then there are $(p, \infty, s)$ -atoms $a_{B}$ and coefficients $λ_{B} > 0$ with $f = \sum_{B} λ_{B} a_{B}$ in $S^{'}$ and $\sum_{B} λ_{B}^{p} \leq C_{1} ∥ f ∥_{H^{p}}^{p}$ , where $C_{1} = C (n, p, N, K, φ, Φ)$ includes the auxiliary flat reproducing kernel $Φ$ (Level decomposition of an $H^{p}$ distribution produces atoms). For the norm bound, use Countable Choice over the integer pairs $(n, K)$ to fix once one admissible kernel $Φ_{n, K}$ from Calderon reproducing pair and the telescoping identity in $S^{'}$ . With this fixed family, $C (n, p, N, K, φ, Φ_{n, K})$ is a function of $n, p, N, K, φ$ . No uniformity over all admissible reproducing kernels is asserted or needed: the atomic class and the infimum over representations do not depend on the auxiliary kernel.

[F2]

$ℓ^{p}$ sums: if $(a_{j})$ are $(p, \infty, s)$ -atoms and $(λ_{j}) \in ℓ^{p}$ , then $g = \sum_{j} λ_{j} a_{j}$ converges absolutely in $S^{'}$ , lies in $H^{p}$ and satisfies $∥ g ∥_{H^{p}} \leq C_{2} (\sum_{j} ∣ λ_{j} ∣^{p})^{1 / p}$ with $C_{2}$ depending on $n, p, s, N, φ$ ; the tail bound of that lemma gives convergence in the $H^{p}$ quasi-norm ( $ℓ^{p}$ sums of atoms converge in $S^{'}$ and in $H^{p}$ ).

Proof technique: the two implications supplied by the level decomposition and the $ℓ^{p}$ -summation lemma, then the infimum.

Proof

technique · direct

1.1F2algebra

2 $\Rightarrow$ 1 and the upper norm bound. Let $f = \sum_{j} λ_{j} a_{j}$ with $(λ_{j}) \in ℓ^{p}$ and atoms $a_{j}$ . By [F2], $f \in H^{p}$ and $∥ f ∥_{H^{p}} \leq C_{2} (\sum_{j} ∣ λ_{j} ∣^{p})^{1 / p}$ ; taking the infimum over all representations gives the inequality $∥ f ∥_{H^{p}} \leq C_{2} \inf (\sum_{j} ∣ λ_{j} ∣^{p})^{1 / p}$ .

1.2F1algebra

1 $\Rightarrow$ 2 and the lower norm bound. Let $f \in H^{p}$ and apply [F1] using the auxiliary kernel $Φ_{n, K}$ fixed there, obtaining $f = \sum_{B} λ_{B} a_{B}$ . Then $(λ_{B}) \in ℓ^{p}$ with $\sum_{B} λ_{B}^{p} \leq C_{1} ∥ f ∥_{H^{p}}^{p}$ , so $f$ has an atomic representation and $\inf (\sum_{j} ∣ λ_{j} ∣^{p})^{1 / p} \leq C_{1}^{1 / p} ∥ f ∥_{H^{p}}$ .

2.1F2step 1.2

$H^{p}$ convergence. If $f = \sum_{j} λ_{j} a_{j}$ with $(λ_{j}) \in ℓ^{p}$ , the tail estimate of [F2] applied to the partial sums gives $∥ f - \sum_{j \leq J} λ_{j} a_{j} ∥_{H^{p}} \leq C_{2} (\sum_{j > J} ∣ λ_{j} ∣^{p})^{1 / p} \to 0$ ; hence the series converges in the $H^{p}$ quasi-norm. This applies in particular to the level-decomposition representation of [F1] and to any atomic representation of $f \in H^{p}$ .

3.1step 1.1step 1.2step 2.1∎

Conclusion. Steps 1.1-1.2 prove the equivalence and the two-sided norm bound, and step 2.1 gives the quasi-norm convergence. This proves the theorem.

Depends on

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Sources

Shai Dekel, Gerard Kerkyacharian, George Kyriazis, Pencho Petrushev, A New Proof of the Atomic Decomposition of Hardy Spaces, Constructive Theory of Functions (Sozopol 2016), pp. 59-73 (standard reference, not scraped)
Li-An Daniel Wang, Multiplier Theorems on Anisotropic Hardy Spaces (PhD dissertation, University of Oregon, 2012) (standard reference, not scraped)
Stefano Meda, Peter Sjogren, Maria Vallarino, Atomic decompositions and operators on Hardy spaces, Revista de la Union Matematica Argentina 50 (2009), no. 2, 15-22 (standard reference, not scraped)

Atomic characterisation of real HpHp for 0<p≤10<p≤1