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$ℓ^{p}$ sums of atoms converge in $S^{'}$ and in $H^{p}$

Statement

Assume Countable Choice. Let $n \geq 1$ , $0 < p \leq 1$ , $s \geq ⌊ n (1 / p - 1) ⌋$ , fix the admissible kernel $φ$ defining $H^{p}$ , and let $N$ be an admissible order for the grand maximal function with $N \geq \max (N_{0} (n, p, φ), n + s + 1)$ . Let $(a_{j})$ be a sequence of $(p, \infty, s)$ -atoms and $(λ_{j}) \in ℓ^{p}$ . Then the series $\sum_{j} λ_{j} a_{j}$ converges absolutely in $S^{'} (R^{n})$ to an element $g \in S^{'} (R^{n})$ ; the partial sums converge to $g$ in the $H^{p}$ quasi-norm of The real Hardy space $H^{p}$ defined by a radial maximal function; $g \in H^{p}$ ; and with $C_{p} = C (n, p, s, N, φ) < \infty$ independent of the atoms and coefficients, $∥ g ∥_{H^{p}} \leq C_{p} (\sum_{j} ∣ λ_{j} ∣^{p})^{1 / p}, ∥ g - \sum_{j \leq J} λ_{j} a_{j} ∥_{H^{p}} \leq C_{p} (\sum_{j > J} ∣ λ_{j} ∣^{p})^{1 / p} .$

Facts & Assumptions

Given: Countable Choice, $n \geq 1$ , $0 < p \leq 1$ , $s \geq ⌊ n (1 / p - 1) ⌋$ , the fixed admissible kernel $φ$ , an admissible order $N \geq \max (N_{0} (n, p, φ), n + s + 1)$ , atoms $a_{j}$ , coefficients $(λ_{j}) \in ℓ^{p}$ .

[F1]

Uniform atom bound: there is $C = C (n, p, s)$ with $∥ M_{N} a_{j} ∥_{L^{p}} \leq C$ and $∣ ⟨ a_{j}, ψ ⟩ ∣ \leq C ∥ ψ ∥_{C^{s + 1} (R^{n})}, ∥ ψ ∥_{C^{s + 1} (R^{n})} : = \max_{∣ β ∣ \leq s + 1} \sup_{x \in R^{n}} ∣ \partial^{β} ψ (x) ∣,$ for every $ψ \in S$ . The pairing estimate follows from assertion 2 of Atoms have uniformly bounded $H^{p}$ quasi-norm and uniformly bounded test pairings because its minimum cube-volume factor is at most one; in particular this is a uniform bound by a continuous Schwartz seminorm.

[F2]

Domination: $M_{φ}^{0} g \leq 2^{N} P_{N} (φ) M_{N} g$ for every $g \in S^{'}$ (The grand maximal function dominates every admissible radial and nontangential maximal function).

[F3]

Convergence in $S^{'}$ of the partial sums implies pointwise convergence of the convolutions: if $g_{J} \to g$ in $S^{'}$ , then $(g_{J} * φ_{t}) (y) \to (g * φ_{t}) (y)$ for every $t > 0$ and $y$ , since $(g_{J} * φ_{t}) (y) = ⟨ g_{J}, φ_{t} (y - \cdot) ⟩$ and $φ_{t} (y - \cdot) \in S$ (Convolution of a tempered distribution with a schwartz function, Tempered distribution).

[F4]

$M_{N}$ is Borel measurable and the $p$ -th power inequality $∣ \sum_{j} z_{j} ∣^{p} \leq \sum_{j} ∣ z_{j} ∣^{p}$ holds for $0 < p \leq 1$ ; monotone convergence applies to the nonnegative measurable partial sums (Measurability and lower semicontinuity of the smooth maximal functions, Monotone convergence for the integral).

Proof technique: absolute convergence of pairings, monotone maximal control and monotone convergence.

Proof

technique · direct

1.1F1givenalgebra

Absolute convergence in $S^{'}$ . Fix $ψ \in S$ . By [F1], $∣ λ_{j} ⟨ a_{j}, ψ ⟩ ∣ \leq C ∥ ψ ∥_{C^{s + 1} (R^{n})} ∣ λ_{j} ∣$ , and $\sum_{j} ∣ λ_{j} ∣ < \infty$ because $(λ_{j}) \in ℓ^{p}$ and $p \leq 1$ . Thus the scalar series $\sum_{j} λ_{j} ⟨ a_{j}, ψ ⟩$ converges absolutely for every $ψ$ . Its limit defines a linear functional $g$ satisfying $∣ ⟨ g, ψ ⟩ ∣ \leq C ∥ ψ ∥_{C^{s + 1} (R^{n})} \sum_{j} ∣ λ_{j} ∣$ ; this continuous-seminorm bound proves $g \in S^{'}$ , and the partial sums converge to $g$ on every Schwartz test.

1.2F3givenalgebra

Maximal control of the sum. For every $J$ put $g_{J} = \sum_{j \leq J} λ_{j} a_{j}$ . For fixed $Ψ \in F_{N}$ , $t > 0$ and $y$ with $∣ y - x ∣ \leq t$ , [F3] gives $(g * Ψ_{t}) (y) = \lim_{J} (g_{J} * Ψ_{t}) (y)$ , so $∣ (g * Ψ_{t}) (y) ∣ \leq {lim sup}_{J} \sum_{j \leq J} ∣ λ_{j} ∣ ∣ (a_{j} * Ψ_{t}) (y) ∣ \leq \sum_{j} ∣ λ_{j} ∣ ∣ (a_{j} * Ψ_{t}) (y) ∣$ . Taking the defining suprema and using $∣ (a_{j} * Ψ_{t}) (y) ∣ \leq M_{N} a_{j} (x)$ for each such $Ψ, t, y$ gives $M_{N} g (x) = \sup_{Ψ \in F_{N}} \sup_{t > 0} \sup_{∣ y - x ∣ \leq t} ∣ (g * Ψ_{t}) (y) ∣ \leq \sum_{j} ∣ λ_{j} ∣ M_{N} a_{j} (x)$ pointwise.

2.1step 1.1step 1.2F1F2F4algebra

$H^{p}$ bound and convergence. By [F4] and the $p$ -power inequality for finite sums, letting the number of terms increase in the display of step 1.2 gives $(M_{N} g)^{p} \leq \sum_{j} ∣ λ_{j} ∣^{p} (M_{N} a_{j})^{p} .$ The nonnegative partial sums on the right are measurable; monotone convergence and [F1] therefore give $∥ M_{N} g ∥_{L^{p}}^{p} \leq C^{p} \sum_{j} ∣ λ_{j} ∣^{p}$ , so $∥ g ∥_{H^{p}} \leq 2^{N} P_{N} (φ) C (\sum_{j} ∣ λ_{j} ∣^{p})^{1 / p}$ by [F2]. Applying the same argument to the tail $g - \sum_{j \leq J} λ_{j} a_{j} = \sum_{j > J} λ_{j} a_{j}$ gives the stated tail bound; in particular the partial sums converge to $g$ in the $H^{p}$ quasi-norm.

3.1step 1.1step 1.2step 2.1∎

Conclusion. Steps 1.1 and 1.2 establish the absolute convergence in $S^{'}$ , and step 2.1 establishes the membership $g \in H^{p}$ , the quasi-norm bound and the tail bound. This proves the lemma.

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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)
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)
Li-An Daniel Wang, Multiplier Theorems on Anisotropic Hardy Spaces (PhD dissertation, University of Oregon, 2012) (standard reference, not scraped)

ℓpℓp sums of atoms converge in S′S′ and in HpHp