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Fourier transform decay of real $H^{p}$ elements

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 of the atomic decomposition, and an admissible grand-maximal order $N \geq \max {N_{0} (n, p, φ), n + s + 1}$ . There is $C = C (n, p, N, K, φ) < \infty$ such that every $f \in H^{p} (R^{n})$ has a Fourier transform that is a continuous function on $R^{n} ∖ {0}$ and satisfies $∣ \hat{f} (ξ) ∣ \leq C ∥ f ∥_{H^{p}} ∣ ξ ∣^{n (1 / p - 1)}, ξ \neq 0,$ and moreover $\lim_{ξ \to 0} \frac{∣ \hat{f} (ξ) ∣}{∣ ξ ∣^{n (1 / p - 1)}} = 0.$ Here $\hat{f}$ is the tempered-distribution Fourier transform (Fourier transform of a tempered distribution), identified with a continuous function off the origin by the estimate.

Facts & Assumptions

Given: Countable Choice, $n \geq 1$ , $0 < p \leq 1$ , the fixed kernel $φ$ , reproducing order $K$ and order $N$ , $s = ⌊ n (1 / p - 1) ⌋$ , $f \in H^{p}$ , and multi-indices as in $C^{k}$ maps and multi-index derivative notation in Euclidean space.

[F1]

Atomic characterisation: $f = \sum_{j} λ_{j} a_{j}$ in $S^{'}$ with $(p, \infty, s)$ -atoms $a_{j}$ and $(λ_{j}) \in ℓ^{p}$ ; the representation may be chosen with $\sum_{j} ∣ λ_{j} ∣ \leq C_{n, p, N, K, φ} ∥ f ∥_{H^{p}}$ (Atomic characterisation of real $H^{p}$ for $0 < p \leq 1$ ).

[F2]

For an $L^{1}$ atom the distributional transform agrees with the integral transform: the absolute double integral against a Schwartz test $χ$ is bounded by $∥ a ∥_{1} ∥ χ ∥_{1}$ , so Fubini identifies $⟨ a, \hat{χ} ⟩$ with $\int \hat{a} χ$ (Fubini's theorem for L^1 functions on a sigma-finite product). Fourier transform is continuous on $S^{'}$ : if $g_{J} \to g$ in $S^{'}$ then $\hat{g_{J}} \to \hat{g}$ in $S^{'}$ (Fourier transform of a tempered distribution).

[F3]

For an atom $a$ supported in a cube $Q$ with centre $c_{Q}$ , $∥ a ∥_{L^{\infty}} \leq ∣ Q ∣^{- 1 / p}$ and moments vanishing through order $s$ ( $H^{p}$ atoms with a prescribed moment order), the Taylor expansion of $x \mapsto e^{- 2 π i x \cdot ξ}$ about $c_{Q}$ through order $s$ gives $∣ \hat{a} (ξ) ∣ \leq C \min (∣ Q ∣^{1 - 1 / p}, ∣ ξ ∣^{s + 1} ∣ Q ∣^{1 - 1 / p + (s + 1) / n}) (ξ \neq 0),$ with $C = C (n, s)$ : the first bound is $∥ a ∥_{1} \leq ∥ a ∥_{\infty} ∣ Q ∣ \leq ∣ Q ∣^{1 - 1 / p}$ , and the second uses the vanishing moments, the Taylor remainder bound $∣ \partial^{β} e^{- 2 π i x \cdot ξ} ∣ \leq (2 π ∣ ξ ∣)^{∣ β ∣}$ and $\int_{Q} ∣ x - c_{Q} ∣^{s + 1} d x \leq C ℓ (Q)^{s + 1 + n}$ . Consequently $∣ \hat{a} (ξ) ∣ \leq C^{'} ∣ ξ ∣^{n (1 / p - 1)}$ for $ξ \neq 0$ (split at $∣ ξ ∣ ℓ (Q) ≍ 1$ and use $s + 1 > n (1 / p - 1)$ ) and $∣ \hat{a} (ξ) ∣ / ∣ ξ ∣^{n (1 / p - 1)} \to 0$ as $ξ \to 0$ for each fixed atom. [def-multidimensional-rectangle-and-volume, def-ck-and-multi-index-notation-in-several-variables, algebra]

Proof technique: the atomic representation, termwise Fourier transformation and dominated summation.

Proof

technique · direct

1.1F1F2F3algebra

Continuity and decay off the origin. Let $f = \sum_{j} λ_{j} a_{j}$ be the representation of [F1]. By [F2], $\hat{f} = \sum_{j} λ_{j} \hat{a_{j}}$ in $S^{'}$ ; since each $\hat{a_{j}}$ is a continuous function (the atoms are integrable) and, by [F3], $∣ λ_{j} \hat{a_{j}} (ξ) ∣ \leq C ∣ λ_{j} ∣ ∣ ξ ∣^{n (1 / p - 1)}$ for $ξ \neq 0$ , the numerical series $\sum_{j} λ_{j} \hat{a_{j}}$ converges absolutely and locally uniformly on $R^{n} ∖ {0}$ . Its sum is therefore a continuous function off the origin and agrees with $\hat{f}$ there as a distribution. There is no additional distribution supported at the origin: define the sum to be zero there. The uniform atom bound holds globally after this assignment, and $∣ ξ ∣^{n (1 / p - 1)} ∣ χ (ξ) ∣$ is integrable for every Schwartz test $χ$ . Dominated convergence therefore identifies the regular distribution of this sum with the distributional limit of the transformed partial sums on all of $R^{n}$ ; this identifies $\hat{f}$ with that continuous function on $R^{n} ∖ {0}$ and gives $∣ \hat{f} (ξ) ∣ \leq C (\sum_{j} ∣ λ_{j} ∣) ∣ ξ ∣^{n (1 / p - 1)} \leq C^{'} ∥ f ∥_{H^{p}} ∣ ξ ∣^{n (1 / p - 1)}$ .

2.1step 1.1F1F3algebra

The little- $o$ statement. Fix $η > 0$ . Choose $J$ so large that $C \sum_{j > J} ∣ λ_{j} ∣ < η / 2$ , possible because $(λ_{j}) \in ℓ^{p} \subseteq ℓ^{1}$ ; then by [F3] the tail satisfies $\sum_{j > J} ∣ λ_{j} ∣ ∣ \hat{a_{j}} (ξ) ∣ \leq (η / 2) ∣ ξ ∣^{n (1 / p - 1)}$ for every $ξ \neq 0$ . The finite sum $\sum_{j \leq J} λ_{j} \hat{a_{j}}$ is a finite combination of continuous functions each vanishing faster than $∣ ξ ∣^{n (1 / p - 1)}$ at the origin, so there is $δ > 0$ with $∣ \sum_{j \leq J} λ_{j} \hat{a_{j}} (ξ) ∣ < (η / 2) ∣ ξ ∣^{n (1 / p - 1)}$ for $0 < ∣ ξ ∣ < δ$ . Hence $∣ \hat{f} (ξ) ∣ \leq η ∣ ξ ∣^{n (1 / p - 1)}$ for $0 < ∣ ξ ∣ < δ$ , which is the stated little- $o$ relation since $η > 0$ was arbitrary.

3.1step 1.1step 2.1∎

Conclusion. Steps 1.1 and 2.1 give the identification of $\hat{f}$ with a continuous function off the origin, the decay estimate and the little- $o$ refinement. This proves the theorem.

Depends on

Used by

Weighted-integrable Hᵖ functions have vanishing moments in the atomic range Corollary

Dependency tree · two levels

61 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Marcin Bownik, Li-An Daniel Wang, Fourier transform of anisotropic Hardy spaces, Proc. Amer. Math. Soc. 141 (2013), 2299-2308 (author offprint) (standard reference, not scraped)
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)

Fourier transform decay of real HpHp elements