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The square function of a low-frequency-localised function

Example

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $f \in S (R^{n})$ have Fourier transform supported in ${∣ ξ ∣ \leq 1}$ . Then $Δ_{j} f = 0$ for every $j \geq 1$ and $Δ_{0} f = f$ , so $S f = ∣ Δ_{0} f ∣ = ∣ f ∣$ and hence $∥ S f ∥_{p} = ∥ f ∥_{p}$ for every $1 \leq p < \infty$ . The example records that in the region where exactly one piece of the partition is nonzero the square function degenerates to the modulus of that piece, so for nonzero functions in this class the strict-range coefficients satisfy $c_{p} \leq 1 \leq C_{p}$ .

Verification

Given: Countable Choice and $f \in S (R^{n})$ with $supp \hat{f} \subset {∣ ξ ∣ \leq 1}$ .

[L1] The partition satisfies $φ_{0} = ψ$ with $ψ = 1$ on ${∣ ξ ∣ \leq 1}$ , and for $j \geq 1$ the piece $φ_{j}$ vanishes on ${∣ ξ ∣ \leq 2^{j - 1}}$ (Existence of a smooth inhomogeneous dyadic frequency partition, The inhomogeneous dyadic frequency partition and its Littlewood-Paley operators).

[L2] For $f \in S$ one has $\hat{Δ_{j} f} = φ_{j} \hat{f}$ , and Fourier inversion gives $g = F^{- 1} (\hat{g})$ for Schwartz $g$ (The inhomogeneous dyadic frequency partition and its Littlewood-Paley operators, Fourier inversion on Schwartz space); the square function is $S f = (\sum_{j \geq 0} ∣ Δ_{j} f ∣^{2})^{1 / 2}$ (The Littlewood-Paley square function).

1.1L1givenalgebra

The pieces on the low-frequency ball. Since $supp \hat{f} \subset {∣ ξ ∣ \leq 1}$ and $ψ = 1$ there, $φ_{0} \hat{f} = ψ \hat{f} = \hat{f}$ . For $j \geq 1$ , $supp \hat{f}$ is contained in the vanishing region ${∣ ξ ∣ \leq 2^{j - 1}}$ of $φ_{j}$ (because $2^{j - 1} \geq 1$ ), so $φ_{j} \hat{f} = 0$ .

2.1L2step 1.1algebra∎

The square function. By step 1.1 and [L2], $Δ_{0} f = F^{- 1} (\hat{f}) = f$ and $Δ_{j} f = F^{- 1} (0) = 0$ for every $j \geq 1$ ; hence only the $j = 0$ term of the square function survives, $S f = ∣ Δ_{0} f ∣ = ∣ f ∣$ , and therefore $∥ S f ∥_{p} = ∥ ∣ f ∣ ∥_{p} = ∥ f ∥_{p}$ for every $1 \leq p < \infty$ . For a nonzero function in this class, $c_{p} ∥ f ∥_{p} \leq ∥ S f ∥_{p} \leq C_{p} ∥ f ∥_{p}$ therefore forces $c_{p} \leq 1 \leq C_{p}$ .

Depends on

Used by

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Dependency tree · two levels

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Sources

Loukas Grafakos, Classical Fourier Analysis, 3rd ed. (Springer GTM 249) (standard reference, not scraped)