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The choice of admissible dyadic partition does not change the Lp square-function space

Statement

Assume Countable Choice. Call an inhomogeneous dyadic frequency partition admissible when it is obtained as in Existence of a smooth inhomogeneous dyadic frequency partition from some radial cutoff satisfying the standing hypotheses. Let $(φ_{j})$ and $(ψ_{j})$ be two admissible partitions with associated square functions $S^{(φ)}$ and $S^{(ψ)}$ . Then for every $1 < p < \infty$ there are constants $0 < c \leq C < \infty$ , depending only on $n, p$ and the two cutoffs, such that for every $f \in L^{p} (R^{n}; C)$ $c ∥ S^{(ψ)} f ∥_{p} \leq ∥ S^{(φ)} f ∥_{p} \leq C ∥ S^{(ψ)} f ∥_{p} .$ Moreover the mixed pieces are almost orthogonal: $Δ_{j}^{(φ)} Δ_{k}^{(ψ)} = 0$ whenever $∣ j - k ∣ \geq 3$ , and each mixed operator $Δ_{j}^{(φ)} Δ_{k}^{(ψ)}$ is the Fourier multiplier with symbol $φ_{j} ψ_{k}$ supported in $supp φ_{j} \cap supp ψ_{k}$ . For $j, k \geq 1$ these are intersections of the corresponding annular supports; if either index is zero, use the actual low-frequency ball support of that block.

Facts & Assumptions

Given: Countable Choice, two admissible partitions $(φ_{j})$ , $(ψ_{j})$ with cutoffs $ψ_{φ}, ψ_{ψ}$ and operators $Δ_{j}^{(φ)}, Δ_{k}^{(ψ)}$ ; a real $1 < p < \infty$ ; a function $f \in L^{p} (R^{n}; C)$ .

[F1]

Littlewood-Paley square-function equivalence on Lp for 1<p<infinity applies to each admissible partition separately: for the partition $(φ_{j})$ there are constants $0 < c_{1} \leq C_{1} < \infty$ depending only on $n, p$ and the cutoff $ψ_{φ}$ with $c_{1} ∥ g ∥_{p} \leq ∥ S^{(φ)} g ∥_{p} \leq C_{1} ∥ g ∥_{p}$ for all $g \in L^{p}$ , and for the partition $(ψ_{j})$ there are constants $0 < c_{2} \leq C_{2} < \infty$ depending only on $n, p$ and $ψ_{ψ}$ with $c_{2} ∥ g ∥_{p} \leq ∥ S^{(ψ)} g ∥_{p} \leq C_{2} ∥ g ∥_{p}$ .

[F2]

For each of the two admissible partitions, every fixed dyadic piece is a bounded operator on $L^{p}$ and agrees with its convolution representative; the composition on $S$ has symbol $φ_{j} ψ_{k}$ (Dyadic pieces are uniformly Mihlin multipliers and uniformly Lp-bounded, The inhomogeneous dyadic frequency partition and its Littlewood-Paley operators). Also $C_{c}^{\infty}$ is dense in $L^{p}$ for $1 < p < \infty$ ( $C_{c}^{\infty} (R^{n})$ is dense in $L^{p} (R^{n})$ for $1 \leq p < \infty$ ). For $j, k \geq 1$ the supports satisfy $supp φ_{j} \subset {2^{j - 1} \leq ∣ ξ ∣ \leq 2^{j + 1}}$ and $supp ψ_{k} \subset {2^{k - 1} \leq ∣ ξ ∣ \leq 2^{k + 1}}$ ; the low blocks satisfy $supp φ_{0} \subset {∣ ξ ∣ \leq 2}$ and $supp ψ_{0} \subset {∣ ξ ∣ \leq 2}$ (The inhomogeneous dyadic frequency partition and its Littlewood-Paley operators, Existence of a smooth inhomogeneous dyadic frequency partition).

Proof

technique · direct

1.1F1algebra

Comparability of the two square functions. By [F1] applied to $f$ , $c_{1} ∥ f ∥_{p} \leq ∥ S^{(φ)} f ∥_{p} \leq C_{1} ∥ f ∥_{p}$ and $c_{2} ∥ f ∥_{p} \leq ∥ S^{(ψ)} f ∥_{p} \leq C_{2} ∥ f ∥_{p}$ ; eliminating $∥ f ∥_{p}$ gives $\frac{c_{1}}{C_{2}} ∥ S^{(ψ)} f ∥_{p} \leq ∥ S^{(φ)} f ∥_{p} \leq \frac{C_{1}}{c_{2}} ∥ S^{(ψ)} f ∥_{p}$ , with constants depending only on $n, p$ and the two cutoffs.

1.2F2algebra

The mixed pieces and their supports. For $f \in S$ the composition rule [F2] gives $Δ_{j}^{(φ)} Δ_{k}^{(ψ)} f = T_{φ_{j} ψ_{k}} f$ , whose symbol is supported in $supp φ_{j} \cap supp ψ_{k}$ . If $∣ j - k ∣ \geq 3$ , say $k \geq j + 3$ , then $supp φ_{j} \subset {∣ ξ ∣ \leq 2^{j + 1}}$ (including $j = 0$ , since its support lies in the radius-two ball) and $supp ψ_{k} \subset {∣ ξ ∣ \geq 2^{k - 1}}$ , with $2^{j + 1} < 2^{k - 1}$ , so the supports are disjoint and $φ_{j} ψ_{k} \equiv 0$ . Thus the mixed operator vanishes on $S$ ; by [F2] each fixed dyadic piece is bounded on $L^{p}$ , so its composition is bounded, and $C_{c}^{\infty} \subset S$ is dense in $L^{p}$ . Hence the identity $Δ_{j}^{(φ)} Δ_{k}^{(ψ)} = 0$ extends to all of $L^{p}$ . The same argument applies when $j \geq k + 3$ after interchanging the partitions. When $j, k \geq 1$ the support intersection is the intersection of their annuli; if a low block occurs it is the intersection with that block's actual ball support from [F2].

2.1step 1.1step 1.2∎

Conclusion. Step 1.1 is the stated two-sided comparability with constants depending only on $n, p$ and the two cutoffs, and step 1.2 is the almost-orthogonality and support statement for the mixed pieces.

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Sources

Terence Tao, Math 247A Lecture Notes 4 (UCLA, Fall 2006) (standard reference, not scraped)
Loukas Grafakos, Classical Fourier Analysis, 3rd ed. (Springer GTM 249) (standard reference, not scraped)
Mark Williams, Notes on Harmonic Analysis (January 11, 2022) (standard reference, not scraped)