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Finite lacunary exponential sums belong to BMO

Example

Let $0 < a \leq b < \infty$ , and let $(ξ_{n})_{n \in Z}$ be nonzero real frequencies satisfying $a 2^{n} \leq ∣ ξ_{n} ∣ \leq b 2^{n}$ for every $n$ . Let $(c_{n})_{n \in Z}$ have finite support and put $f (x) : = \sum_{n \in Z} c_{n} e^{2 π i ξ_{n} x} .$ Then $∥ f ∥_{B M O (R)} \leq C_{a, b, ϕ} (\sum_{n} ∣ c_{n} ∣^{2})^{1 / 2} .$ More precisely, fix a nonnegative $ϕ \in C_{c}^{\infty} (R)$ with $ϕ \geq 1$ on $[- \frac{1}{2}, \frac{1}{2}]$ . For an interval $I$ of length $L$ and centre $x_{I}$ , set $ϕ_{I} (x) : = ϕ ((x - x_{I}) / L)$ . There is a constant $c_{I} \in C$ such that $\int_{R} ϕ_{I} (x) ∣ f (x) - c_{I} ∣^{2} d x \leq C_{a, b, ϕ} L \sum_{n} ∣ c_{n} ∣^{2} .$

Facts & Assumptions

Given: The finite exponential sum $f$ , the frequency bounds above, a nonnegative compactly supported smooth bump $ϕ$ with $ϕ \geq 1$ on $[- \frac{1}{2}, \frac{1}{2}]$ , and a nondegenerate interval $I$ .

[L1]

For every locally integrable $g$ and every constant $c$ , $∣ I ∣^{- 1} \int_{I} ∣ g - g_{I} ∣ \leq 2 ∣ I ∣^{- 1} \int_{I} ∣ g - c ∣,$ and the BMO seminorm is the supremum of the mean oscillations over intervals in dimension one (BMO seminorm and the quotient by constants).

[L2]

Since $ϕ \in C_{c}^{\infty}$ , both $\int ϕ (y) d y$ and $\int ϕ (y) y^{2} d y$ are finite. For every integer $M \geq 1$ , integration by parts $M$ times gives $∣ \int_{R} ϕ (y) e^{2 π i ζ y} d y ∣ \leq C_{M, ϕ} (1 + ∣ ζ ∣)^{- M} .$ For $∣ ζ ∣ \leq 1$ this follows instead from $∥ ϕ ∥_{1}$ ; combining the two estimates gives the displayed bound.

Verification

technique · direct low- and high-frequency split

Fix $I$ , write $L = ∣ I ∣$ , and let $n_{0}$ be the least integer with $2^{n_{0}} \geq L^{- 1}$ . Thus $1 \leq L 2^{n_{0}} < 2$ . Split $f = f_{<} + f_{\geq}$ at $n_{0}$ and choose $c_{I} : = f_{<} (x_{I}) = \sum_{n < n_{0}} c_{n} e^{2 π i ξ_{n} x_{I}} .$

1.1L2algebra

Low frequencies. For every $n < n_{0}$ , $∣ e^{2 π i ξ_{n} x} - e^{2 π i ξ_{n} x_{I}} ∣ \leq 2 π ∣ ξ_{n} ∣ ∣ x - x_{I} ∣$ . After $x = x_{I} + L y$ , ${(\int ϕ_{I} (x) ∣ e^{2 π i ξ_{n} x} - e^{2 π i ξ_{n} x_{I}} ∣^{2} d x)}^{1 / 2} \leq C_{ϕ} ∣ ξ_{n} ∣ L^{3 / 2} .$ The triangle inequality in $L^{2} (ϕ_{I} d x)$ , Cauchy--Schwarz, and the upper frequency bound therefore give $\begin{aligned} ∥ f_{<} - c_{I} ∥_{L^{2} (ϕ_{I} d x)} & \leq C_{ϕ} L^{3 / 2} \sum_{n < n_{0}} ∣ c_{n} ∣ ∣ ξ_{n} ∣ \\ \leq C_{b, ϕ} L^{3 / 2} (\sum_{n} ∣ c_{n} ∣^{2})^{1 / 2} (\sum_{n < n_{0}} 4^{n})^{1 / 2} \\ \leq C_{b, ϕ} L^{1 / 2} (\sum_{n} ∣ c_{n} ∣^{2})^{1 / 2}, \end{aligned}$ because $2^{n_{0}} < 2 L^{- 1}$ .

1.2L2algebra

High frequencies. Define $G_{n m} : = L^{- 1} \int_{R} ϕ_{I} (x) e^{2 π i (ξ_{n} - ξ_{m}) x} d x (n, m \geq n_{0}) .$ The change of variables $x = x_{I} + L y$ and [L2] give, for every fixed integer $M \geq 1$ , $∣ G_{n m} ∣ \leq C_{M, ϕ} (1 + L ∣ ξ_{n} - ξ_{m} ∣)^{- M} .$ Choose an integer $K \geq 1$ such that $b 2^{- K} \leq a / 2$ . If $∣ n - m ∣ > K$ , the frequency comparability implies $∣ ξ_{n} - ξ_{m} ∣ \geq ∣ ∣ ξ_{n} ∣ - ∣ ξ_{m} ∣ ∣ \geq (a / 2) 2^{\max (n, m)} .$ For each fixed $n \geq n_{0}$ , the terms with $∣ m - n ∣ \leq K$ contribute at most a constant to $\sum_{m \geq n_{0}} ∣ G_{n m} ∣$ . For $m > n + K$ their sum is bounded by $C \sum_{r \geq 1} (1 + L 2^{n + r})^{- M} \leq C \sum_{r \geq 1} 2^{- M r} < \infty,$ using $L 2^{n} \geq 1$ . For $n_{0} \leq m < n - K$ , there are at most $n - n_{0}$ terms, each bounded by $C (1 + L 2^{n})^{- M}$ ; their total is bounded uniformly because $L 2^{n} \geq 2^{n - n_{0}}$ and $\sup_{r \geq 0} r (1 + 2^{r})^{- M} < \infty$ . Hence the absolute row sums of $(G_{n m})_{n, m \geq n_{0}}$ are uniformly bounded. Since $∣ G_{n m} ∣ = ∣ G_{m n} ∣$ , the inequality $2 ∣ c_{n} ∣ ∣ c_{m} ∣ \leq ∣ c_{n} ∣^{2} + ∣ c_{m} ∣^{2}$ yields $\begin{aligned} \int ϕ_{I} ∣ f_{\geq} ∣^{2} & \leq L \sum_{n, m \geq n_{0}} ∣ c_{n} ∣ ∣ c_{m} ∣ ∣ G_{n m} ∣ \\ \leq C_{a, b, ϕ} L \sum_{n} ∣ c_{n} ∣^{2} . \end{aligned}$

2.1step 1.1step 1.2L1algebra∎

Combining the two pieces with $∣ f - c_{I} ∣^{2} \leq 2 ∣ f_{<} - c_{I} ∣^{2} + 2 ∣ f_{\geq} ∣^{2}$ gives $\int_{R} ϕ_{I} ∣ f - c_{I} ∣^{2} \leq C_{a, b, ϕ} L \sum_{n} ∣ c_{n} ∣^{2} .$ Since $ϕ_{I} \geq 1$ on $I$ , Cauchy--Schwarz and [L1] imply $\begin{aligned} L^{- 1} \int_{I} ∣ f - f_{I} ∣ & \leq 2 L^{- 1} \int_{I} ∣ f - c_{I} ∣ \\ \leq 2 L^{- 1 / 2} {(\int_{I} ∣ f - c_{I} ∣^{2})}^{1 / 2} \\ \leq C_{a, b, ϕ} (\sum_{n} ∣ c_{n} ∣^{2})^{1 / 2} . \end{aligned}$ Taking the supremum over all nondegenerate intervals proves the BMO bound. The proof uses finite sums only and no choice principle.

Depends on

BMO seminorm and the quotient by constants

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

Terence Tao, Math 247A Lecture Notes 4 (UCLA, Fall 2006) (standard reference, not scraped)