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BMO oscillation norms in Lq are equivalent

Statement

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Let $1 \leq q < \infty$ and for $b \in L_{l o c}^{1} (R^{n})$ put $∥ b ∥_{B M O, q} : = \sup_{Q} (∣ Q ∣^{- 1} \int_{Q} ∣ b - b_{Q} ∣^{q})^{1 / q} \in [0, \infty]$ , the supremum over all cubes. Then there are constants $0 < c_{n, q} \leq C_{n, q} < \infty$ such that $c_{n, q} ∥ b ∥_{B M O} \leq ∥ b ∥_{B M O, q} \leq C_{n, q} ∥ b ∥_{B M O}$ for every $b \in B M O (R^{n})$ ; indeed $(∣ Q ∣^{- 1} \int_{Q} ∣ b - b_{Q} ∣^{q})^{1 / q} \leq C_{n, q} ∥ b ∥_{B M O}$ for every cube $Q$ . In particular every $B M O$ function lies in $L_{l o c}^{q} (R^{n})$ .

Facts & Assumptions

Given: Countable Choice, $1 \leq q < \infty$ , a function $b \in L_{l o c}^{1} (R^{n})$ , a cube $Q$ , and the mean and seminorm of BMO seminorm and the quotient by constants.

[F1]

The mean is $b_{Q} = ∣ Q ∣^{- 1} \int_{Q} b$ and $∥ b ∥_{B M O} = \sup_{Q} ∣ Q ∣^{- 1} \int_{Q} ∣ b - b_{Q} ∣$ ; the seminorm vanishes exactly on the almost-everywhere constants (BMO seminorm and the quotient by constants).

[F2]

Holder's inequality on the finite-measure cube $Q$ , applied to the nonnegative functions $∣ b - b_{Q} ∣$ and $1$ , gives $∣ Q ∣^{- 1} \int_{Q} ∣ b - b_{Q} ∣ \leq (∣ Q ∣^{- 1} \int_{Q} ∣ b - b_{Q} ∣^{q})^{1 / q}$ for $1 \leq q < \infty$ (Holder's inequality for integrals, including the endpoint cases); if the right-hand side is infinite the inequality is immediate, and $q = 1$ is equality.

[F3]

The layer-cake formula gives $∣ Q ∣^{- 1} \int_{Q} ∣ b - b_{Q} ∣^{q} = q \int_{0}^{\infty} λ^{q - 1} ∣ Q ∣^{- 1} ∣ {x \in Q : ∣ b - b_{Q} ∣ > λ} ∣ d λ$ (For 0 < p < infinity, the layer-cake formula computes the integral of |f|^p from the distribution function).

[F4]

There are constants $c_{n}, C_{n} \in (0, \infty)$ with $∣ {x \in Q : ∣ b - b_{Q} ∣ > λ} ∣ \leq C_{n} ∣ Q ∣ e^{- c_{n} λ / ∥ b ∥_{B M O}}$ for every $λ > 0$ , the zero-seminorm case giving the value $0$ (John-Nirenberg exponential inequality).

Proof

technique · direct

1.1F1F2

Lower bound. By [F2], $∣ Q ∣^{- 1} \int_{Q} ∣ b - b_{Q} ∣ \leq (∣ Q ∣^{- 1} \int_{Q} ∣ b - b_{Q} ∣^{q})^{1 / q}$ for every cube $Q$ ; taking the supremum over all cubes gives $∥ b ∥_{B M O} \leq ∥ b ∥_{B M O, q}$ , so $c_{n, q} = 1$ is admissible.

1.2F3F4F1algebra

Upper bound for a fixed cube. If $∥ b ∥_{B M O} > 0$ , [F4] inserted into the layer-cake formula [F3] gives $∣ Q ∣^{- 1} \int_{Q} ∣ b - b_{Q} ∣^{q} \leq q C_{n} \int_{0}^{\infty} λ^{q - 1} e^{- c_{n} λ / ∥ b ∥_{B M O}} d λ = q C_{n} ∥ b ∥_{B M O}^{q} \int_{0}^{\infty} μ^{q - 1} e^{- c_{n} μ} d μ$ after the substitution $λ = μ ∥ b ∥_{B M O}$ ; the last integral is finite because $μ^{q - 1} e^{- c_{n} μ}$ is integrable on $(0, \infty)$ . If $∥ b ∥_{B M O} = 0$ the left-hand side is $0$ by [F1], so the same estimate holds with either side zero.

2.1step 1.2step 1.1algebra

With $K_{n, q} : = (q C_{n} \int_{0}^{\infty} μ^{q - 1} e^{- c_{n} μ} d μ)^{1 / q} < \infty$ , step 1.2 gives $(∣ Q ∣^{- 1} \int_{Q} ∣ b - b_{Q} ∣^{q})^{1 / q} \leq K_{n, q} ∥ b ∥_{B M O}$ for every cube $Q$ , and taking the supremum over $Q$ gives $∥ b ∥_{B M O, q} \leq K_{n, q} ∥ b ∥_{B M O}$ ; so $C_{n, q} = K_{n, q}$ is admissible. Both constants depend only on $n$ and $q$ .

3.1step 2.1F1∎

Finally, for $b \in B M O (R^{n})$ and a cube $Q$ one has $∣ b ∣ \leq ∣ b - b_{Q} ∣ + ∣ b_{Q} ∣$ , hence $\int_{Q} ∣ b ∣^{q} \leq 2^{q - 1} (\int_{Q} ∣ b - b_{Q} ∣^{q} + ∣ Q ∣ ∣ b_{Q} ∣^{q}) < \infty$ by step 2.1 and the finiteness of the local mean $b_{Q}$ ; every compact set is covered by finitely many cubes, so $b \in L_{l o c}^{q} (R^{n})$ .

Depends on

Used by

Integrating the John-Nirenberg tail recovers the Lq oscillation bound Example

Dependency tree · two levels

32 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

Mark Williams, Notes on Harmonic Analysis (January 11, 2022) (standard reference, not scraped)
Juha Kinnunen, Harmonic Analysis (Aalto University lecture notes) (standard reference, not scraped)
Terence Tao, Math 247A Lecture Notes 4 (UCLA, Fall 2006) (standard reference, not scraped)