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The BMO seminorm is unchanged by adding a constant

Example

For $b \in L_{l o c}^{1} (R^{n})$ and $c \in C$ one has $(b + c)_{Q} = b_{Q} + c$ for every cube $Q$ and hence $∥ b + c ∥_{B M O} = ∥ b ∥_{B M O}$ ; so the BMO seminorm descends to the quotient $B M O / C$ and is a norm there.

Facts & Assumptions

Given: $b \in L_{l o c}^{1} (R^{n})$ , a constant $c \in C$ and a cube $Q$ , with the mean, the seminorm and the quotient of BMO seminorm and the quotient by constants.

[L1]

The mean is $b_{Q} = ∣ Q ∣^{- 1} \int_{Q} b$ , the seminorm is $∥ b ∥_{B M O} = \sup_{Q} ∣ Q ∣^{- 1} \int_{Q} ∣ b - b_{Q} ∣$ , and $∥ b ∥_{B M O} = 0$ exactly when $b$ is constant almost everywhere (BMO seminorm and the quotient by constants).

Verification

technique · direct

1.1L1algebra

Linearity of the integral over the cube $Q$ gives $(b + c)_{Q} = ∣ Q ∣^{- 1} \int_{Q} (b + c) = ∣ Q ∣^{- 1} \int_{Q} b + ∣ Q ∣^{- 1} \int_{Q} c = b_{Q} + c$ .

2.1step 1.1L1

By step 1.1, $∣ (b + c) - (b + c)_{Q} ∣ = ∣ b - b_{Q} ∣$ pointwise on $Q$ , so $∣ Q ∣^{- 1} \int_{Q} ∣ (b + c) - (b + c)_{Q} ∣ = ∣ Q ∣^{- 1} \int_{Q} ∣ b - b_{Q} ∣$ for every cube; taking the supremum over all cubes gives $∥ b + c ∥_{B M O} = ∥ b ∥_{B M O}$ , including the value $+ \infty$ .

3.1step 2.1L1∎

The identity of step 2.1 shows that the seminorm is constant on each equivalence class modulo constants, so it descends to $B M O (R^{n}) / C$ ; on classes it is a norm because $∥ [b] ∥ = 0$ holds exactly when $b$ is almost everywhere constant by [L1], that is exactly for the zero class. No choice principle is used.

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

Juha Kinnunen, Harmonic Analysis (Aalto University lecture notes) (standard reference, not scraped)