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Proportional functions realise the equality case of Holder

Example

Fix $1 < p < \infty$ , let $q$ be its conjugate exponent, and work on $[0, 1]$ with Lebesgue measure. For positive constants $a, b$ , define

$f : = a χ_{[0, 1]}, g : = b χ_{[0, 1]} .$

Then

$\int_{0}^{1} ∣ f g ∣ d λ = ∥ f ∥_{p} ∥ g ∥_{q} .$

Facts & Assumptions

Given: Constants $a, b > 0$ , an exponent $1 < p < \infty$ , and its conjugate $q$ .

[L1]

Equality in Holder holds when $∣ f ∣^{p}$ and $∣ g ∣^{q}$ are proportional almost everywhere (Equality in Holder's inequality for $1 < p < \infty$ ).

Verification

Proof technique: Choose nonnegative functions with $∣ f ∣^{p}$ and $∣ g ∣^{q}$ proportional almost everywhere and invoke the equality theorem.

1.1

The functions satisfy [given] $∣ f ∣^{p} = a^{p} χ_{[0, 1]}, ∣ g ∣^{q} = b^{q} χ_{[0, 1]},$ so $∣ f ∣^{p} = (a^{p} / b^{q}) ∣ g ∣^{q}$ almost everywhere.

2.1

Applying [L1] gives equality in Holder: [L1, step 1.1] $\int_{0}^{1} ∣ f g ∣ d λ = ∥ f ∥_{p} ∥ g ∥_{q} .$ ∎

Depends on

Equality in Holder's inequality for $1 < p < \infty$

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

Sheldon Axler, Measure, Integration & Real Analysis, Holder's Inequality (standard reference, not scraped)