Layer cake representation

For a nonnegative measurable function

g

,

\int g = \int_{0}^{\infty} μ {g > t} d t .

On a graph, applying this identity to

∣ g (x) - g (y) ∣

expresses its total variation as the integral of the edge boundaries of the superlevel sets.

More generally, if

φ : [0, \infty) \to [0, \infty)

is differentiable, increasing, and satisfies

φ (0) = 0

, then Tonelli theorem and the fundamental theorem of calculus give

\int φ (g) d μ = \int_{0}^{\infty} φ^{'} (s) μ {g > s} d s .

Table of contents
- Lp bound from a tail domination inequality Layer cake representation

Lp bound from a tail domination inequality ( $a μ {F > a} \leq \int_{{F > a}} G ⟹ ∥ F ∥_{p} \leq \frac{p}{p - 1} ∥ G ∥_{p}$ )

Let

F, G \geq 0

,

1 < p < \infty

, and

G \in L^{p}

. Assume the displayed tail inequality and finite positive-threshold superlevel measures. First split

G

at

a /2

to obtain

a μ {F > 2 a} \leq \int_{{G > a}} G

. Layer cake representation and Tonelli theorem imply

F \in L^{p}

. Integrating the original inequality against

p a^{p - 2}

then gives

∥ F ∥_{p}^{p} \leq q \int G F^{p - 1} \leq q ∥ G ∥_{p} ∥ F ∥_{p}^{p - 1}

, with

q = p / (p - 1)

. This is the analytic estimate behind the Doob Lp maximal inequality.

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Layer cake representation is a concept often used in various fields, including geography, data visualization, and computer science, to illustrate the arrangement of different layers or components in a structured way. The term is commonly associated with two main contexts: 1. **Geology and Geography**: In this context, a layer cake representation illustrates the stratification of geological layers. Each "layer" represents different materials, sediments, or rock formations that have accumulated over time.

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