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How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Weighted norm of an interval indicator

Example

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

Let $n = 1$ , let $1 < p < \infty$ , let $- 1 < α < p - 1$ and let $r > 0$ . For the admissible power weight $w (x) = ∣ x ∣^{α}$ one has $∥ 1_{(0, r)} ∥_{L^{p} (w)} = (\frac{r^{α + 1}}{α + 1})^{1 / p},$ which tends to $0$ as $r \to 0^{+}$ and grows like $r^{(α + 1) / p}$ as $r \to \infty$ . At the endpoint $α = - 1$ the integral $\int_{0}^{r} x^{- 1} d x$ diverges logarithmically for every $r > 0$ , and for each fixed $r > 0$ the displayed norm diverges as $α ↓ - 1$ .

Facts & Assumptions

Given: Countable Choice; $1 < p < \infty$ , $- 1 < α < p - 1$ , $r > 0$ , and $w (x) = ∣ x ∣^{α}$ on $R$ .

[F1]

$∥ f ∥_{L^{p} (w)}^{p} = \int_{R} ∣ f ∣^{p} w d λ$ and $L^{p} (w)$ is the corresponding space of classes (Weights, their associated measures, and the spaces L^p(w)); the weight $∣ x ∣^{α}$ is admissible for $α > - 1$ and lies in $A_{p}$ for $- 1 < α < p - 1$ (The A_p range of a power weight, Muckenhoupt A_p and A_1 weights).

[F2]

For $α > - 1$ the power function has antiderivative $t^{α + 1} / (α + 1)$ on $(0, \infty)$ , and $\int_{0}^{r} t^{- 1} d t = + \infty$ for every $r > 0$ , the divergence being logarithmic (Continuity and derivatives of positive-base real powers, The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t, Comparison tests for improper integrals).

Verification

technique · direct

1.1F1F2givenalgebra

Direct computation: $∥ 1_{(0, r)} ∥_{L^{p} (w)}^{p} = \int_{0}^{r} ∣ x ∣^{α} d x = \int_{0}^{r} x^{α} d x = r^{α + 1} / (α + 1)$ for $α > - 1$ by [F2], since $∣ x ∣ = x$ on $(0, r)$ ; taking $p$ -th roots gives the displayed formula.

2.1F2step 1.1givenalgebra∎

As $r \to 0^{+}$ the expression $(r^{α + 1} / (α + 1))^{1 / p}$ tends to $0$ because $α + 1 > 0$ ; as $r \to \infty$ it grows like the constant multiple $r^{(α + 1) / p}$ of the power function. At $α = - 1$ one has $\int_{0}^{r} x^{- 1} d x = + \infty$ by [F2], so for fixed $r > 0$ the full expression $(r^{α + 1} / (α + 1))^{1 / p}$ diverges as $α ↓ - 1$ , since $r^{α + 1} \to 1$ . The density $∣ x ∣^{- 1}$ does not satisfy this page's local-integrability definition of a weight; its integral of the indicator is nevertheless well defined and infinite.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Loukas Grafakos, Classical Fourier Analysis, 3rd ed. (Springer GTM 249, 2014) (standard reference, not scraped)
Juha Kinnunen, Harmonic Analysis (Aalto University lecture notes) (standard reference, not scraped)