Alphabeta MathDefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-31
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.
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- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
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Conjugate exponents, including the endpoint conventions
Definition
Numbers are conjugate exponents when
with the convention . Thus the endpoint pairs and are conjugate, and if then the conjugate of is
No other endpoint pair is conjugate: gives , while gives .
Used by
- Hᵖ equals Lᵖ with equivalent norms for 1<p<∞ Corollary
- Interpolate L1 to Linfinity and L2 to L2 bounds Corollary
- Stein-Tomas for compact hypersurfaces with nonzero curvature Corollary
- The Aₚ classes are open in the exponent Corollary
- Knapp rules out extension below the Tomas exponent Counterexample
- Weak closure can exceed sequential weak closure Counterexample
- Hᵖ atoms with a prescribed moment order Definition
- The Sobolev conjugate exponent and the scaling identity Definition
- Coordinate vectors converge weakly to zero in ell p Example
- The right-translation semigroup on Lp has the weak derivative as generator Example
- The Stein-Tomas exponents on the circle Example
- Clarkson inequalities in both exponent ranges Lemma
- Complex Lq norm recovery from finite simple dual tests Lemma
- Difference-quotient calculus: integration by parts, product rule, commutation Lemma
- Duality and nesting of the Aₚ classes Lemma
- Finite simple analytic families and their exact endpoint norms Lemma
- For 1 < p < ∞, the Lᵖ norm of a nonnegative function is the supremum of its pairings with L^q unit vectors Lemma
- Graph-patch extension family: dispersive and L2 slice bounds Lemma
- Stein-Tomas TT-star bound from fractional integration Lemma
- The Radon-Nikodym density of a bounded Lᵖ functional belongs to L^q Lemma
- TT-star reduces extension to convolution with the surface-measure transform Lemma
- Uniformly bounded difference quotients represent a weak derivative Lemma
- The functional Λ_g has norm ‖g‖_q; for q=∞ assume μ is semifinite Proposition
- Finite-measure Lʳ includes into Lᵖ for p < r Theorem
- Generalized Holder inequality puts products into Lʳ Theorem
- Holder's inequality for integrals, including the endpoint cases Theorem
- Knapp necessary condition for spherical L2 restriction Theorem
- Lyapunov interpolation inequality for Lᵖ norms Theorem
- Stein-Tomas spherical restriction theorem Theorem
- The Caccioppoli inequality for weak elliptic solutions Theorem
- The difference-quotient characterisation of W^1,p for 1<p<∞ Theorem
- Young's convolution inequality under Countable Choice Theorem
Dependency tree · 0 levels
Nothing. This result depends on no other item in the library.
Sources
- John K. Hunter, Measure Theory, Definition 7.6 (standard reference, not scraped)
- Sheldon Axler, Measure, Integration & Real Analysis, Section 7A (standard reference, not scraped)