The class $L^1(\mu )$ of integrable functions

Depends on

• Integrable real and complex functions, and their integrals

Used by

• Infinite propagation speed for nonnegative heat data Corollary
• Mass conservation and positivity of the heat flow Corollary
• A radial Poisson limit does not control a tangential path Counterexample
• The Hardy-Littlewood maximal operator is not strong type (1,1) Counterexample
• The zero-countable / infinity-cocountable measure space breaks the p=1 endpoint of duality Counterexample
• An L¹ approximate identity on ℝⁿ Definition
• Complex Lp classes and Euclidean test-function conventions Definition
• Conditional expectation given a sigma algebra Definition
• Convergence in L¹(mu) Definition
• Convolution of two functions on ℝⁿ Definition
• Fourier transform on complex L1 classes Definition
• Integration against a signed or complex measure, and the class L¹(nu) = L¹(|nu|) Definition
• The circle maximal function and nontangential approach regions Definition
• The Poisson integral of a finite complex boundary measure Definition
• The square of the Volterra operator has zero trace Example
• Maximal dyadic cubes above a level Lemma
• The circle maximal function is weak type one one for finite measures Lemma
• The integral of the divergence of an integrable C1 field vanishes Lemma
• The Lp range: interpolation below two and adjoint duality above two Lemma
• Calderón–Zygmund operators are of weak type (1,1) Theorem
• Fubini's theorem for L¹ functions on a sigma-finite product Theorem
• hᵖ is the Poisson image of Lp for 1<p<=infinity Theorem
• Maximal truncations: weak (1,1) and strong Lp bounds Theorem
• On a finite measure space, uniform integrability is equivalent to L¹-boundedness plus uniform absolute continuity Theorem
• Positive harmonic boundary measures and compact normalized families Theorem
• The centered Hardy-Littlewood maximal operator is weak type (1,1) Theorem
• The indefinite integral of an L¹ function is differentiable almost everywhere Theorem
• Uniqueness of the scalar Laplace transform in the exponential-growth class Theorem

Dependency tree · two levels

4 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

• Gerald B. Folland, Real Analysis, 2nd ed., §2.3 (standard reference, not scraped)