The centered and uncentered Hardy-Littlewood maximal functions

Depends on

• The Axiom of Countable Choice ($\mathrm{AC}_\omega$)
• The average of a locally integrable function over a Euclidean ball

Used by

• Hᵖ equals Lᵖ with equivalent norms for 1<p<∞ Corollary
• The Hardy-Littlewood maximal operator is not strong type (1,1) Counterexample
• The maximal function of 1_[0,1] is not integrable Counterexample
• Axis-parallel cubes, their averages, and cube maximal functions Definition
• Muckenhoupt Aₚ and A₁ weights Definition
• The centered maximal function of 1_[0,1] on ℝ Example
• Ball and cube maximal functions are pointwise comparable Lemma
• Cotlar's inequality for maximal truncations Lemma
• Dyadic annulus far-field estimates for the maximal function Lemma
• Hedberg pointwise inequality for Riesz potentials Lemma
• L2-normalised H1 atoms have uniformly bounded H1 norm Lemma
• Near and far bounds for a Riesz potential Lemma
• Radially decreasing kernels are dominated by the maximal function Lemma
• The tangential maximal function is controlled by the aperture-one nontangential maximal function in Lᵖ Lemma
• The two defining forms of A₁ agree Lemma
• Truncated maximal functions: finiteness, comparison estimates and the good-set bound Lemma
• Unweighted local good-lambda estimate for maximal truncations Lemma
• Weighted weak (1,1) bound for the maximal function under A₁ Lemma
• The centered and uncentered maximal functions are pointwise comparable Proposition
• The centered maximal operator is bounded on L^∞ Proposition
• Hardy–Littlewood–Sobolev fractional integration inequality Theorem
• Maximal truncations: weak (1,1) and strong Lp bounds Theorem
• The centered Hardy-Littlewood maximal function is Borel measurable Theorem
• The centered Hardy-Littlewood maximal operator is weak type (1,1) Theorem
• Weighted L-p bounds for standard Calderon-Zygmund maximal truncations Theorem

Dependency tree · two levels

9 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: Modern Techniques and Their Applications, 2nd ed., Theorem 3.17 and Exercise 22 (standard reference, not scraped)
• Terence Tao, An Introduction to Measure Theory, Theorem 1.6.20 (standard reference, not scraped)