The average of a locally integrable function over a Euclidean ball

Depends on

• The Axiom of Countable Choice ($\mathrm{AC}_\omega$)
• A locally integrable function on $\mathbb{R}^n$
• Euclidean balls have positive finite Lebesgue measure

Used by

• One centred ball-mean identity does not force harmonicity Counterexample
• Poincare-Wirtinger fails on disconnected bounded domains Counterexample
• The Harnack estimate needs an additive forcing term Counterexample
• The Harnack inequality requires nonnegativity Counterexample
• Axis-parallel cubes, their averages, and cube maximal functions Definition
• Lebesgue points and the Lebesgue set of an L¹_loc class Definition
• Spherical averages and local ball means in Rn Definition
• Spherical means and the weighted ball integral of space-dependent data Definition
• The centered and uncentered Hardy-Littlewood maximal functions Definition
• A locally integrable function can fail to differentiate on a null set Example
• Hilbert transform of the line Poisson kernel Example
• Piecewise-affine approximation of a measurable coefficient Example
• Worked oscillation decay and its Hölder modulus Example
• A finite interior ball chain propagates weak Harnack bounds Lemma
• Ball means and sphere means are related by a radial derivative Lemma
• Ball-mean oscillation bound by the Riesz potential of the gradient Lemma
• Continuous compactly supported functions are recovered by small ball averages Lemma
• De Giorgi oscillation reduction: one half-level set is small Lemma
• Geometric oscillation decay implies a Hölder modulus Lemma
• Moser iteration for positive supersolutions: negative-power and logarithmic comparison Lemma
• Sobolev level-set step: energy decay with explicit level gap and radius loss Lemma
• The Euler–Poisson–Darboux equation for spherical means Lemma
• Zero-set propagation for a nonnegative Holder weak solution Lemma
• Ball averages vary continuously with the centre and radius Proposition
• De Giorgi local boundedness of homogeneous subsolutions Theorem
• De Giorgi-Nash interior Holder regularity for divergence-form equations Theorem
• Harnack inequality for nonnegative weak solutions Theorem
• Interior Schauder estimate for uniformly elliptic equations Theorem
• Lebesgue differentiation theorem on ℝⁿ Theorem
• Morrey's inequality for p>n Theorem
• Poincare inequality on a ball Theorem
• The mean-zero Poincare inequality on bounded John domains Theorem
• The Poincare inequality with a positive-measure zero set Theorem
• W^1,∞ functions on convex domains have Lipschitz representatives Theorem
• Weak Harnack inequality for nonnegative supersolutions Theorem

Dependency tree · two levels

17 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., Section 3.4 (standard reference, not scraped)