Sphere and ball measures scale in Rn

Depends on

• Spherical averages and local ball means in Rn
• Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma
• The Axiom of Countable Choice ($\mathrm{AC}_\omega$)

Used by

• Ball mean-value property for harmonic functions under Countable Choice Corollary
• Energy uniqueness for the wave Cauchy problem Corollary
• Harmonic Cauchy estimates in supremum norm Corollary
• Interior gradient bound for Poisson solutions Corollary
• Finite speed of propagation does not imply strong Huygens Counterexample
• Replacing the sphere measure by the ball measure in Kirchhoff's formula Counterexample
• Spherical means and the weighted ball integral of space-dependent data Definition
• Wave equation, Cauchy data and wave speed Definition
• A point source produces a uniform expanding sphere Example
• A two-dimensional pulse has a tail inside the cone Example
• Poisson extension fixes coordinate functions Example
• Quantitative concentration of the ball Poisson kernel Example
• Zero wave energy means a spatial constant, fixed by the displacement datum Example
• Ball means and sphere means are related by a radial derivative Lemma
• Ball-mean oscillation bound by the Riesz potential of the gradient Lemma
• Bounded-overlap ball chains in a bounded John domain Lemma
• Calderon reproducing pair and the telescoping identity in S' Lemma
• Cap and complement estimate for the ball Poisson integral Lemma
• L2-normalised H1 atoms have uniformly bounded H1 norm Lemma
• Radial derivative of a spherical average Lemma
• Radial mollification fixes local mean-value functions Lemma
• Smooth sphere data have a harmonic replacement under Countable Choice Lemma
• Sphere integrals of a cylindrical function project to weighted ball integrals Lemma
• The ball Poisson kernel is positive and has unit mass Lemma
• The tangential maximal function is controlled by the aperture-one nontangential maximal function in Lᵖ Lemma
• Truncated maximal functions: finiteness, comparison estimates and the good-set bound Lemma
• Truncated wave cones: convexity, piecewise C1 presentation and outward normals Lemma
• Whitney-type ball cover with disjoint small balls and bounded overlap Lemma
• Continuous ball-mean-value functions are harmonic Theorem
• Continuous Dirichlet problem on a ball Theorem
• Energy uniqueness for homogeneous Dirichlet waves on bounded domains Theorem
• Finite propagation speed for the wave equation Theorem
• Harmonic functions are real analytic Theorem
• Harnack inequality on a ball Theorem
• Interior derivative estimates for harmonic functions Theorem
• Interior estimate for the Poisson equation with Hölder data Theorem
• Kirchhoff's formula in three dimensions Theorem
• Morrey's inequality for p>n Theorem
• Poisson kernel and bounded Dirichlet problem on a half-space Theorem
• The forced three-dimensional version as a retarded potential Theorem

…and 2 more results.

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

• PDE source treatment (standard reference, not scraped)