The polar surface set function on the unit sphere

Depends on

• The Borel product of R^m and R^n is the Borel sigma-algebra of R^{m+n}
• The Borel sigma-algebra of a subspace is the trace of the ambient Borel sigma-algebra
• Assuming countable choice, every Borel subset of $\mathbb{R}^n$ is Lebesgue measurable
• A continuous map has Borel preimages of Borel sets
• The Borel sigma-algebra of a topological space
• The Axiom of Countable Choice ($\mathrm{AC}_\omega$)

Used by

• Polar integration may discard the cut locus Corollary
• Morrey's inequality has no p=n endpoint Counterexample
• The fundamental Hessian is not absolutely locally integrable Counterexample
• Fourier restriction and adjoint extension operators Definition
• Fundamental solution for the positive operator minus Laplacian Definition
• Model space radial area and ball volume Definition
• Spherical averages and local ball means in Rn Definition
• Spherical means and the weighted ball integral of space-dependent data Definition
• Wave equation, Cauchy data and wave speed Definition
• A compact-support Cauchy–Pompeiu calculation Example
• A Dirac mass has precisely sufficiently negative Sobolev order Example
• An explicit ∂̄ solution with an L² estimate Example
• Bishop gromov ratio is constant in the model space Example
• Hörmander estimate with a Gaussian weight Example
• Newton shell theorem from harmonic mean values Example
• Projective-plane curvature via a hemisphere Example
• Radial powers approach the Morrey borderline exponent Example
• Sharp Sobolev threshold for a radial power Example
• Volume growth in euclidean and hyperbolic space Example
• Archimedean product region, volume and norm bound Lemma
• Ball-mean oscillation bound by the Riesz potential of the gradient Lemma
• Heat-ball representation formula Lemma
• Monomial integrals on the sphere and orthonormality on the distinguished torus Lemma
• Pointwise potential bound for compactly supported smooth functions Lemma
• Radial derivative of a spherical average Lemma
• Reflection invariance and vanishing first moment of the sphere measure Lemma
• Sphere graph charts, surface density, and a finite partition Lemma
• Sphere integrals of a cylindrical function project to weighted ball integrals Lemma
• The level-set kernel measure estimate for the Slobodeckij kernel Lemma
• The mean-value L² bound for holomorphic functions on a polydisc Lemma
• The truncated Riesz kernel is bounded on Lᵖ of a bounded set Lemma
• The unit sphere is Lebesgue null Lemma
• Weighted monomial integrals and monomial norms for the disc, ball and polydisc Lemma
• Bishop gromov volume comparison Theorem
• Calderon-Zygmund operators map L-infinity to BMO Theorem
• Polar coordinates decompose Lebesgue measure into rⁿ⁻¹ dr d sigma Theorem
• Strong parabolic maximum principle Theorem
• The critical Sobolev embedding into every finite L^q Theorem

Dependency tree · two levels

26 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., Theorem 2.49 (standard reference, not scraped)