OAI.SignedDisk.CenteredDiskEndpoint.main_signed_finite_band

The theorem states that the defined proposition SignedFiniteBandStatement holds (the proof is admitted, not verified). On the Euclidean plane ℝ², write diskAverage(f,x,r) for the integral of f over the open ball B(x,r) divided by πr², and signedBand(f,a,b)(x) for the supremum of diskAverage(f,x,r) over all real r in the closed interval [a,b], endpoints included, with no absolute value taken. The statement asserts that there is a constant C ≥ 0 such that for every smooth (C^∞) compactly supported f : ℝ² → ℝ, every a > 0 and every natural number m, the function signedBand(f,a,2^m a) is locally integrable and has an integrable vector-valued weak gradient G : ℝ² → ℝ². Weak gradient means that for every smooth compactly supported test function φ and each coordinate i, ∫ signedBand·∂ᵢφ = −∫ Gᵢ φ. Moreover the L¹ norm of G, ∫‖G‖ with the Euclidean norm, is at most C times the L¹ norm of the gradient of f. The constant C does not depend on f, a or m.