Higher-dimensional equiangular sector square-function bounds

Establish L^p bounds, uniform up to at most subpolynomial or logarithmic factors in the number of sectors N, for the Littlewood–Paley square function built from an equiangular partition of Fourier space into sectors Δ_j = {ξ ∈ ℝ^n : 2π j/N ≤ arctan(|ξ_2|/|ξ_1|) < 2π (j+1)/N} (and their smooth variants) in dimensions n ≥ 3 for the natural window 2 ≤ p ≤ 2n/(n−1).

Background

In ℝ^2, Córdoba proved square-function bounds for equiangular sector decompositions in the range 2 ≤ p ≤ 4. In higher dimensions, pure bounds independent of the number of sectors N fail due to Kakeya-type phenomena, but there is a natural range 2 ≤ p ≤ 2n/(n−1) where one expects bounds with at most subpolynomial or logarithmic dependence on N.

The paper contrasts this equiangular setting with lacunary angular sectors, which are accessible via parabolic dilations and Marcinkiewicz-type multiplier theory. For equiangular sectors, rotational symmetry and Kakeya counterexamples create significant obstacles that are unresolved in n ≥ 3.