An improved lower bound is given for the decay of conical averages of Fourier transforms of measures, in dimensions d ≥ 4. The proof uses a weighted version of the broad restriction inequality, a narrow decoupling inequality for the cone, and some techniques of Du and Zhang originally developed for the Schrödinger equation.

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An improved a.e. lower bound is given for Hausdorff dimension under vertical projections in the first Heisenberg group.

Let γ : [0, 1] → S 2 be a non-degenerate curve in R 3 , that is to say, det γ(θ), γ ′ (θ), γ ′′ (θ) = 0. For each θ ∈ [0, 1], let V θ = γ(θ) ⊥ and let π θ : R 3 → V θ be the orthogonal projections. We prove that if A ⊂ R 3 is a Borel set, then for a.e. θ ∈ [0, 1] we have dim(π θ (A)) = min{2, dimA}.More generally, we prove an exceptional set estimate. For A ⊂ R 3 and 0 ≤ s ≤ 2, define Es(A) := {θ ∈ [0, 1] : dim(π θ (A)) < s}. We have dim(Es(A)) ≤ 1 + s − dim(A). We also prove that if dim(A) > 2, then for a.e. θ ∈ [0, 1] we have H 2 (π θ (A)) > 0.2020 Mathematics Subject Classification. 42B15, 42B20.

It is shown that every homogeneous gradient Young measure supported on matrices of the form a 1,1 · · · a 1,n−1 a 1,n 0 · · · 0 a 2,n is a laminate. This is used to prove the same result on the 3-dimensional nonlinear submanifold of M 2×2 defined by det X = 0 and X 12 > 0.