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Terence L. J. Harris

Publications and source records attributed to Terence L. J. Harris.

23 records · Page 2Linked to original sources
arXiv2020

Fourier decay of fractal measures on hyperboloids

Let $μ$ be an $α$-dimensional probability measure. We prove new upper and lower bounds on the decay rate of hyperbolic averages of the Fourier transform $\widehatμ$. More precisely, if $\mathbb{H}$ is a truncated hyperbolic paraboloid in $\mathbb{R}^d$ we study the optimal $β$ for which $$\int_{\mathbb{H}} |\hatμ(Rξ)|^2 \, d σ(ξ)\leq C(α, μ) R^{-β}$$ for all $R > 1$. Our estimates for $β$ depend on the minimum between the number of positive and negative principal curvatures of $\mathbb{H}$; if this number is as large as possible our estimates are sharp in all dimensions.

math.CA↗
arXiv2019

Improved decay of conical averages of the Fourier transform

An improved lower bound is given for the decay of conical averages of Fourier transforms of measures, for cones of dimension $d \geq 4$. 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.

math.AP↗
arXiv2019

Two-by-two upper triangular matrices and Morrey's conjecture

It is shown that every homogeneous gradient Young measure supported on matrices of the form $\begin{pmatrix} a_{1,1} & \cdots & a_{1,n-1} & a_{1,n} \\ 0 & \cdots & 0 & a_{2,n} \end{pmatrix}$ is a laminate. This is used to prove the same result on the 3-dimensional nonlinear submanifold of $\mathbb{M}^{2 \times 2}$ defined by $\det X = 0$ and $X_{12}>0$.

math.AP↗
arXiv2019

Upper semicontinuity of the lamination hull

Let $K \subseteq \mathbb{R}^{2 \times 2}$ be a compact set, let $K^{rc}$ be its rank-one convex hull, and let $L(K)$ be its lamination convex hull. It is shown that the mapping $K \to \overline{L(K)}$ is not upper semicontinuous on the diagonal matrices in $\mathbb{R}^{2 \times 2}$, which was a problem left by Kolář. This is followed by an example of a 5-point set of $2 \times 2$ symmetric matrices with non-compact lamination hull. Finally, an example of another 5-point set $K$ is given, which has $L(K)$ connected, compact and strictly smaller than $K^{rc}$.

math.AP↗