arXiv subjects
Publications and source records attributed to Alex Barron.
We show that $$\bigg\|\sup_{0 < t < 1} \big|\sum_{n=1}^{N} e^{2\pi i (n(\cdot) + n^2 t)}\big| \bigg\|_{L^{4}([0,1])} \leq C_{\epsilon} N^{3/4 + \epsilon}$$ and discuss some applications to the theory of large values of Weyl sums. This estimate is sharp for quadratic Weyl sums, up to the loss of $N^{\epsilon}$.
We prove a sharp, global-in-time Strichartz estimate for the Schr\"odinger equation on the cylinder $\mathbb{R}\times\mathbb{T}$.
Let $\mu$ be an $\alpha$-dimensional probability measure. We prove new upper and lower bounds on the decay rate of hyperbolic averages of the Fourier transform $\widehat{\mu}$. More precisely, if $\mathbb{H}$ is a truncated hyperbolic paraboloid in $\mathbb{R}^d$ we study the optimal $\beta$ for which $$\int_{\mathbb{H}} |\hat{\mu}(R\xi)|^2 \, d \sigma (\xi)\leq C(\alpha, \mu) R^{-\beta}$$ for all $R > 1$. Our estimates for $\beta$ 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.
Let $\mathbb{H}$ be a $(d-1)$-dimensonal hyperbolic paraboloid in $\mathbb{R}^d$ and let $Ef$ be the Fourier extension operator associated to $\mathbb{H},$ with $f$ supported in $B^{d-1}(0,2)$. We prove that $\|Ef\|_{L^p (B(0,R))} \leq C_{\epsilon}R^{\epsilon}\|f\|_{L^p}$ for all $p \geq \frac{2(d+2)}{d}$ whenever $ \frac{d}{2} \geq m + 1$, where $m$ is the minimum between the number of positive and negative principal curvatures of $\mathbb{H}$. Bilinear restriction estimates for $\mathbb{H}$ proved by S. Lee and Vargas play an important role in our argument.
We study the problem of dominating the dyadic strong maximal function by $(1, 1)$-type sparse forms based on rectangles with sides parallel to the axes, and show that such domination is impossible. Our proof relies on an explicit construction of a pair of maximally separated point sets with respect to an appropriately defined notion of distance