Taryn C. Flock

Taryn C. Flock
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Taryn C. Flock

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Mathematics - Classical Analysis and ODEs (5)

Publications Authored By Taryn C. Flock

We prove that $$ \|X(|u|^2)\|_{L^3_{t,\ell}}\leq C\|f\|_{L^2(\mathbb{R}^2)}^2, $$ where $u(x,t)$ is the solution to the linear time-dependent Schr\"odinger equation on $\mathbb{R}^2$ with initial datum $f$, and $X$ is the (spatial) X-ray transform on $\mathbb{R}^2$. In particular, we identify the best constant $C$ and show that a datum $f$ is an extremiser if and only if it is a gaussian. We also establish bounds of this type in higher dimensions $d$, where the X-ray transform is replaced by the $k$-plane transform for any $1\leq k\leq d-1$. Read More

Recent progress in multilinear harmonic analysis naturally raises questions about the local behaviour of the best constant (or bound) in the general Brascamp--Lieb inequality as a function of the underlying linear transformations. In this paper we prove that this constant is continuous, but is not in general differentiable. Read More

We prove that the best constant in the general Brascamp-Lieb inequality is a locally bounded function of the underlying linear transformations. As applications we deduce certain very general Fourier restriction, Kakeya-type, and nonlinear variants of the Brascamp-Lieb inequality which have arisen recently in harmonic analysis. Read More

Certain rearrangement inequalities of a type considered by Hardy, Riesz, and Brascamp-Lieb-Luttinger are studied. Subsets of the real line that extremize these inequalities are characterized. Our results apply only to special cases, and complement earlier results by Burchard treating other cases. Read More

The $k$-plane transform is a bounded operator from $\lp$ to $L^q$ of the Grassmann manifold of all affine $k$-planes in $\R^n$ for certain exponents depending on $k$ and $n$. In the endpoint case $q=n+1$, we identify all extremizers of the associated inequality for the general $k$-plane transform. Read More