arXiv subjects

Jeremy Schwend

Publications and source records attributed to Jeremy Schwend.

3 recordsLinked to original sources

arXiv2020

Near Optimal $L^p\rightarrow L^q$ Estimates for Euclidean Averages Over Prototypical Hypersurfaces in $\mathbb{R}^3$

We find the precise range of $(p,q)$ for which local averages along graphs of a class of two-variable polynomials in $\mathbb{R}^3$ are of restricted weak type $(p,q)$, given the hypersurfaces have Euclidean surface measure. We derive these results using non-oscillatory, geometric methods, for a model class of polynomials bearing a strong connection to the general real-analytic case.

math.CA↗

arXiv2019

Fourier restriction above rectangles

In this article, we study the problem of obtaining Lebesgue space inequalities for the Fourier restriction operator associated to rectangular pieces of the paraboloid and perturbations thereof. We state a conjecture for the dependence of the operator norms in these inequalities on the sidelengths of the rectangles, prove that this conjecture follows from (a slight reformulation of the) restriction conjecture for elliptic hypersurfaces, and prove that, if valid, the conjecture is essentially sharp. Such questions arise naturally in the study of restriction inequalities for degenerate hypersurfaces; we demonstrate this connection by using our positive results to prove new restriction inequalities for a class of hypersurfaces having some additive structure.

math.CA↗

arXiv2019

$\ell^2$ Decoupling in $\mathbb R^2$ for curves with Vanishing Curvature

We expand the class of curves $(φ_1(t),φ_2(t)),\ t\in[0,1]$ for which the $\ell^2$ decoupling conjecture holds for $2\leq p\leq 6$. Our class of curves includes all real-analytic regular curves with isolated points of vanishing curvature and all curves of the form $(t,t^{1+ν})$ for $ν\in (0,\infty)$.

math.CA↗