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Melissa Tacy

Publications and source records attributed to Melissa Tacy.

16 recordsLinked to original sources
arXiv2025

The principle of simultaneous saturation: Application to the $k$-linear restriction/extension problem

This paper develops a new framework, \emph{simultaneous saturation}, designed to quantify the size of sets whose elements are simultaneously large. The framework establishes a correspondence between the magnitude of such sets and a system of interdependent conditions linking their points. We first prove a general theorem establishing the correspondence and then apply the framework to multilinear restriction-type estimates. From this perspective, we obtain a new proof (independent of Bennett-Carbery-Tao \cite{BCT}) of the $d$-linear restriction/extension theorem, and establish the $λ^ε$ loss conjectured bounds for the $k$-linear $L^{2}\to L^{p/k}$ extension problem under mixed transversality/curvature conditions $(k<d)$.

math.CA↗
arXiv2025

$L^p$ estimates for joint quasimodes of two pseudodifferential operators whose characteristic sets have $k$-th order contact

On a smooth, compact, $n$-dimensional Riemannian manifold, we consider functions $u_h$ that are joint quasimodes of two semiclassical pseudodifferential operators $p_1(x,hD)$ and $p_2(x,hD)$. We develop $L^p$ estimates for $u_h$ when the characteristic sets of $p_1$ and $p_2$ meet with $k$-th order contact. This paper is the natural extension of the two-dimensional results from arXiv:1909.12559 to $n$ dimensions.

math.AP↗
arXiv2023

Filament structure of random waves

We investigate the small scale equidistribution properties of random waves in $\mathbb{R}^{n}$. Numerical evidence suggests that such objects display a fine scale filament structure. We show that the X-ray along any line segment is uniformly equidistributed so any limiting behaviour must be weaker than $L^{2}$ scaring. On the other hand, we show that at Planck scale in phase space there are (with high probability) logarithmic fluctuations above what would be expected given equidistribution. Taken together these results suggest that the filament structure may be a configuration space echo of the phase space concentrations.

math.AP↗
arXiv2023

$L^{p}$ estimates for joint quasimodes of semiclassical pseudodifferential operators whose characteristic sets have $k$th order contact

In this paper we develop $L^{p}$ estimates for functions $u$ which are joint quasimodes of semiclassical pseudodifferential operators $p_{1}(x,hD)$ and $p_{2}(x,hD)$ whose characteristic sets meet with $k$th order contact, $k\geq{}1$. As part of the technical development we use Fourier integral operators to adapt a flat wavelet analysis to the curved level sets of $p_{1}(x,ξ)$.

math.AP↗
arXiv2021

Small scale equidistribution of random waves generated by an unfair coin flip

In this paper we study the small scale equidistribution property of random waves whose coefficients are determined by an unfair coin. That is the coefficients take value $+1$ with probability $p$ and $-1$ with probability $1-p$. Random waves whose coefficients are associated with a fair coin are known to equidistribute down to the wavelength scale. We obtain explicit requirements on the deviation from the fair ($p=0.5$) coin to retain equidistribution.

math.AP↗
arXiv2020

Stationary phase type estimates for low symbol regularity

The well-known stationary phase formula gives us a way to precisely compute oscillating integrals so long as the symbol is regular enough (in comparison to the large parameter controlling the oscillation). However in a number of applications we find ourselves with symbols that are not suitably regular. In this paper we obtain decay bounds for such oscillatory integrals.

math.AP↗
arXiv2019

Equidistribution of random waves on small balls

In this paper, we investigate the small scale equidistribution properties of randomised sums of Laplacian eigenfunctions (i.e. random waves) on a compact manifold. We prove small scale expectation and variance results for random waves on all compact manifolds. Here, "small scale" refers to balls of radius $r(λ)\to 0$ such that $r/r_{\text{Planck}}\to\infty$, where $r_{\text{Planck}}$ is the Planck scale. For balls at a larger scale (although still $r(λ)\to 0$) we also obtain estimates showing that the probability that a random wave fails to equidistribute decays exponentially with the eigenvalue.

math.SP↗
arXiv2018

Lp bilinear quasimode estimates

In this paper, we investigate the $L^p$ bilinear quasimode estimates on compact Riemannian manifolds. We obtain results in the full range $p\ge2$ on all $n$-dimensional manifolds with $n\ge2$. This in particular implies the $L^p$ bilinear eigenfunction estimates. We further show that all of these estimates are sharp by constructing various quasimodes and eigenfunctions that saturate our estimates.

math.AP↗
arXiv2017

A note on constructing sharp examples for $L^{p}$ norms of eigenfunctions and quasimodes near submanifolds

In this note we analyse $L^{p}$ estimates for Laplacian eigenfunctions and quasimodes and their associated sharp examples. In particular we use previously determined estimates to produce a new set of estimates for restriction to thickened neighbourhoods of submanifolds. In addition we produce a family flat model quasimode examples that can be used to determine sharpness of estimates on Laplacian eigenfunctions restricted to subsets. For each quasimode in the family we show that there is a corresponding spherical harmonic that displays the same growth properties. Therefore it is enough to check $L^{p}$ growth estimates against the simple flat model examples. Finally we present a heuristic that for any subset determines which quasimode in the family is expected to produce sharp examples.

math.AP↗
arXiv2017

The quantisation of normal velocity does not concentrate on hypersurfaces

We seek to extend work by Christianson-Hassell-Toth \cite{CHT} on restrictions of Neumann data of Laplacian eigenfunctions to interior hypersurfaces to a general semiclassical setting. In the semiclassical regime the appropriate generalisation is to study the restrictions of the function $v=ν(x,hD)u$ where $ν(x,hD)$ is the operator defined by quantising the normal velocity observable. For the Laplacian $ν(x,hD)=\frac{1}{2}hD_ν$ where $ν$ is the normal to the hypersurface. We find that $||ν(x,hD)u||_{L^{2}(H)}\lesssim||u||_{L^{2}(M)}$ provided $u$ is an $O_{L^{2}}(h)$ quasimode of the semiclassical pseudodifferential operator $p(x,hD)$. This statement should be interpreted as a statement of non-concentration for the quantisation of normal velocity.

math.AP↗
arXiv2015

Comparable upper and lower bounds for boundary values of Neumann eigenfunctions and tight inclusion of eigenvalues

For smooth bounded domains in $\mathbb{R}$, we prove upper and lower $L^2$ bounds on the boundary data of Neumann eigenfunctions, and prove quasi-orthogonality of this boundary data in a spectral window. The bounds are tight in the sense that both are independent of eigenvalue; this is achieved by working with an appropriate norm for boundary functions, which includes a `spectral weight', that is, a function of the boundary Laplacian. This spectral weight is chosen to cancel concentration at the boundary that can happen for `whispering gallery' type eigenfunctions. These bounds are closely related to wave equation estimates due to Tataru. Using this, we bound the distance from an arbitrary Helmholtz parameter $E>0$ to the nearest Neumann eigenvalue, in terms of boundary normal-derivative data of a trial function $u$ solving the Helmholtz equation $(\Delta-E)u=0$. This `inclusion bound' improves over previously known bounds by a factor of $E^{5/6}$. It is analogous to a recently improved inclusion bound in the Dirichlet case, due to the first two authors. Finally, we apply our theory to present an improved numerical implementation of the method of particular solutions for computation of Neumann eigenpairs on smooth planar domains. We show that the new inclusion bound improves the relative accuracy in a computed Neumann eigenvalue (around the $42000$th) from 9 digits to 14 digits, with little extra effort.

math.AP↗
arXiv2014

Sharp norm estimates of layer potentials and operators at high frequency

In this paper, we investigate single and double layer potentials mapping boundary data to interior functions of a domain at high frequency $λ^2\to\infty$. For single layer potentials, we find that the $L^{2}(\partialΩ)\to{}L^{2}(Ω)$ norms decay in $λ$. The rate of decay depends on the curvature of $\partialΩ$: The norm is $λ^{-3/4}$ in general domains and $λ^{-5/6}$ if the boundary $\partialΩ$ is curved. The double layer potential, however, displays uniform $L^{2}(\partialΩ)\to{}L^{2}(Ω)$ bounds independent of curvature. By various examples, we show that all our estimates on layer potentials are sharp. The appendix by Galkowski gives bounds $L^{2}(\partialΩ)\to{}L^{2}(\partialΩ)$ for the single and double layer operators at high frequency that are sharp modulo $\log λ$. In this case, both the single and double layer operator bounds depend upon the curvature of the boundary.

math.AP↗
arXiv2013

Improvement of eigenfunction estimates on manifolds of nonpositive curvature

Let $(M,g)$ be a compact, boundaryless manifold of dimension $n$ with the property that either (i) $n=2$ and $(M,g)$ has no conjugate points, or (ii) the sectional curvatures of $(M,g)$ are nonpositive. Let $Δ$ be the positive Laplacian on $M$ determined by $g$. We study the $L^{2}\to{}L^{p}$ mapping properties of a spectral cluster of $\sqrtΔ$ of width $1/\logλ$. Under the geometric assumptions above, \cite{berard77} Bérard obtained a logarithmic improvement for the remainder term of the eigenvalue counting function which directly leads to a $(\logλ)^{1/2}$ improvement for Hörmander's estimate on the $L^{\infty}$ norms of eigenfunctions. In this paper we extend this improvement to the $L^p$ estimates for all $p>\frac{2(n+1)}{n-1}$.

math.AP↗
arXiv2010

Semiclassical L^p Estimates of Quasimodes on Curved Hypersurfaces

Let M be a compact manifold of dimension n, P = P(h) a semiclassical pseudodifferential operator on M, and u = u(h) an L^2 normalised family of functions such that Pu is O(h) in L^2(M) as h goes to 0. Let H be a compact submanifold of M. In a previous article, the second-named author proved estimates on the L^p norms, p > 2, of u restricted to H, under the assumption that the u are semiclassically localised and under some natural structural assumptions about the principal symbol of P. These estimates are of the form Ch^d(n;k;p) where k=dimH (except for a logarithmic divergence in the case k = n-2; p = 2). When H is a hypersurface, i.e. k = n-1, we have d(n;n-1;2)=1/4, which is sharp when M is the round n-sphere and H is an equator. In this article, we assume that H is a hypersurface, and make the additional geometric assumption that H is curved with respect to the bicharacteristic flow of P. Under this assumption we improve the estimate from d=1/4 to 1/6, generalising work of Burq-Gerard-Tzvetkov and Hu for Laplace eigenfunctions. To do this we apply the Melrose-Taylor theorem, as adapted by Pan and Sogge, for Fourier integral operators with folding canonical relations.

math.AP↗
arXiv2009

Semiclassical L^p Estimates of Quasimodes on Submanifolds

Let M be a compact manifold and P = P(h) a semiclassical pseudodifferential operator on M . Suppose that u(h) is a L^2 normalised family of functions such that P(h)u(h) is O(h) in L^2, as h goes to 0. Then, for any compact submanifold Y contained in M, we obtain estimates on the L^p norm of u(h) restricted to Y, with exponents that are sharp for h goes to 0. As part of the technical development we prove some extensions of the abstract Strichartz estimates of Keel and Tao.

math.AP↗