Searcharxiv⌕ Search

arXiv subjects

Matthew D. Blair

Publications and source records attributed to Matthew D. Blair.

At least 19 recordsLinked to original sources
arXiv2026

Nonconcentration of eigenfunctions in Microlocal Kakeya-Nikodym norms: a phase space approach

Previous works of the author and Sogge [BS17], [BS18] showed the significance of microlocal Kakeya-Nikodym averages in improving $L^p$ bounds on (approximate) eigenfunctions of the Laplacian in the high frequency limit. These averages are formed by taking the $L^2$ norm of an eigenfunction when localized in phase space to a small, frequency-dependent tube about a geodesic segment via a pseudodifferential operator. The former work showed that for values of $p$ beneath the Stein-Tomas exponent, $L^p$ norms are controlled by a supremum over these averages. The latter work then showed that when $(M,g)$ has nonpositive sectional curvatures, there is a logarithmic gain in the averages. In combination, these two works improved the $L^p$ theory for eigenfunctions over the universal bounds of Sogge in this geometric setting. In the present work, we develop sufficient conditions for improving these averages which are more general than nonpositive curvature. Instead our sufficient conditions are rooted in the dynamics of the geodesic flow on the tangent bundle, considering cases where the flow expands and contracts tangent vectors in at least some directions, e.g. partially hyperbolic flows. We make use of Gaussian wave packet (phase space) transforms on the manifold in order to fully appreciate the gain these hypotheses impart on the microlocal averages. In the process, we further develop Gaussian beam approximations to the wave equation in a coordinate invariant manner.

math.AP↗
arXiv2024

$L^q$ Estimates on the Restriction of Schrödinger Eigenfunctions with singular potentials

We consider eigenfunction estimates in $L^p$ for Schrödinger operators, $H_V=-Δ_g+V(x)$, on compact Riemannian manifolds $(M, g)$. Eigenfunction estimates over the full manifolds were already obtained by Sogge \cite{Sogge1988concerning} for $V\equiv 0$ and the first author, Sire, and Sogge \cite{BlairSireSogge2021Quasimode}, and the first author, Huang, Sire, and Sogge \cite{BlairHuangSireSogge2022UniformSobolev} for critically singular potentials $V$. For the corresponding restriction estimates for submanifolds, the case $V\equiv 0$ was considered in Burq, Gérard, and Tzvetkov \cite{BurqGerardTzvetkov2007restrictions}, and Hu \cite{Hu2009lp}. In this article, we will handle eigenfunction restriction estimates for some submanifolds $Σ$ on compact Riemannian manifolds $(M, g)$ with $n:=\dim M\geq 2$, where $V$ is a singular potential.

math.AP↗
arXiv2024

Improved spectral projection estimates

We obtain new improved spectral projection estimates on manifolds of non-positive curvature, including sharp ones for relatively large spectral windows for general tori. Our results are stronger than those in an earlier work of the first and third authors [6], and the arguments have been greatly simplified. We more directly make use of pointwise estimates that are implicit in the work of Berard [2] and avoid the use of weak-type spaces that were used in the previous works [6] and [22]. We also simplify and strengthen the bilinear arguments by exploiting the use of microlocal $L^2\to L^{q_c}$ Kakeya-Nikodym estimates and avoiding the of $L^2\to L^2$ ones as in earlier results. This allows us to prove new results for manifolds of negative curvature and some new sharp estimates for tori. We also have new and improved techniques in two dimensions for general manifolds of non-positive curvature.

math.AP↗
arXiv2023

Strichartz estimates for the Schrödinger equation on negatively curved compact manifolds

We obtain improved Strichartz estimates for solutions of the Schrödinger equation on negatively curved compact manifolds which improve the classical universal results results of Burq, Gérard and Tzvetkov [11] in this geometry. In the case where the spatial manifold is a hyperbolic surface we are able to obtain no-loss $L^{q_c}_{t,x}$-estimates on intervals of length $\log λ\cdot λ^{-1} $ for initial data whose frequencies are comparable to $λ$, which, given the role of the Ehrenfest time, is the natural analog of the universal results in [11]. We are also obtain improved endpoint Strichartz estimates for manifolds of nonpositive curvature, which cannot hold for spheres.

math.AP↗
arXiv2021

The Van Vleck Formula on Ehrenfest time scales and stationary phase asymptotics for frequency-dependent phases

The Van Vleck formula is a semiclassical approximation to the integral kernel of the propagator associated to a time-dependent Schrödinger equation. Under suitable hypotheses, we present a rigorous treatment of this approximation which is valid on "Ehrenfest time scales", i.e. $\hbar$-dependent time intervals which most commonly take the form $|t| \leq c|\log\hbar|$. Our derivation is based on an approximation to the integral kernel often called the "Herman-Kluk approximation", which realizes the kernel as an integral superposition of Gaussians parameterized by points in phase space. As was shown by Robert, this yields effective approximations over Ehrenfest time intervals. In order to derive the Van Vleck approximation from the Herman-Kluk approximation, we are led to develop stationary phase asymptotics where the phase functions depend on the frequency parameter in a nontrivial way, a result which may be of independent interest.

math.AP↗
arXiv2021

Uniform Sobolev Estimates on compact manifolds involving singular potentials

We obtain generalizations of the uniform Sobolev inequalities of Kenig, Ruiz and the fourth author \cite{KRS} for Euclidean spaces and Dos Santos Ferreira, Kenig and Salo \cite{DKS} for compact Riemannian manifolds involving critically singular potentials $V\in L^{n/2}$. We also obtain the analogous improved quasimode estimates of the the first, third and fourth authors \cite{BSS} , Hassell and Tacy \cite{HassellTacy}, the first and fourth author \cite{SBLog}, and Hickman \cite{Hickman} as well as analogues of the improved uniform Sobolev estimates of \cite{BSSY} and \cite{Hickman} involving such potentials. Additionally, on $S^n$, we obtain sharp uniform Sobolev inequalities involving such potentials for the optimal range of exponents, which extend the results of S. Huang and the fourth author \cite{SHSo}. For general Riemannian manifolds we improve the earlier results in \cite{BSS} by obtaining quasimode estimates for a larger (and optimal) range of exponents under the weaker assumption that $V\in L^{n/2}$.

math.AP↗
arXiv2019

Quasimode, eigenfunction and spectral projection bounds for Schrödinger operators on manifolds with critically singular potentials

We obtain quasimode, eigenfunction and spectral projection bounds for Schrödinger operators, $H_V=-Δ_g+V(x)$, on compact Riemannian manifolds $(M,g)$ of dimension $n\ge2$, which extend the results of the third author~\cite{sogge88} corresponding to the case where $V\equiv 0$. We are able to handle critically singular potentials and consequently assume that $V\in L^{\tfrac{n}2}(M)$ and/or $V\in {\mathcal K}(M)$ (the Kato class). Our techniques involve combining arguments for proving quasimode/resolvent estimates for the case where $V\equiv 0$ that go back to the third author \cite{sogge88} as well as ones which arose in the work of Kenig, Ruiz and this author~\cite{KRS} in the study of "uniform Sobolev estimates" in ${\mathbb R}^n$. We also use techniques from more recent developments of several authors concerning variations on the latter theme in the setting of compact manifolds. Using the spectral projection bounds we can prove a number of natural $L^p\to L^p$ spectral multiplier theorems under the assumption that $V\in L^{\frac{n}2}(M)\cap {\mathcal K}(M)$. Moreover, we can also obtain natural analogs of the original Strichartz estimates~\cite{Strichartz77} for solutions of $(\partial_t^2-Δ+V)u=0$. We also are able to obtain analogous results in ${\mathbb R}^n$ and state some global problems that seem related to works on absence of embedded eigenvalues for Schrödinger operators in ${\mathbb R}^n$ (e.g., \cite{IonescuJerison}, \cite{JK}, \cite{KenigNar}, \cite{KochTaEV} and \cite{iRodS}.)

math.AP↗
arXiv2018

Logarithmic improvements in $L^{p}$ bounds for eigenfunctions at the critical exponent in the presence of nonpositive curvature

We consider the problem of proving $L^p$ bounds for eigenfunctions of the Laplacian in the high frequency limit in the presence of nonpositive curvature and more generally, manifolds without conjugate points. In particular, we prove estimates at the "critical exponent" $p_c = \frac{2(d+1)}{d-1}$, where a spectrum of scenarios for phase space concentration must be ruled out. Our work establishes a gain of an inverse power of the logarithm of the frequency in the bounds relative to the classical $L^p$ bounds of the second author.

math.AP↗
arXiv2017

Refined and Microlocal Kakeya-Nikodym Bounds of Eigenfunctions in Higher Dimensions

We prove a Kakeya-Nikodym bound on eigenfunctions and quasimodes, which sharpens a result of the authors and extends it to higher dimensions. As in the prior work, the key intermediate step is to prove a microlocal version of these estimates, which involves a phase space decomposition of these modes which is essentially invariant under the bicharacteristic/geodesic flow. In a companion paper, it will be seen that these sharpened estimates yield improved $L^q(M)$ bounds on eigenfunctions in the presence of nonpositive curvature when $2 < q < \frac{2(d+1)}{d-1}$.

math.AP↗
arXiv2016

On logarithmic improvements of critical geodesic restriction bounds in the presence of nonpositive curvature

We consider upper bounds on the growth of $L^p$ norms of restrictions of eigenfunctions and quasimodes to geodesic segments in a nonpositively curved manifold in the high frequency limit. This sharpens results of Chen and Sogge as well as Xi and Zhang, which showed that the crux of the problem is to establish bounds on the mixed partials of the distance function on the covering manifold restricted to geodesic segments. The innovation in this work is the development of a formula for the third variation of arc length on the covering manifold, which allows for a coordinate free expressions of these mixed partials.

math.AP↗
arXiv2016

$L^p$-bounds on spectral clusters associated to polygonal domains

We look at the $L^p$ bounds on eigenfunctions for polygonal domains (or more generally Euclidean surfaces with conic singularities) by analysis of the wave operator on the flat Euclidean cone $C(\mathbb{S}^1_ρ) := \mathbb{R}_+ \times \left(\mathbb{R} \big/ 2πρ\mathbb{Z}\right)$ of radius $ρ> 0$ equipped with the metric $h(r,θ) = d r^2 + r^2 \, dθ^2$. Using explicit oscillatory integrals and relying on the fundamental solution to the wave equation in geometric regions related to flat wave propagation and diffraction by the cone point, we can prove spectral cluster estimates equivalent to those in works on smooth Riemannian manifolds.

math.AP↗
arXiv2015

Concerning Toponogov's Theorem and logarithmic improvement of estimates of eigenfunctions

We use Toponogov's triangle comparison theorem from Riemannian geometry along with quantitative scale oriented variants of classical propagation of singularities arguments to obtain logarithmic improvements of the Kakeya-Nikodym norms introduced in \cite{SKN} for manifolds of nonpositive sectional curvature. Using these and results from our paper \cite{BS15} we are able to obtain log-improvements of $L^p(M)$ estimates for such manifolds when $2<p<\tfrac{2(n+1)}{n-1}$. These in turn imply $(\logλ)^{σ_n}$, $σ_n\approx n$, improved lower bounds for $L^1$-norms of eigenfunctions of the estimates of the second author and Zelditch~\cite{SZ11}, and using a result from Hezari and the second author~\cite{HS}, under this curvature assumption, we are able to improve the lower bounds for the size of nodal sets of Colding and Minicozzi~\cite{CM} by a factor of $(\log λ)^μ$ for any $μ<\tfrac{2(n+1)^2}{n-1}$, if $n\ge3$.

math.AP↗
arXiv2014

Strichartz and Localized Energy Estimates for the Wave Equation in Strictly Concave Domains

We prove localized energy estimates for the wave equation in domains with a strictly concave boundary when homogeneous Dirichlet or Neumann conditions are imposed. By restricting the solution to small, frequency dependent, space time collars of the boundary, it is seen that a stronger gain in regularity can be obtained relative to the usual energy estimates. Mixed norm estimates of Strichartz and square function type follow as a result, using the energy estimates to control error terms which arise in a wave packet parametrix construction. While the latter estimates are not new for Dirichlet conditions, the present approach provides an avenue for treating these estimates when Neumann conditions are imposed. The method also treats Schrödinger equations with time independent coefficients.

math.AP↗
arXiv2013

On Kakeya-Nikodym averages, $L^p$-norms and lower bounds for nodal sets of eigenfunctions in higher dimensions

We extend a result of the second author \cite[Theorem 1.1]{soggekaknik} to dimensions $d \geq 3$ which relates the size of $L^p$-norms of eigenfunctions for $2<p<\frac{2(d+1)}{d-1}$ to the amount of $L^2$-mass in shrinking tubes about unit-length geodesics. The proof uses bilinear oscillatory integral estimates of Lee \cite{leebilinear} and a variable coefficient variant of an "$\veps$ removal lemma" of Tao and Vargas \cite{tv1}. We also use Hörmander's \cite{HorOsc} $L^2$ oscillatory integral theorem and the Cartan-Hadamard theorem to show that, under the assumption of nonpositive curvature, the $L^2$-norm of eigenfunctions $e_\la$ over unit-length tubes of width $\la^{-\frac12}$ goes to zero. Using our main estimate, we deduce that, in this case, the $L^p$-norms of eigenfunctions for the above range of exponents is relatively small. As a result, we can slightly improve the known lower bounds for nodal sets in dimensions $d\ge3$ of Colding and Minicozzi \cite{CM} in the special case of (variable) nonpositive curvature.

math.AP↗
arXiv2012

$L^q$ bounds on restrictions of spectral clusters to submanifolds for low regularity metrics

We prove $L^q$ bounds on the restriction of spectral clusters to submanifolds in Riemannian manifolds equipped with metrics of $C^{1,α}$ regularity for $0 \leq α\leq 1$. Our results allow for Lipschitz regularity when $α=0$, meaning they give estimates on manifolds with boundary. When $0< α\leq 1$, the scalar second fundamental form for a codimension 1 submanifold can be defined, and we show improved estimates when this form is negative definite. This extends results of Burq-Gérard-Tzvetkov and Hu to manifolds with low regularity metrics.

math.AP↗
arXiv2011

Strichartz estimates and the nonlinear Schrödinger equation on manifolds with boundary

We establish Strichartz estimates for the Schrödinger equation on Riemannian manifolds $(Ω,\g)$ with boundary, for both the compact case and the case that $Ω$ is the exterior of a smooth, non-trapping obstacle in Euclidean space. The estimates for exterior domains are scale invariant; the range of Lebesgue exponents $(p,q)$ for which we obtain these estimates is smaller than the range known for Euclidean space, but includes the key $L^4_tL^\infty_x$ estimate, which we use to give a simple proof of well-posedness results for the energy critical Schrödinger equation in 3 dimensions. Our estimates on compact manifolds involve a loss of derivatives with respect to the scale invariant index. We use these to establish well-posedness for finite energy data of certain semilinear Schrödinger equations on general compact manifolds with boundary.

math.AP↗