arXiv subjects

Shohei Nakamura

Publications and source records attributed to Shohei Nakamura.

At least 19 recordsLinked to original sources

arXiv2026

Rectangles, triangles and Schr\"{o}dinger waves

Can a finite set of lattice points determine many rectangles and few isosceles triangles? This turns out to be a surprisingly interesting question in combinatorial geometry that we answer using basic analytic number theory combined with a finite-field construction. The result is useful because it gives obstructions to Mizohata--Takeuchi-type estimates in the setting of the paraboloid. Specifically, we establish transference between Euclidean and periodic weighted $\mathrm{L}^2$ estimates for solutions to the Schr\"{o}dinger equation, and then relate the failure of the latter to quantities tied to combinatorial problems, such as the one above. By completing this programme we give new explicit combinatorial counterexamples to the paraboloid case of the Mizohata--Takeuchi conjecture, which was recently shown to be false by Cairo for curved hypersurfaces.

math.CA↗

arXiv2026

The Gaussian Conjugate Rogers-Shephard Inequality

We fuse between the Rogers-Shephard inequality for the Lebesgue measure and Royen's Gaussian Correlation Inequality, simultaneously extending both into a single sharp inequality for the Gaussian measure $γ$ on $\mathbb{R}^n$, stating that \[ γ(K) γ(L) \leq γ(K\cap L) γ(K+L) \] whenever $K$ and $L$ are origin-symmetric convex sets in $\mathbb{R}^n$. This confirms a conjecture of M. Tehranchi [https://doi.org/10.1214/17-ECP89]. In fact, we show that the inequality remains valid whenever the Gaussian barycenters of $K$ and $L$ are at the origin, and characterize the equality cases. After rescaling, this also yields the following new inequality for convex sets with (Lebesgue) barycenters at the origin: \[ |K| |L| \leq |K \cap L| |K + L | ; \] this can be seen as a conjugate counterpart to Spingarn's extension of the Rogers-Shephard inequality (where $K+L$ is replaced by $K-L$ above). We also derive an additional conjugate version of a Gaussian inequality due to V. Milman and Pajor, as well as several extensions. Our main tool is a new Gaussian Forward-Reverse Brascamp-Lieb inequality for centered log-concave functions, of independent interest, which is crucially applicable to degenerate Gaussian covariances.

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arXiv2025

The Gaussian correlation inequality for centered convex sets and the case of equality

Inspired by Milman's recent observation, we prove that the Gaussian correlation inequality holds for convex sets having the same barycenter, and especially for centered ones. This gives an affirmative answer to the problem proposed by Szarek and Werner. We also characterize the equality case. The study of the equality case in the non-symmetric Gaussian correlation inequality relates to the following question: Let $X$ be a standard Gaussian random vector in $\mathbb{R}^n$. For which convex sets $K_1,K_2 \subset \mathbb{R}^n$, are the two events $\{X\in K_1\}$ and $\{X\in K_2\}$ independent? By imposing an additional normalization that $K_1$ and $K_2$ have the same barycenter, we give the necessary and sufficient conditions for this independence. The conditions also identify when $\|X\|_{K_1}$ and $\|X\|_{K_2}$ are independent as random variables.

math.FA↗

arXiv2025

A phase-space approach to weighted Fourier extension inequalities

The purpose of this paper is to expose and investigate natural phase-space formulations of two longstanding problems in the restriction theory of the Fourier transform. These problems, often referred to as the Stein and Mizohata--Takeuchi conjectures, assert that Fourier extension operators associated with rather general (codimension 1) submanifolds of Euclidean space, may be effectively controlled by the classical X-ray transform via weighted $L^2$ inequalities. Our phase-space formulations, which have their origins in recent work of Dendrinos, Mustata and Vitturi, expose close connections with a conjecture of Flandrin from time-frequency analysis, and rest on the identification of an explicit ``geometric" Wigner transform associated with an arbitrary (smooth strictly convex) submanifold $S$ of $\mathbb{R}^n$. Our main results are certain natural ``Sobolev variants" of the Stein and Mizohata--Takeuchi conjectures, and involve estimating the Sobolev norms of such Wigner transforms by geometric forms of classical bilinear fractional integrals. Our broad geometric framework allows us to explore the role of the curvature of the submanifold in these problems, and in particular we obtain bounds that are independent of any lower bound on the curvature; a feature that is uncommon in the wider restriction theory of the Fourier transform. Finally, we provide a further illustration of the effectiveness of our analysis by establishing a form of Flandrin's conjecture in the plane with an $\varepsilon$-loss. While our perspective comes primarily from Euclidean harmonic analysis, the procedure used for constructing phase-space representations of extension operators is well-known in optics.

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arXiv2025

Mizohata-Takeuchi inequalities for orthonormal systems

We establish some weighted $L^2$ inequalities for Fourier extension operators in the setting of orthonormal systems. In the process we develop a direct approach to such inequalities based on generalised Wigner distributions, complementing the Schatten space approach that is prevalent in the wider context of estimates for such orthonormal systems. Our results are set within a broader family of tentatively suggested ($L^p$) inequalities of Mizohata--Takeuchi type. For $p$ an even integer we see that such weighted inequalities may be recast as questions of co-positivity of tensor forms, and for $p\leq 1$ we provide some evidence that they may hold in reverse provided the orthonormal sequence is complete.

math.CA↗

arXiv2025

Duality and Heat flow

We reveal the relation between the Legendre transform of convex functions and heat flow evolution, and how it applies to the functional Blaschke-Santalo inequality. We also describe local maximizers in this inequality.

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arXiv2024

Regularized Brascamp--Lieb inequalities

Given any (forward) Brascamp--Lieb inequality on euclidean space, a famous theorem of Lieb guarantees that gaussian near-maximizers always exist. Recently, Barthe and Wolff used mass transportation techniques to establish a counterpart to Lieb's theorem for all non-degenerate cases of the inverse Brascamp--Lieb inequality. Here we build on work of Chen--Dafnis--Paouris and employ heat-flow techniques to understand the inverse Brascamp--Lieb inequality for certain regularized input functions, in particular extending the Barthe--Wolff theorem to such a setting. Inspiration arose from work of Bennett, Carbery, Christ and Tao for the forward inequality, and we recover their generalized Lieb's theorem using a clever limiting argument of Wolff. In fact, we use Wolff's idea to deduce regularized inequalites in the broader framework of the forward-reverse Brascamp--Lieb inequality, in particular allowing us to recover the gaussian saturation property in this framework first obtained by Courtade, Cuff, Liu and Verdú.

math.CA↗

arXiv2024

A generalized Legendre duality relation and Gaussian saturation

Motivated by the barycenter problem in optimal transportation theory, Kolesnikov--Werner recently extended the notion of the Legendre duality relation for two functions to the case for multiple functions. We further generalize the duality relation and then establish the centered Gaussian saturation property for a Blaschke--Santaló type inequality associated with it. Our approach to the understanding such a generalized Legendre duality relation is based on our earlier observation that directly links Legendre duality with the inverse Brascamp--Lieb inequality. More precisely, for a large family of degenerate Brascamp--Lieb data, we prove that the centered Gaussian saturation property for the inverse Brascamp--Lieb inequality holds true when inputs are restricted to even and log-concave functions. As an application to convex geometry, we establish the most important case of a conjecture of Kolesnikov and Werner about the Blaschke--Santaló inequality for multiple even functions as well as multiple symmetric convex bodies. Furthermore, in the direction of information theory and optimal transportation theory, this provides an affirmative answer to another conjecture of Kolesnikov--Werner about a Talagrand type inequality for multiple even probability measures that involves the Wasserstein barycenter.

math.FA↗

arXiv2024

Tomographic Fourier Extension Identities for Submanifolds of $\mathbb{R}^n$

We establish identities for the composition $T_{k,n}(|\widehat{gdσ}|^2)$, where $g\mapsto \widehat{gdσ}$ is the Fourier extension operator associated with a general smooth $k$-dimensional submanifold of $\mathbb{R}^n$, and $T_{k,n}$ is the $k$-plane transform. Several connections to problems in Fourier restriction theory are presented.

math.CA↗

arXiv2024

The functional volume product under heat flow

We prove that the functional volume product for even functions is monotone increasing along the Fokker--Planck heat flow. This in particular yields a new proof of the functional Blaschke--Santaló inequality by K. Ball and also Artstein-Avidan--Klartag--Milman in the even case. This result is the consequence of a new understanding of the regularizing property of the Ornstein--Uhlenbeck semigroup. That is, we establish an improvement of Borell's reverse hypercontractivity inequality for even functions and identify the sharp range of the admissible exponents. As another consequence of successfully identifying the sharp range for the inequality, we derive the sharp $L^p$-$L^q$ inequality for the Laplace transform for even functions. The best constant of the inequality is attained by centered Gaussians, and thus this provides an analogous result to Beckner's sharp Hausdorff--Young inequality. Our technical novelty in the proof is the use of the Brascamp--Lieb inequality for log-concave measures and Cramér--Rao's inequality in this context.

math.FA↗

arXiv2023

Decoupling inequality for paraboloid under shell type restriction and its application to the periodic Zakharov system

In this paper, we establish local well-posedness for the Zakharov system on $\mathbb{T}^d$, $d\ge3$ in a low regularity setting. Our result improves the work of Kishimoto. Moreover, the result is sharp up to $\varepsilon$-loss of regularity when $d=3$ and $d\ge5$ as long as one utilizes the iteration argument. We introduce ideas from recent developments of the Fourier restriction theory. The key element in the proof of our well-posedness result is a new trilinear discrete Fourier restriction estimate involving paraboloid and cone. We prove this trilinear estimate by improving Bourgain--Demeter's range of exponent for the linear decoupling inequality for paraboloid under the constraint that the input space-time function $f$ satisfies ${\rm supp}\, \hat{f} \subset \{ (ξ,τ) \in \mathbb{R}^{d+1}: 1- \frac1N \le |ξ| \le 1 + \frac1N,\; |τ- |ξ|^2| \le \frac1{N^{2}} \} $ for large $N\ge1$.

math.AP↗

arXiv2022

Hypercontractivity beyond Nelson's time and its applications to Blaschke--Santaló inequality and inverse Santaló inequality

We explore an interplay between an analysis of diffusion flows such as Ornstein--Uhlenbeck flow and Fokker--Planck flow and inequalities from convex geometry regarding the volume product. More precisely, we introduce new types of hypercontractivity for the Ornstein--Uhlenbeck flow and clarify how these imply the Blaschke--Santaló inequality and the inverse Santaló inequality, also known as Mahler's conjecture. Motivated the link, we establish two types of new hypercontractivity in this paper. The first one is an improvement of Borell's reverse hypercontractivity inequality in terms of Nelson's time relation under the restriction that the inputs have an appropriate symmetry. We then prove that it implies the Blaschke--Santaló inequality. At the same time, it also provides an example of the inverse Brascamp--Lieb inequality due to Barthe--Wolff beyond their non-degenerate condition. The second one is Nelson's forward hypercontractivity inequality with exponents below 1 for the inputs which are log-convex and semi-log-concave. This yields new lower bounds of the volume product for convex bodies whose boundaries are well curved. This consequence provides a quantitative result of works by Stancu and Reisner--Schütt--Werner where they observed that a convex body with well curved boundary is not a local minimum of the volume product.

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arXiv2022

Stability of hypercontractivity, the logarithmic Sobolev inequality, and Talagrand's cost inequality

We provide deficit estimates for Nelson's hypercontractivity inequality, the logarithmic Sobolev inequality, and Talagrand's transportation cost inequality under the restriction that the inputs are semi-log-subharmonic, semi-log-convex, or semi-log-concave. In particular, our result on the logarithmic Sobolev inequality complements a recently obtained result by Eldan, Lehec and Shenfeld concerning a deficit estimate for inputs with small covariance. Similarly, our result on Talagrand's transportation cost inequality complements and, for a large class of semi-log-concave inputs, improves a deficit estimate recently proved by Mikulincer. Our deficit estimates for hypercontractivity will be obtained by using a flow monotonicity scheme built on the Fokker--Planck equation, and our deficit estimates for the logarithmic Sobolev inequality will be derived as a corollary. For Talagrand's inequality, we use an optimal transportation argument. An appealing feature of our framework is robustness and this allows us to derive deficit estimates for the hypercontracivity inequality associated with the Hamilton--Jacobi equation, the Poincaré inequality, and for Beckner's inequality.

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arXiv2020

Strichartz estimates for orthonormal families of initial data and weighted oscillatory integral estimates

We establish new Strichartz estimates for orthonormal families of initial data in the case of the wave, Klein-Gordon and fractional Schrödinger equations. Our estimates extend those of Frank-Sabin in the case of the wave and Klein-Gordon equations, and generalize work of Frank-Lewin-Lieb-Seiringer and Frank-Sabin for the Schrödinger equation. Due to a certain technical barrier, except for the classical Schrödinger equation, the Strichartz estimates for orthonormal families of initial data have not previously been established up to the sharp summability exponents in the full range of admissible pairs. We obtain the optimal estimates in various notable cases and improve the previous results. The main novelty of this paper is the use of estimates for weighted oscillatory integrals which we combine with an approach due to Frank and Sabin. This strategy also leads us to proving new estimates for weighted oscillatory integrals with optimal decay exponents which we believe to be of wider independent interest. Applications to the theory of infinite systems of Hartree type, weighted velocity averaging lemmas for kinetic transport equations, and refined Strichartz estimates for data in Besov spaces are also provided.

math.AP↗

arXiv2020

Tomography bounds for the Fourier extension operator and applications

We explore the extent to which the Fourier transform of an $L^p$ density supported on the sphere in $\mathbb{R}^n$ can have large mass on affine subspaces, placing particular emphasis on lines and hyperplanes. This involves establishing bounds on quantities of the form $X(|\widehat{gdσ}|^2)$ and $\mathcal{R}(|\widehat{gdσ}|^2)$, where $X$ and $\mathcal{R}$ denote the X-ray and Radon transforms respectively; here $dσ$ denotes Lebesgue measure on the unit sphere $\mathbb{S}^{n-1}$, and $g\in L^p(\mathbb{S}^{n-1})$. We also identify some conjectural bounds of this type that sit between the classical Fourier restriction and Kakeya conjectures. Finally we provide some applications of such tomography bounds to the theory of weighted norm inequalities for $\widehat{gdσ}$, establishing some natural variants of conjectures of Stein and Mizohata--Takeuchi from the 1970s. Our approach, which has its origins in work of Planchon and Vega, exploits cancellation via Plancherel's theorem on affine subspaces, avoiding the conventional use of wave-packet and stationary-phase methods.

math.CA↗

arXiv2019

Maximal estimates for the Schrödinger equation with orthonormal initial data

For the one-dimensional Schrödinger equation, we obtain sharp maximal-in-time and maximal-in-space estimates for systems of orthonormal initial data. The maximal-in-time estimates generalize a classical result of Kenig--Ponce--Vega and allow us obtain pointwise convergence results associated with systems of infinitely many fermions. The maximal-in-space estimates simultaneously address an endpoint problem raised by Frank--Sabin in their work on Strichartz estimates for orthonormal systems of data, and provide a path toward proving our maximal-in-time estimates.

math.AP↗

arXiv2019

Finite difference scheme for two-dimensional periodic nonlinear Schrödinger equations

A nonlinear Schrödinger equation (NLS) on a periodic box can be discretized as a discrete nonlinear Schrödinger equation (DNLS) on a periodic cubic lattice, which is a system of finitely many ordinary differential equations. We show that in two spatial dimensions, solutions to the DNLS converge strongly in $L^2$ to those of the NLS as the grid size $h>0$ approaches zero. As a result, the effectiveness of the finite difference method (FDM) is justified for the two-dimensional periodic NLS.

math.AP↗

arXiv2018

The orthonormal Strichartz inequality on torus

In this paper, motivated by recent important works due to Frank-Lewin-Lieb-Seiringer \cite{FLLS} and Frank-Sabin \cite{frank-sabin-1}, we study the Strichartz inequality on torus with the orthonormal system input and obtain sharp estimates in certain sense. An application of the inequality shows the well-posedness to the periodic Hartree equation describing the infinitely many quantum particles with the power type interaction.

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