Related papers: Strichartz inequalities: some recent developments

Complexes of discrete distributional differential forms are introduced into finite element exterior calculus. Thus we generalize a notion of Braess and Sch\"oberl, originally studied for a posteriori error estimation. We construct…

Lower and upper bounds for a given function are important in many mathematical and engineering contexts, where they often serve as a base for both analysis and application. In this short paper, we derive piecewise linear and quadratic…

This paper analyzes Sch\"odinger operators from viewpoint of correlation inequalities. We construct Griffiths inequalities for the ground state expectations by applying operator-theoretic correlation inequalities. As an example of such an…

We prove certain mixed-norm Strichartz estimates on manifolds with boundary. Using them we are able to prove new results for the critical and subcritical wave equation in 4-dimensions with Dirichlet or Neumann boundary conditions. We obtain…

We prove a sharpened version of the Strichartz inequality for radial solutions of the Schr\"odinger equation in $\mathbb{R}^2\times \mathbb{R}$. We establish an improved upper bound for functions that nearly extremize the inequality, with a…

The analytic properties of the Jost functions are fundamental in quantum scattering theory and in the analytic continuation of the scattering matrix into the complex energy plane. In this work, the analyticity of the Jost functions is…

This work concerns generalized backward stochastic differential equations, which are coupled with a family of reflecting diffusion processes. First of all, we establish the large deviation principle for forward stochastic differential…

These Notes are intended for graduate or undergraduate students who have familiarity with Lebesgue measure theory, partial differential equations, and functional analysis. The main topics covered in this work are the study of the Cauchy…

One way to define the concentration of measure phenomenon is via Talagrand inequalities, also called transportation-information inequalities. That is, a comparison of the Wasserstein distance from the given measure to any other absolutely…

These are intended to be review notes on emergent symmetries, i.e., symmetries which manifest themselves in specific sectors of energy in many systems. The emphasis is on the physical aspects rather than computation methods. We include some…

In this paper, we present a different proof on the discrete Fourier restriction. The proof recovers Bourgain's level set result on Strichartz estimates associated with Schr\"odinger equations on torus. Some sharp estimates on…

These are the lecture notes for a course at the Summer School on "Applied Analysis" at the Technical University Chemnitz in September 2011. We start with the definition of a fractal algebra and show that the fractal property is enormously…

In this paper we prove some new Strichartz estimates related to the Cauchy problem for the Bessel operator on the half-line and we establish a fractal version of the Tomas-Stein restriction theorem for the Hankel transform. Then we use the…

Differential complexes such as the de Rham complex have recently come to play an important role in the design and analysis of numerical methods for partial differential equations. The design of stable discretizations of systems of partial…

In this paper, we first show that there exists a maximizer for the non-endpoint Strichartz inequalities for the Schr\"odinger equation in all dimensions based on the recent linear profile decomposition results. We then present a new proof…

In this paper we survey some recent results on the central limit theorem and its weak invariance principle for stationary sequences. We also describe several maximal inequalities that are the main tool for obtaining the invariance…

Let $\Delta_{k,a}$ and $\Delta_k $ be the $(k,a)$-generalized Laguerre operator and the Dunkl Laplacian operator on $\mathbb{R}^n$, respectively. The aim of this article is twofold. First, we prove a restriction theorem for the…

Opial's inequality and its ramifications play an important role in the theory of differential and difference equations. A sharp unifying generalization of Opial's inequality is presented that contains both its continuous and discrete…

Stanley's inequalities for partially ordered sets establish important log-concavity relations for sequences of linear extensions counts. Their extremals however, i.e., the equality cases of these inequalities, were until now poorly…

We obtain refined Strichartz estimates for the sub-Riemannian Schr\"{o}dinger equation on $H$-type Carnot groups using Fourier restriction techniques. In particular, we extend the previously known Strichartz estimates previously obtained…