相关论文: On Bourgain's counterexample for the Schr\"odinger…

We study the inverse backscattering problem for the Schr\"odinger equation in two dimensions. We prove that, for a non-smooth potential in 2D the main singularities up to 1/2 of the derivative of the potential are contained in the Born…

In this paper, we study the extremal problem for the Strichartz inequality for the Schr\"{o}dinger equation on $\mathbb{R}^2$. We show that the solutions to the associated Euler-Lagrange equation are exponentially decaying in the Fourier…

In this paper we consider the pointwise convergence to the initial data for the Schr\"{o}dinger-Dirac equation $i\tfrac{\partial u}{\partial t}=D^{\beta}u$ with $u(x,0)=u^0$ in a dyadic Besov space. Here $D^{\beta}$ denotes the fractional…

We obtain (essentially sharp) boundedness results for certain generalized local maximal operators between fractional weighted Sobolev spaces and their modifications. Concrete boundedness results between well known fractional Sobolev spaces…

We present sharp estimates for the extremal eigenvalues of the Schur complements arising in saddle point problems. These estimates are derived using the auxiliary space theory, in which a given iterative method is interpreted as an…

We study weighted Sobolev inequalities on open convex cones endowed with $\alpha$-homogeneous weights satisfying a certain concavity condition. We establish a so-called reduction principle for these inequalities and characterize optimal…

We consider non-gauge-invariant cubic nonlinear Schr\"odinger equations in one space dimension. We show that initial data of size $\varepsilon$ in a weighted Sobolev space lead to solutions with sharp $L_x^\infty$ decay up to time…

In this short article we show a particular version of the Hedberg inequality which can be used to derive, in a very simple manner, functional inequalities involving Sobolev and Besov spaces in the general setting of Lebesgue spaces of…

We show a pointwise estimate for the Fourier transform on the line involving the number of times the function changes monotonicity. The contrapositive of the theorem may be used to find a lower bound to the number of local maxima of a…

The purpose of this expository note is to give the proof of a theorem of Bourgain with some additional details and updated notation. The theorem first appeared as an appendix to the breakthrough paper by Friedgut, \emph{Sharp Thresholds of…

We present an alternative way to recover the recent result from \cite{LR} using the pseudoconformal transformation.

We study the defocusing nonlinear Schr\"odinger equation in three space dimensions. We prove that any radial solution that remains bounded in the critical Sobolev space must be global and scatter. In the energy-supercritical setting, we…

The Riesz-Sobolev inequality provides an upper bound, in integral form, for the convolution of indicator functions of subsets of Euclidean space. We formulate and prove a sharper form of the inequality. This can be equivalently phrased as a…

We utilize the theory of de Branges spaces to show when certain Schr\"odinger operators with strongly singular potentials are uniquely determined by their associated spectral measure. The results are applied to obtain an inverse uniqueness…

We develop a new method suitable for establishing lower bounds on the ball measure of noncompactness of operators acting between considerably general quasinormed function spaces. This new method removes some of the restrictions…

We derive a Maximum Principle for optimal control problems with constraints given by the coupling of a system of ODEs and a PDE of Vlasov-type. Such problems arise naturally as ${\Gamma}$-limits of optimal control problems subject to ODE…

We consider the PDEs version of the Carleson problem in the context of the cubic nonlinear Klein-Gordon equation. This means that we aim to establish the lowest regularity class for which one has almost everywhere pointwise convergence of…

In this article we prove martingale type pointwise convergence theorems pertaining to tensor product splines defined on $d$-dimensional Euclidean space ($d$ is a positive integer), where conditional expectations are replaced by their…

The inverse problems about fractional Calder\'on problem and fractional Schr\"odinger equations are of interest in the study of mathematics. In this paper, we propose the inverse problem to simultaneously reconstruct potentials and sources…

In this article, we re-examine some of the classical pointwise multiplication theorems in Sobolev-Slobodeckij spaces, in part motivated by a simple counter-example that illustrates how certain multiplication theorems fail in…