Related papers: Bourgain's counterexample in the sequential conver…
This paper provides a rigorous derivation of a counterexample of Bourgain, related to a well-known question of pointwise a.e. convergence for the solution of the linear Schr\"odinger equation, for initial data in a Sobolev space. This…
It is shown that Schr\"odinger maximal inequalities over fractals are equivalent to the $L^2$ decay rates of Fourier transforms of fractal measures over the paraboloid. A similar connection is shown between the wave equation and cone…
We investigate the pointwise convergence of the solution to the fractional Schr\"odinger equation in $\mathbb R^2$. By establishing $H^s(\mathbb R^2)-L^3(\mathbb R^2)$ estimates for the associated maximal operator provided that $s>1/3$, we…
We construct counterexamples for the fractal Schr\"odinger convergence problem by combining a fractal extension of Bourgain's counterexample and the intermediate space trick of Du--Kim--Wang--Zhang. We confirm that the same regularity as…
In this paper, we study the almost everywhere convergence results of Schr\"odinger operator with complex time along curves. We also consider the fractional cases. All results are sharp up to the endpoints.
We consider the pointwise convergence problem for the solution of Schr\"odinger-type equations along directions determined by a given compact subset of the real line. This problem contains Carleson's problem as the most simple case and was…
The purpose of this note is to provide a summary of the recent work of the authors on two variations of the pointwise convergence problem for the solutions to the fractional Schr\"odinger equations; convergence along a tangential line and…
In this paper we study the pointwise convergence problem along a tangential curve for the fractional Schr\"odinger equations in one spatial dimension and estimate the capacitary dimension of the divergence set. We extend a prior paper by…
We consider Carleson's problem regarding pointwise convergence for the Schr\"odinger equation. Bourgain recently proved that there is initial data, in $H^s(\mathbb{R}^n)$ with $s<\frac{n}{2(n+1)}$, for which the solution diverges on a set…
We study the pointwise convergence of Landau type Schr\"odinger operators within the fractional Sobolev space $W^{s,p}(\mathbb R)$. Our results extend those established by Bailey (Rev. Mat. Iberoam., 29 (2): 531-546, 2013) and Yuan, Zhao…
We consider a fractal refinement of the Carleson problem for the Schr\"odinger equation, that is to identify the minimal regularity needed by the solutions to converge pointwise to their initial data almost everywhere with respect to the…
We study pointwise convergence of the fractional Schr\"odinger means along sequences $t_n$ which converge to zero. Our main result is that bounds on the maximal function $\sup_{n} |e^{it_n(-\Delta)^{\alpha/2}} f| $ can be deduced from those…
Almost everywhere convergence on the solution of Schr\"odinger equation is an important problem raised by Carleson, which was essentially solved by Du-Guth-Li and Du-Zhang. In this note, we obtain the sharp pointwise convergence on the…
The local maximal inequality for the Schr\"{o}dinger operators of order $\a>1$ is shown to be bounded from $H^s(\R^2)$ to $L^2$ for any $s>\frac38$. This improves the previous result of Sj\"{o}lin on the regularity of solutions to…
In this paper, we prove new multilinear Strichartz estimates, which are obtained by using techniques of Bourgain. These estimates lead to new critical well-posedness results for the nonlinear Schr\"odinger equation on irrational tori in two…
We prove finite-field analogs of Bourgain's projection theorem in higher dimensions. In particular, for a certain range of parameters we improve on an exceptional set estimate by Chen in all dimensions and codimensions.
In a seminal paper (1996), Bourgain proved invariance of the Gibbs measure for the defocusing cubic nonlinear Schr\"odinger equation on the two-dimensional torus by constructing local-in-time solutions in a probabilistic manner. In this…
We give new lower bounds for $L^p$ estimates of the Schr\"odinger maximal function by generalizing an example of Bourgain.
Multiparameter maximal estimates are considered for operators of Schr\"odinger type. Sharp and almost sharp results, that extend work by Rogers and Villarroya, are obtained. We provide new estimates via the integrability of the kernel which…
We prove time-pointwise quantitative unique continuation estimates for the evolution operators associated to (fractional powers of) the Baouendi--Grushin operators on the cylinder $\mathbb{R}^d \times \mathbb{T}^d$. Corresponding spectral…