arXiv subjects
Publications and source records attributed to Sanghyuk Lee.
We prove a weighted norm inequality for the maximal Bochner--Riesz operator and the associated square-function. This yields new $L^p(R^d)$ bounds on classes of radial Fourier multipliers for $p\ge 2+4/d$ with $d\ge 2$, as well as space-time regularity results for the wave and Schrödinger equations.
We consider the mass concentration phenomenon for the $L^2$-critical nonlinear Schrödinger equations of higher orders. We show that any solution $u$ to $iu_{t} + (-Δ)^{\fracα2} u =\pm |u|^\frac{2α}{d}u$, $u(0,\cdot)\in L^2$ for $α>2$, which blows up in a finite time, satisfies a mass concentration phenomenon near the blow-up time. We verify that as $α$ increases, the size of region capturing a mass concentration gets wider due to the stronger dispersive effect.
In this paper we establish mixed norm estimates of interactive Schrödinger waves and apply them to study smoothing properties and global well-posedness of the nonlinear Schrödinger equations with mass critical nonlinearity.
We study pointwise convergence of the solutions to Schrödinger equations with initial datum $f\in H^s(\mathbb R^n)$. The conjecture is that the solution $e^{itΔ}f$ converges to $f$ almost everywhere for all $f\in H^s(\mathbb R^n)$ if and only if $s\ge 1/4$. The conjecture is known true for one spatial dimension and the convergence when $s>1/2$ was verified for $n\ge 2$. Recently, concrete progresses have been made in $\mathbb R^2$ for some $s<1/2$. However, when $n\ge 3$ no positive result is known for the initial datum $f\in H^s(\mathbb R^n)$, $s\le 1/2$. We show that $\lim_{t\to 0} e^{itΔ}f= f$ a.e. for $f\in H^s(\mathbb R^3)$ whenever $s>1/2-1/{24}$.