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Sunggeum Hong

Publications and source records attributed to Sunggeum Hong.

4 recordsLinked to original sources
arXiv2023

Hörmander type theorem for multilinear Pseudo-differential operators

We establish a Hörmander type theorem for the multilinear pseudo-differential operators, which is also a generalization of the results in \cite{MR4322619} to symbols depending on the spatial variable. Most known results for multilinear pseudo-differential operators were obtained by assuming their symbols satisfy pointwise derivative estimates(Mihlin-type condition), that is, their symbols belong to some symbol classes $n$-$\mathcal{S}^m_{ρ, δ}(\mathbb{R}^d)$, $0 \le δ\le ρ\le1$, $0 \le δ<1$ for some $m \le 0$. In this paper, we shall consider multilinear pseudo-differential operators whose symbols have limited smoothness described in terms of function space and not in a pointwise form(Hörmander type condition). Our conditions for symbols are weaker than the Mihlin-type conditions in two senses: the one is that we only assume the first-order derivative conditions in the spatial variable and lower-order derivative conditions in the frequency variable, and the other is that we make use of $L^2$-average condition rather than pointwise derivative conditions for the symbols. As an application, we obtain some mapping properties for the multilinear pseudo-differential operators associated with symbols belonging to the classes $n$-$\mathcal{S}^{m}_{ρ,δ}(\mathbb{R}^{d})$, $0 \le ρ\le 1$, $0 \le δ<1$, $m \le 0$. Moreover, it can be pointed out that our results can be applied to wider classes of symbols which do not belong to the traditional symbol classes $n$-$\mathcal{S}^{m}_{ρ,δ}(\mathbb{R}^{d})$.

math.AP↗
arXiv2009

Mass concentration for the $L^2$-critical Nonlinear Schrödinger equations of higher orders

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.

math.AP↗
arXiv2005

Weak type estimates on certain Hardy spaces for smooth cone type multipliers

Let $\varrho\in C^{\infty} ({\Bbb R}^d\setminus\{0\})$ be a non-radial homogeneous distance function satisfying $\varrho(tξ)=t\varrho(ξ)$. For $f\in\frak S ({\Bbb R}^{d+1})$ and $δ>0$, we consider convolution operator ${\Cal T}^δ$ associated with the smooth cone type multipliers defined by $$\hat {{\Cal T}^δ f}(ξ,τ)= (1-\frac{\varrho(ξ)}{|τ|} )^δ_+\hat f (ξ,τ), (ξ,τ)\in {\Bbb R}^d \times \Bbb R.$$ If the unit sphere $Σ_{\varrho}\fallingdotseq\{ξ\in {\Bbb R}^d : \varrho(ξ)=1\}$ is a convex hypersurface of finite type and $\varrho$ is not radial, then we prove that ${\Cal T}^{δ(p)}$ maps from $H^p({\Bbb R}^{d+1})$, $0 0} λ^p|\{(x,t)\in \bar{{\Bbb R}^{d+1}\setminusΓ_γ} : |{\Cal T}_{\varrho}^{δ(p)}f(x,t)|>λ\}|=\infty.$$

math.CA↗