양찬우 교수

In this paper, we give $L^p-L^q$ estimates and the $L^p$ regularizing estimate of Radon transforms associated to real analytic functions, and we also give estimates of the decay rate of the $L^p$ operator norm of corresponding oscillatory integral operators. For $L^p-L^q$ estimates and estimates of the decay rate of the $L^p$ operator norm we obtain sharp results except for extreme points; however, for $L^p$ regularity we allow some restrictions on the phase function.

In this paper, we study decay estimates for a two-dimensional scalar oscillatory integral with degenerate real-analytic phase and amplitude. Integrals such as these form a model for certain higher-dimensional degenerate oscillatory integrals, for which it is known that many of the two-dimensional results fail. We define an analogue of the Newton distance in the weighted case, and prove that this gives the optimal rate of decay for the weighted oscillatory integral under certain generic hypothese

We give sharp endpoint estimates for the decay rates of L p operator norms of oscillatory integral operators with some real homogeneous polynomial phases.

We consider the mass concentration phenomenon for the L 2-critical nonlinear Schrödinger equations. We show the mass concentration of blow-up solutions contained in space near the finite time. The new ingredient in this paper is a refinement of Strichartz's estimates with the mixed norm for 2 < q ≤ r.

We study double Hilbert transforms and maximal functions along surfaces of the form (t_1,t_2,\gamma_1(t_1)\gamma_2(t_2)) . The L^p(\mathbb{R}^3) boundedness of the maximal operator is obtained if each \gamma_i is a convex increasing and \gamma_i(0)=0 . The double Hilbert transform is bounded in L^p(\mathbb{R}^3) if both \gamma_i 's above are extended as even functions. If \gamma_1 is odd, then we need an additional comparability condition on \gamma_2 . This result is extended to higher dimension

We prove the global existence and scattering for the Hartree-type equation in H s (ℝ3) the low regularity space s < 1. We follow the ideas in Colliander et al. (2004 Colliander , J. , Keel , M. , Staffilani , G. , Takaoka , H. , Tao , T. ( 2004 ). Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on ℝ3 . Comm. Pure Appl. Math. 57 : 987 – 1014 .[Crossref], [Web of Science ®] , [Google Scholar]) to the Hartree-type nonlinearity, and also develop the theory of

In this paper, we formulate necessary conditions for decay rates of L^p operator norms of weighted oscillatory integral operators on \mathbb{R} and give sharp L^2 estimates and nearly sharp L^p estimates.