이정진 교수

The purpose of this paper is to improve the known estimates for Mockenhaupt's square function in $\mathbb R^3$ and for Sogge's local smoothing in $\mathbb R^{2+1}$ spacetime. For this we use the trilinear approach of S. Lee and A. Vargas for the cone multiplier with some trilinear estimates obtained from the $\ell^2$ decoupling theorem and multilinear restriction theorem.

The purpose of this paper is to improve the known estimates for Mockenhaupt's\nsquare function in $\\mathbb R^3$ and for Sogge's local smoothing in $\\mathbb\nR^{2+1}$ spacetime. For this we use the trilinear approach of S. Lee and A.\nVargas for the cone multiplier with some trilinear estimates obtained from the\n$\\ell^2$ decoupling theorem and multilinear restriction theorem.\n

We prove an endpoint estimate for oscillatory integral operators whose phase function satisfies the cinematic curvature condition. It is a generalization of a result due to Tao (2001).

In Fourier restriction problems, a cone and a paraboloid are model surfaces. The sharp bilinear cone restriction estimate was first shown by Wolff, and later the endpoint was obtained by Tao. For a paraboloid, the sharp $L^2$ bilinear restriction estimate was obtained by Tao, but the endpoint was remained open. In this paper we prove the endpoint $L^2$ bilinear restriction estimate for a paraboloid.

Recently, L. Guth improved the restriction estimate for the surfaces with strictly positive Gaussian curvature in R3. In this paper we generalize his restriction estimate to the surfaces with strictly negative Gaussian curvature.

In this paper we consider the k ‐plane Nikodym maximal estimates in the variable Lebesgue spaces . We first formulate the problem about the boundedness of the k ‐plane Nikodym maximal and show that the maximal estimate in is equivalent to that in for . So, the optimal Nikodym maximal estimate in follows from Cordoba's estimate.

In this paper we show that the local Kato type smoothing estimates are\nessentially equivalent to the global Kato type smoothing estimates for some\nclass of dispersive equations including the Schr\\"odinger equation. From this\nwe immediately have two results as follows. One is that the known local Kato\nsmoothing estimates are sharp. The sharp regularity ranges of the global Kato\nsmoothing estimates are already known, but those of the local Kato smoothing\nestimates are not. Recently, Sun, Tr

We will show that a local space-time estimate implies a global space-time estimate for dispersive operators. In order for this implication we consider a Littlewood-Paley type square function estimate for dispersive operators in a time variable and a generalization of Tao's epsilon removal lemma in mixed norms. By applying this implication to the fractional Schrodinger equation in R^{2+1} we obtain the sharp global space-time estimates with optimal regularity from the previous known local ones.