Chan Woo Yang

Scattering Theory Below Energy for a Class of Hartree Type Equations

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) to the Hartree-type nonlinearity, and also develop the theory of the classical multilinear operator modifying the L p estimate in Coifman and Meyer (1978).

An endpoint estimate for the cone multiplier

In this paper we consider an endpoint estimate for high-dimensional cone multipliers.

Maximal Operators Associated with Vector Polynomials of Lacunary Coefficients

We prove the Lp(Rd) (1 < p ≤ ∞) boundedness of the maximal operators associated with a family of vector polynomials given by the form {(2 k1 p1(t), . . . ,2 kd pd(t)) : t ∈ R}. Furthermore, by using the lifting argument, we extend this result to a general class of vector polynomials whose coefficients are of the form constant times 2 k .

Multiparameter singular integrals and maximal operators along flat surfaces

We study double Hilbert transforms and maximal functions along surfaces of the form (t1, t2, γ1(t1)γ2(t2)). The Lp(R3) boundedness of the maximal operator is obtained if each γi is a convex increasing and γi(0) = 0. The double Hilbert transform is bounded in Lp(R3) if both γi’s above are extended as even functions. If γ1 is odd, then we need an additional comparability condition on γ2. This result is extended to higher dimensions and the general hyper-surfaces of the form (t1, . . . , tn,Γ(t1, . . . , tn)) on Rn+1.

L p -SOBOLEV REGULARITY FOR INTEGRAL OPERATORS OVER CERTAIN HYPERSURFACES

In this paper we establish sharp L p -regularity estimates for averaging operators with convolution kernel associated to hypersurfaces in Rd(d ≥ 2) of the form y 7→(y,(y)) where y ∈ Rd−1 and (y) = P d−1 i=1 ±|yi| mi with 2 ≤ m1 ≤ ··· ≤ m d−1.