Tran Vu Khanh

Lp estimates for the

We prove Lp estimates, 1 ≤ p ≤ ∞, for solutions to the Cauchy–Riemann equations

\bar\partial

\bar\partial u = \phi

-equation on a class of infinite type domains

on a class of infinite type domains in ℂ2. The domains under consideration are a class of convex ellipsoids, and we show that if ϕ is a

Loss of derivatives for systems of complex vector fields and sums of squares

\bar\partial

A decomposition approach via Fourier sine transform for valuing American knock-out options with rebates

-closed (0, 1)-form with coefficients in Lp and u is the Henkin kernel solution to

Compactness estimate for the -Neumann problem on a Q-pseudoconvex domain

\bar\partial u = \phi

Bergman-Toeplitz operators on weakly pseudoconvex domains

, then ‖u‖p ≤ C‖ϕ‖p where the constant C is independent of ϕ. In particular, we prove L1 estimates and obtain Lp estimates by interpolation.

A general method of weights in the d-bar-Neumann problem

We discuss, both for systems of complex vector fields and for sums of squares, the phenomenon discovered by Kohn of hypoellipticity with loss of derivatives.

Regularity of the ¯ ∂-Neumann problem at point of infinite type

We present an innovative decomposition approach for computing the price and the hedging parameters of American knock-out options with a time-dependent rebate. Our approach is built upon: (i) the Fourier sine transform applied to the partial differential equation with a finite time-dependent spatial domain that governs the option price, and (ii) the decomposition technique that partitions the price of the option into that of the European counterpart and an early exercise premium. Our analytic representations can generalize a number of existing decomposition formulas for some European-style and American-style options. A complexity analysis of the method, together with numerical results, show that the proposed approach is significantly more efficient than the state-of-the-art adaptive finite difference methods, especially in dealing with spot prices near the barrier. Numerical results are also examined in order to provide new insight in the significant effects of the rebate on the option price, the hedging parameters, and the optimal exercise boundary.

Necessary geometric and analytic conditions for general estimates in the \bar{\partial}-Neumann problem

The purpose of this article is to discuss compactness estimate for the -Neumann problem at a boundary with mixed Levi signature. We consider a domain D ⊂⊂ ℂ n which is q-pseudoconvex and introduce the ‘(q − P) property’ which is the natural variant of the classical ‘P property’ by Catlin adapted to the new class of domains. In Section 1, we prove that (q − P) property implies compactness estimate. Next, in Section 2, we introduce the notion of ‘weak q regularity’ of ∂D, the natural variant of the classical ‘weak regularity’ by Catlin and prove that it implies (q − P) property. In Section 3, we recall how compactness yields Sobolev estimates. In Section 4, we give a criterion for weak q regularity of a real-analytic boundary and finally, in Section 5, we exhibit a class of weakly q regular domains.