The $$\bar \partial $$ -problem with support conditions on some weakly pseudoconvex domainswith support conditions on some weakly pseudoconvex domains

We consider a domain Ω with Lipschitz boundary, which is relatively compact in ann-dimensional Kähler manifold and satisfies some “logδ-pseudoconvexity” condition. We show that the\(\bar \partial \)-equation with exact support in ω admits a solution in bidegrees (p, q), 1≤q≤n?1. Moreover, the range of\(\bar \partial \) acting on smooth (p, n?1)-forms with support in\(\bar \Omega \) is closed. Applications are given to the solvability of the tangential Cauchy-Riemann equations for smooth forms and currents for all intermediate bidegrees on boundaries of weakly pseudoconvex domains in Stein manifolds and to the solvability of the tangential Cauchy-Riemann equations for currents on Levi flatCR manifolds of arbitrary codimension.     

Semiclassical $$\bar \partial $$ -Method: Generating Equations for Dispersionless Integrable Hierarchies

We use the semiclassical -dressing method to derive compact generating equations for dispersionless hierarchies. The considered illustrative examples are the dispersionless Kadomtsev–Petviashvili and two-dimensional Toda lattice hierarchies.     

Sharp lipschitz estimates for operator $$\bar \partial _M $$ on a q-pseudoconcave CR manifoldon a q-pseudoconcave CR manifold

We construct integral operators R_r and H_r on a regular q-pseudoconcave CR manifoldM such that

On the matrix equation $$ X\bar X = \bar XX $$

It is shown that every solution to the equation $ X\bar X = \bar XX $ X\bar X = \bar XX can be reduced by a real orthogonal similarity transformation to a block triangular form with diagonal blocks of orders one and two. If the solution X is a normal matrix, then its block triangular form is actually a block diagonal. In this case, the form of the diagonal blocks is found, yielding new proof of the recent results of Goodson and Horn.     

Horizontal $$\bar \partial $$ -Laplacian on complex Finsler manifolds-Laplacian on complex Finsler manifolds

A horizontal\(\bar \partial \)-Laplacian is defined on strongly pseudoconvex complex Finsler manifolds, first for functions and then for horizontal differential forms of type (p, q). The principal part of the\(\bar \partial \)-Laplacian is computed in local coordinates. As an application, the\(\bar \partial \)-Laplacian on strongly Kähler Finsler manifold is obtained explicitly in terms of the horizontal covariant derivatives of the Chern-Finsler conncetion.

     

The Quasi-Canonical Solution Operator to $${\bar \partial }$$ Restricted to the Fock-Space

We consider the solution operator S: ℱ_μ,(p,q) → L ²(μ)_{(p, q)} to the -operator restricted to forms with coefficients in ℱ_μ = {f: f is entire and ∫_ℂn |f(z)|² dμ(z) < ∞}. Here ℱ_μ,(p,q) denotes (p,q)-forms with coefficients in ℱ_μ, L ²(μ) is the corresponding L ²-space and μ is a suitable rotation-invariant absolutely continuous finite measure. We will develop a general solution formula S to . This solution operator will have the property Sv ⊥ ℱ_(p,q) ∀v ∈ ℱ_(p,q+1). As an application of the solution formula we will be able to characterize compactness of the solution operator in terms of compactness of commutators of Toeplitz-operators : ℱ_μ → L ²(μ).     

Non-isotropic weighted Lp estimates for the $$ \bar \partial $$-equation on piecewise smooth strictly pseudoconvex domains

In this paper we obtain non-isotropic weighted L ^p estimates with the boundary distance weight function for the -equation on piecewise smooth strictly pseudoconvex domains under a hypothesis of complex transversality in ℂⁿ using the explicit formula of solutions by Berndtsson-Andersson. This work was supported by the Korea Research Foundation Grant funded by Korea Government (MOEHRD, Basic Research Promotion Fund) (Grant No. KRF-2005-070-C00007)     

Homotopy formulas and\bar \partial -equation on localq-convex domains in Stein manifolds-equation on localq-convex domains in Stein manifolds

The homotopy formulas of (r, s) differential forms and the solution of $\bar \partial $ -equation of type (r, s) on localq-convex domains in Stein manifolds are obtained. The homotopy formulas on localq-convex domains have important applications in uniform estimates of $\bar \partial $ -equation and holomorphic extension of CR-manifolds.     

$$\bar\partial$$ -dressing method for a few $$(2+1)$$ -dimensional integrable coupling systems

Theoretical and Mathematical Physics - Several $$(2+1)$$ -dimensional integrable coupling systems are derived from two sets of auxiliary linear problems, including the integrable coupling system of...     

Holomorphic deformations of the Ricci-flat $$\partial \overline \partial $$-manifolds

We present a construction of globally convergent power series of integrable Beltrami differentials on the Ricci-flat \(\partial \overline \partial \)-manifolds and also a construction of global canonical family of holomorphic (n, 0)-forms on the deformation spaces of the Ricci-flat \(\partial \overline \partial \)-manifolds.     

Uniform estimates with weights for the\bar \partial - equation

This paper concernsL ^∞-variants of Hörmanders weightedL ²-estimates for the $\bar \partial - equation$ . In particular, we discuss a conjecture concerning suchL ^∞-estimates which is related to the corona problem in the ball, and show a weaker version of this conjecture. The proof uses a refinedL ²-estimate for the canonical solution to the $\bar \partial - equation$ . An alternative approach based on von Neumann’s Minimax theorem is also given.