Tran Vu Khanh
In this paper, we study supnorm and modified Hölder estimates for the integral solution of the di-bar-equation on a class of convex domains of general type in $\C^2$ that includes many infinite type examples.
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In this paper, we study the $\bar\partial$-equation on some convex domains of infinite type in $\mathbb C^2$. In detail, we prove that supnorm estimates hold for infinite exponential type domains provided the exponent is less than 1.
Let $Ω\subset\mathbb{C}^n$ be a product of one-dimensional open bounded domains with $C^{1,α}$ boundary, where $0<α<1$. Using methods from complex analysis in one variable, we construct an integral operator that solves $\bar\partial$ in $Ω$ with supnorm estimates when the datum is in $C^{n-1,α}(Ω)$.
Suppose that a smooth holomorphic curve $V$ has order of contact $η$ at a point $w_0$ in the boundary of a pseudoconvex domain $Ω$ in $\mathbb{C}^3.$ We show that the maximal gain in Hölder regularity for solutions of the $\bar{\partial}$-equation is at most $\frac{1}η.$
The purpose of this paper is to study Hölder estimates for the $\bar\partial$ problem for $(p,q)$ forms on products of general planar domains. As indicated by an example of Stein and Kerzman, solutions to the $\bar\partial$ problem on product domains in $\mathbb C^n (n\ge 2)$ does not gain regularity in Hölder spaces. Making use of an integral representation of Nijenhuis and Woolf, we show that given a $\bar\partial$-closed $(p,q)$ form with $C^{k,α}$ components, $0\le p\le n, 1\le q\le n$, $k\in \mathbb Z^+\cup \{0\}, 0<α\le 1$, there is a $C^{k, α'}$ solution to the $\bar\partial$ problem on…
The note concerns the $\bar\partial$ problem on product domains in $\mathbb C^2$. We show that there exists a bounded solution operator from $C^{k, α}$ into itself, $k\in \mathbb Z^+\cup \{0\}, 0<α< 1$. The regularity result is optimal in view of an example of Stein-Kerzman.
This note seeks to prove the existence of a canonical solution operator to the $\bar\partial$-equation that preserves Hölder regularity on product domains. It is a well known fact that such solution operators do not in general gain Hölder regularity, and as such, our solution operator is optimal in this regard.
Let $Σ$ be a 2-dimensional subvariety in $C^3$ with an isolated simple (rational double point) singularity at the origin. The main objective of this paper is to solve the $\dbar$-equation on a neighbourhood of the origin in $Σ$, demanding a Holder condition on the solution.
This paper is motivated by a claim in the classical textbook of Muskhelishvili concerning the Cauchy singular integral operator $S$ on Hölder functions with parameters. To the contrary of the claim, a counter example was constructed by Tumanov which shows that $S$ with parameters fails to maintain the same Hölder regularity with respect to the parameters. In view of the example, the behavior of the Cauchy singular integral operator with parameters between a type of Log-Hölder spaces is investigated to obtain the sharp norm estimates. At the end of the paper, we discuss its application to the $…
Hölder regularity of the $\bar\partial-$equation on the polydisc
本文研究了 polydisc 上 $\bar\partial-$ 方程的解算子，证明其保持 Hölder 正则性，并指出该算子在正则性方面是最佳的。
We extend a recent result of Avelin, Hed, and Persson about approximation of functions $u$ that are plurisubharmonic on a domain $Ω$ and continuous on $\barΩ$, with functions that are plurisubharmonic on (shrinking) neighborhoods of $\barΩ$. We show that such approximation is possible if the boundary of $Ω$ is $C^0$ outside a countable exceptional set $E\subset\partial Ω$. In particular, approximation is possible on the Hartogs triangle. For Hölder continuous $u$, approximation is possible under less restrictive conditions on $E$. We next give examples of domains where this kind of approximati…
Let $w_0$ be a bounded, $C^3$, strictly plurisubharmonic function defined on $B_1\subset \mathbb{C}^n$. Then $w_0$ has a neighborhood in $L^{\infty}(B_1)$. Suppose that we have a function $ϕ$ in this neighborhood with $1-ε\le MA(u)\le 1+ε$ and there exists a function $u$ solving the linearized complex Monge-Ampere equation: $det(ϕ_{k\bar{l}})ϕ^{I\bar{j}}u_{I\bar{j}}=0$. Then one has an estimate on $|u|_{C^α(B_{\frac{1}{2}})}$ for some $α>0$ depending on $n$, as long as $ε$ is small depending on $n$. This partially generalizes Caffarelli's estimate for linearized real Monge-Ampere equation to t…
We prove regularity of solutions of the $\bar\partial$-problem in the Hölder-Zygmund spaces of bounded, strongly $\mathbf C$-linearly convex domains of class $C^{1,1}$. The proofs rely on a new, analytic characterization of said domains which is of independent interest, and on techniques that were recently developed by the first-named author to prove estimates for the $\bar\partial$-problem on strongly pseudoconvex domains of class $C^2$.
We construct a family of integral kernels for solving the \bar\partial equation with C^k and Holder estimates in thin tubes around totally real submanifolds in complex Eulidean spaces (theorems 1.1 and 3.1). Combining this with the proof of a theorem of Serre we solve the d-equation with estimates for holomorphic forms in such tubes (theorem 5.1). We apply these techniques and a method of Moser to approximate C^k-diffeomorphisms between totally real submanifolds in C^n in the C^k-topology by biholomorphic mappings in tubes, by unimodular and symplectic biholomorphic mappings, and by holomorphi…
Let $Γ(\cdot,λ)$ be smooth, i.e.\, $\mathcal C^\infty$, embeddings from $\barΩ$ onto $\bar{Ω^λ}$, where $Ω$ and $Ω^λ$ are bounded domains with smooth boundary in the complex plane and $λ$ varies in $I=[0,1]$. Suppose that $Γ$ is smooth on $\barΩ\times I$ and $f$ is a smooth function on $\partialΩ\times I$. Let $u(\cdot,λ)$ be the harmonic functions on $Ω^λ$ with boundary values $f(\cdot,λ)$. We show that $u(Γ(z,λ),λ)$ is smooth on $\barΩ\times I$. Our main result is proved for suitable Hölder spaces for the Dirichlet and Neumann problems with parameter. By observing that the regularity of solu…
Let $Ω$ be a bounded, pseudoconvex domain of $\mathbb C^n$ satisfying the "$f$-Property". The $f$-Property is a consequence of the geometric "type" of the boundary; it holds for all pseudoconvex domains of finite type but may also occur for many relevant classes of domains of infinite type. In this paper, we prove the existence, uniqueness and "weak" Hölder-regularity up to the boundary of the solution to the Dirichlet problem for the complex Monge-Ampère equation $$ \begin{cases} \det\left[\dfrac{\partial^2(u)}{\partial z_i\partial\bar z_j}\right]=h\ge 0 & \text{in}\quadΩ,\\ u=ϕ& \text{on} \q…
The thesis studies Frobenius-type theorems in non-smooth settings. We extend the definition of involutivity to non-Lipschitz subbundles using generalized functions. We prove the real Frobenius Theorem with sharp regularity on log-Lipschitz subbundles. We also develop a singular version of the Frobenius theorem on log-Lipschitz vector fields: if $X_1,\dots,X_m$ are log-Lipschitz vector fields such that $[X_i,X_j]=\sum_{k=1}^mc_{ij}^kX_k$ where $c_{ij}^k$ are the derivatives of log-Lipschitz functions, then for any point $p$ there is a $C^1$-manifold containing $p$ such that $X_1,\dots,X_m$ span…
Gromov hyperbolicity, finite type, and subellipticity on $\mathbb{C}$-convex domains
本文证明有界光滑$\mathbb{C}$-凸域的Kobayashi距离Gromov双曲性、D'Angelo型有限性与$\bar{\partial}$-Neumann问题次椭圆估计三者等价，并给出相关指数与边界等价性结果。
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