On estimates for the $\displaystyle \bar \partial $ equation in Stein manifolds
Eric Amar
备注：
I follow the nice suggestions done by the referee which substantially simplify the proof of theorem 4.1. Also typos are corrected and the presenrtation is slightly modified
We generalize to intersection of strictly $c$ -convex domains in Stein manifold, $ L^{r}-L^{s}$ and Lipschitz estimates for the solutions of the $ \bar \partial $ equation done by Ma and Vassiliadou for domains in $ {\mathbb{C}}^{n}.$ For this we use a Docquier-Grauert holomorphic retraction plus the raising steps method I introduce earlier. This gives results in the case of domains with low regularity, $ {\mathcal{C}}^{3},$ for their boundary.
In order to get estimates on the solutions of the equation $\bar \partial u=ω$ on Stein manifold, we introduce a new method the "raising steps method", to get global results from local ones. In particular it allows us to transfer results form open sets in ${\mathbb{C}}^{n}$ to open sets in a Stein manifold.\ \par Using it we get $\displaystyle L^{r}-L^{s}$ results for solutions of equation $\bar \partial u=ω$ with a gain, $\displaystyle s>r,$ in strictly pseudo convex domains in Stein manifolds.\ \par We also get $\displaystyle L^{r}-L^{s}$ results for domains in ${\mathbb{C}}^{n}$ locally bih…
By a theorem of Andreotti and Grauert if $ω$ is a $(p,q)$ current, $q < n,$ in a Stein manifold $\displaystyle Ω,\ \bar \partial $ closed and with compact support, then there is a solution $u$ to $\bar \partial u=ω$ still with compact support in $\displaystyle Ω.$ The main result of this work is to show that if moreover $\displaystyle ω\in L^{r}(m),$ where $m$ is a suitable Lebesgue measure on the Stein manifold, then we have a solution $u$ with compact support {\sl and} in $L^{s}(m),\ \frac{1}{s}=\frac{1}{r}-\frac{1}{2(n+1)}.$ We prove it by estimates in $L^{r}$ spaces with weights.
Let $X$ be a complete metric space and $\displaystyle Ω$ a domain in $\displaystyle X.$ The Raising Steps Method allows to get from local results on solutions $u$ of a linear equation $\displaystyle Du=ω$ global ones in $\displaystyle Ω.$\ It was introduced in \cite{AmarSt13} to get good estimates on solutions of $\bar \partial $ equation in domains in a Stein manifold. As a simple application we shall get a strong $\displaystyle L^{r}$ Hodge decomposition theorem for $p-$forms in a compact riemannian manifold without boundary, and then we retrieve this known result by an entirely different an…
We use duality in the manner of Serre to generalize a theorem of Hedenmalm on solution of the $\bar \partial $ equation with inverse of the weight in Hörmander $\displaystyle L^{2}$ estimates.\
We study the $\bar \partial $-equation first in Stein manifold then in complete Kähler manifolds. The aim is to get $L^{r}$ and Sobolev estimates on solutions with compact support. In the Stein case we get that for any $(p,q)$-form $ω$ in $L^{r}$ with compact support and $\bar \partial $-closed there is a $(p,q-1)$-form $u$ in $W^{1,r}$ with compact support and such that $\bar \partial u=ω.$ In the case of Kähler manifold, we prove and use estimates on solutions on Poisson equation with compact support and the link with $\bar \partial $ equation is done by a classical theorem stating that the …
We define weak $Z(q)$, a generalization of $Z(q)$ on bounded domains $Ω$ in a Stein manifold $M^n$ that suffices to prove closed range of $\bar\partial$. Under the hypothesis of weak $Z(q)$, we also show (i) that harmonic $(0,q)$-forms are trivial and (ii) if $\partialΩ$ satisfies weak $Z(q)$ and weak $Z(n-1-q)$, then $\dbar_b$ has closed range on $(0,q)$-forms on $\partialΩ$. We provide examples to show that our condition contains examples that are excluded from $(q-1)$-pseudoconvexity and the authors' previous notion of weak $Z(q)$.
One of the most natural and challenging issues in discrete complex analysis is to prove the convergence of discrete holomorphic functions to their continuous counterparts. This article is to solve the open problem in the general setting. To this end we introduce new concepts of discrete surface measure and discrete outer normal vector and establish the discrete Cauchy-Pompeiu integral formula, \begin{eqnarray*} f(ζ)=\displaystyle{\int_{\partial B^h}} \mathcal{K}^h(z,ζ) f(z)dS^h(z)+\displaystyle{\int_{B^h}} E^h(ζ-z) \partial_{\bar z}^h f (z)dV^h(z),\end{eqnarray*} which results in the uniform c…
We prove that the tangential Cauchy-Riemann operator has closed range on Levi-pseudoconvex CR manifolds that are embedded in a q-convex complex manifold $X$. Our result generalizes the known case when $X$ is a Stein manifold.
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.
Combining V. Koziarz's observation about the regularity of some modified section related to the initial extension with J. McNeal--D. Varolin's regularity argument, we generalize two theorems of McNeal--Varolin for the $L^2$ extension of $\bar\partial$-closed high-degree forms on a Stein manifold to the weakly pseudoconvex Kähler case under mixed positivity conditions.
We consider a bounded open Stein subset $Ω$ of a complex Stein manifold $X$ of dimension $n$. We prove that if $f$ is a current on $X$ of bidegree $(p,q+1)$, $\bar\partial$-closed on $Ω$, we can find a current $u$ on $X$ of bidegree $(p,q)$ which is a solution of the equation $\bar\partial u=f$ in $Ω$. In other words, we prove that the Dolbeault complex of temperate currents on $Ω$ (i.e. currents on $Ω$ which extend to currents on $X$) is concentrated in degree $0$. Moreover if $f$ is a current on $X= C^n$ of order $k$, then we can find a solution $u$ which is a current on $C^n$ of order $k+2n…
We show that for any bounded domain $Ω\subset\Cp ^n$ of 1-type $2k $ which is locally convexifiable at $p\in bΩ$, having a Stein neighborhood basis, there is a biholomorphic map $f:\barΩ\rightarrow \Cp ^n $ such that $f(p)$ is a global extreme point of type $2k$ for $f{(\barΩ)}$.
We establish $L^2$ extension theorems for $\bar \partial$-closed $(0,q)$-forms with values in a holomorphic line bundle with smooth Hermitian metric, from a smooth hypersurface on a Stein manifold. Our result extends (and gives a new, perhaps more classical, proof of) a theorem of Berndtsson on compact Kähler manifolds, which itself is a sharpening of the theorem of Koziarz. The proof makes use of the Kohn solution, which is the solution of an (interior) elliptic problem, to handle the well-known regularity issues. As such, our methods require the line bundle to be equipped with a smooth metri…
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…
Let $Ω$ be a complex manifold, and let $X\subset Ω$ be an open submanifold whose closure $\bar X$ is a (not necessarily compact) submanifold with smooth boundary. Let $G$ be a complex Lie group, $Π$ be a differentiable principal $G$-bundle on $Ω$ and $J$ a formally integrable bundle almost complex structure on the restriction $\bar P:= Π|_{\bar X}$. We prove that, if the boundary of $\bar X$ is strictly pseudoconvex, $J$ extends to a holomorphic structure on the restriction of $Π$ to a neighborhood of $\bar X$ in $Ω$. This answers positively and generalizes a problem stated in the article "Bou…
