This is a survey article about $L^2$ estimates for the $\bar \partial$ operator. After a review of the basic approach that has come to be called the "Bochner-Kodaira Technique", the focus is on twisted techniques and their applications to estimates for $\bar \partial$, to $L^2$ extension theorems, and to other problems in complex analysis and geometry, including invariant metric estimates and the $\bar \partial$-Neumann Problem.
We reduce the problem of constructing a linear solution operator to the $\bar{\partial}$-equation on smoothly bounded weakly pseudoconvex domains, $Ω$, in $\mathbb{C}^2$ to the problem of the boundary $\bar{\partial}_b$-equation. We show there is a solution operator to $\bar{\partial}$ which is bounded as a map $W^{s}_{(0,1)}(Ω)\cap{ker} \bar{\partial} \rightarrow W^{s}(Ω)$ for all $s\ge 0$ if there is a corresponding solution operator to the $\bar{\partial}_b$-problem with analogous regularity properties.
We solve the $\partial \bar{\partial}$-problem for the differential forms of class $C^\infty$ with boundary value in currents sense defined on a contractible completely strictly pseudoconvex domain of a complex manifold.
We solve the $\partial \bar{\partial}$-problem for a form with distribution boundary value on a strongly pseudoconvex contractible domain of a complex manifold.
As an application of a new characterization of compactness of the $\bar\partial $-Neumann operator we derive a sufficient condition for compactness of the $\bar\partial $- Neumann operator on $(0,q)$-forms in weighted $L^2$-spaces on $\mathbb{C}^n.$
We show there is a solution operator to $\bar{\partial}$ which is bounded as a map $W^{s}_{(0,1)}(Ω)\cap\mbox{ker }\bar{\partial}\rightarrow W^{s}(Ω)$ for all $s\ge 0$.
Generalized study of the operator $α\partial^k \bar{\partial}^{k} + β\bar{\partial}^k +γ\partial^k + c$ in weighted Hilbert space $L^2(\mathbb{C}, \mathrm{e}^{-|z|^2})$
We prove local hypoellipticity of the complex Laplacian $\Box$ and of the Kohn Laplacian $\Box_b$ in a pseudoconvex boundary when, for a system of cut-off $η$, the gradient $\partial_bη$ and the Levi form $\partial_b\bar\partial_bη^2$ are subelliptic multipliers in the sense of Kohn.
We obtain subelliptic estimates for the $\bar{\partial}$-problem on complex algebraic surfaces embedded in $\mathbb{C}^n$ with isolated singularities. $W^ε$ Sobolev norms of a form, $f$, for $0< ε< 1$ are estimated in terms of weighted $L^2$ norms of $\bar{\partial} f$ and $\bar{\partial}^{\ast}f$, with weights which vanish at the singularities, as well as weighted $L^2$ norms of $f$, with weights which blow up at the singularities.
We solve the $\partial \bar{\partial}$-problem for a form with distribution boundary value on a Levi flat unbounded domain of $\mathbb{C}^n$ with the complementary is also Levi flat and unbounded.
A twisted $\bar{\partial}_f$-Neumann problem associated to a singularity $(\mathscr{O}_n,f)$ is established. By constructing the connection to the Koszul complex for toeplitz $n$-tuples $(f_1,\cdots,f_n)$ on Bergman spaces $B^0(D)$, we can solve this $\bar{\partial}_f$-Neumann problem. Moreover, we can compute the cohomology of the $L^2$ holomorphic Koszul complex $(B^*(D),\partial f\wedge)$ explicitly
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 Dirichlet to Neumann operator of the $\overline{\partial}$-Neumann problem, and the relation between the $\overline{\partial}$-Neumann boundary conditions and the Dirichlet to Neumann operator.
Solution operators for the equation $\bar \partial u=f$ are constructed on general product domains in $\mathbb{C}^n$. When the factors are one-dimensional, the operator is a simple integral operator: it involves specific derivatives of $f$ integrated against iterated Cauchy kernels. For higher dimensional factors, the solution is constructed by solving sub-$\bar \partial$ equations with modified data on the factors. Estimates of the operators in several norms are proved.
The $\bar{\partial}$-Neumann operator (the inverse of the complex Laplacian) is shown to be noncompact on certain domains in complex Euclidean space. These domains are either higher-dimensional analogs of the Hartogs triangle, or have such a generalized Hartogs triangle imbedded appropriately in them.
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.
In this article, we use a Berndtsson-Andersson operator and the Bergman metric in order to solve the $\bar\partial$ equation on convex domains of finite type for forms satisifying a Carleson condition and get norm estimates of the solution in term of the Carleson norm.
This work is a complement of the study on Hörmander's solution of the $\bar\partial$ equation initialised by H. Hedenmalm. Let $φ$ be a strictly plurisubharmonic function of class C 2 in C n, let $c_φ(z)$ be the smallest eigenvalue of $i\partial\bar\partialφ$ then $\forall z\in\mathbb{C}^n$, $c_φ(z)>0$. We denote by $L^2_{p,q}(\mathbb{C}^n, e^φ)$ the $(p, q)$ currents with coefficients in $L^2_{p,q}(\mathbb{C}^n, e^φ)$. We prove that if $ω\in L^2_{p,q}(\mathbb{C}^n,e^φ)$, $\bar\partial$$ω$ = 0 for q <n then there is a solution u $\in L ^2_{p,q-1}(\mathbb{C}^n,c_φe^φ)$ of $\bar\partial$u = $ω$.…
We present a refined, improved $L^2$-theory for the $\bar{\partial}$-operator for $(0,q)$ and $(n,q)$-forms on Hermitian complex spaces of pure dimension $n$ with isolated singularities. The general philosophy is to use a resolution of singularities to obtain a regular model of the $L^2$-cohomology.
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.
In this paper we consider the $\bar\partial$-problem in fiber bundles (fibers biholomorphic to $\mathbb C^k$, $k\geq 1$), namely the equation $\bar\partialσ=ω$ for $(0,1)$-forms $ω$ which decrease along the fibers. The order of decrease is slightly more than one. The important fact is that we do not assume that $ω$ has compact support. The main theorem says that the equation has a solution which also decreases along fibers, however, not necessarily with the order as the original form. Existence of solution of the above mentioned $\bar\partial$-problem can be applied in various situations in Co…
We obtain weighted estimates for the $\bar{\partial}$-Neumann operator on intersections of two smooth strictly pseudoconvex domains in $\mathbb{C}^2$. The regularity estimates are described with the use of Sobolev norms with weights which are powers of the defining functions of the two domains.
For a domain $D$ of $\mathbb{C}^n$ which is weakly $q$-pseudoconvex or $q$-pseudoconcave we give a sufficient condition for subelliptic estimates for the $\bar{\partial}$-Neumann problem. The paper extends to domains which are not necessarily pseudoconvex, the results and the techniques of Catlin. MSC: 32D10, 32U05, 32V25
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 show that Property $(P)$ of $\partialΩ$, compactness of the $\bar{\partial}$-Neumann operators $N_1$, and compactness of Hankel operator on a smooth bounded pseudoconvex Hartogs domain $Ω={\{(z, w_1, w_2,\dots, w_n) \in \mathbb{C}^{n+1} \mid\sum_{k=1}^{n} |w_k|^2 < e^{-2φ(z)}, z\in\mathit{D}\}}$ are equivalent, where $D$ is a smooth bounded connected open set in $\mathbb{C}$.
We give several versions of Siu's $\partial\bar{\partial}$-formula for maps from a strictly pseudoconvex pseudo-Hermitian manifold $(M^{2m+1}, θ)$ into a Kähler manifold $(N^n, g)$. We also define and study the notion of pseudo-Hermitian harmonicity for maps from $M$ into $N$. In particular, we prove a CR version of Siu Rigidity Theorem for pseudo-Hermitian harmonic maps from a pseudo-Hermitian manifold with vanishing Webster torsion into a Kähler manifold having strongly negative curvature.
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…
Dans ce papier, on résout d'abord le $\partial\bar\partial$ pour les courants prolongeables définis dans $\mathbb{C}^n$ privé d'une boule $B$ de $\mathbb{C}^n$, ensuite dans une variété analytique complexe contractile $X$, enfin on le résout pour un domaine $D=X\setminusΩ$, où $Ω$ est un domaine borné de $X$ défini par $\{z\in X\,\ /\,\ φ(z)<0\}$, (avec $φ$ une fonction d'exhaustion strictement plurisouharmonique). In this present paper, we first solve the $\partial\bar\partial$ for extendable currents definite in $\mathbb{C}^n\setminus B$, where $B$ is a ball of $\mathbb{C}^n$, then in a cont…
Let $Ω$ be a product domain in $\mathbb C^n, n\ge 2$, where each slice has smooth boundary. We observe that the canonical solution operator for the $\bar\partial$ equation on $Ω$ is bounded in $W^{k,p}(Ω)$, $k\in \mathbb Z^+, 1<p<\infty$. This Sobolev regularity is sharp in view of Kerzman-type examples.
We prove that the $\bar\partial$-Neumann solution operator is locally regular in a domain which has compactness estimates, is of finite type outside a curve transversal to the CR directions and for which the holomorphic tangential derivatives of a defining function are subelliptic multipliers in the sense of Kohn.
An integral solution operator for $\bar\partial$ is constructed on product domains that include the punctured bidisc. This operator is shown to satisfy $L^p$ estimates for all $1\leq p <\infty$, though with non-standard -- relative to strongly pseudoconvex domains -- bounding term. These estimates imply $L^p$ estimates for $\bar\partial$ on the Hartogs triangle, with greater range of $p$ than the canonical solution satisfies.
We prove that on smooth bounded pseudoconvex Hartogs domains in $\mathbb{C}^2$ compactness of the $\bar{\partial}$-Neumann operator is equivalent to compactness of all Hankel operators with symbols smooth on the closure of the domain.
Using Hörmander $L^2$ method for Cauchy-Riemann equations from complex analysis, we study a simple differential operator $\bar{\partial}^k+a$ of any order (densely defined and closed) in weighted Hilbert space $L^2(\mathbb{C},e^{-|z|^2})$ and prove the existence of a right inverse that is bounded.
In this paper we consider the classical $\bar{\partial}$-problem in the case of one complex variable both for analytic and polyanalytic data. We apply the decomposition property of polyanalytic functions in order to construct particular solutions of this problem and obtain new Hörmander type estimates using suitable powers of the Cauchy-Riemann operator. We also compute particular solutions of the $\bar{\partial}$-problem for specific polyanalytic data such as the Itô complex Hermite polynomials and polyanalytic Fock kernels.
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.
This paper treats subelliptic estimates for the $\bar{\partial}$-Neumann problem on a class of domains known as regular coordinate domains. Our main result is that the largest subelliptic gain for a regular coordinate domain is bounded below by a purely algebraic number, the inverse of twice the multiplicity of the ideal associated to a given boundary point.
We relate the existence and regularity of a solution operator to on smoothly bounded pseudoconvex domains to the existence and regularity of a projection operator onto the kernel of dbar.
The author showed that a sequence in the unit disk is a zero sequence for the Bergman space $A^p$ if and only if a certain weighted space $L^p(W}$ contains a nontrivial analytic function. In this paper it is shown that the sequence is an interpolating sequence for $A^p$ if and only if it is separated in the hyperbolic metric and the $\bar\partial$-equation $(1 - |z|^2)\bar\partial u = f$ has a solution $u$ belonging to $L^p(W)$ for every $f$ in $L^p(W)$.
In this paper, we consider the Cauchy-Riemann equation $\bar\partial u= f$ in a new class of convex domains in $\C^n.$ We prove that under $L^p$ data, we can choose a solution in the Lipschitz space $Λ_α,$ where $α$ is an optimal positive number given explicitly in terms of $p.$
We prove Schwarz-Pick type estimates and coefficient estimates for a class of elliptic partial differential operators introduced by Olofsson. Then we apply these results to obtain a Landau type theorem.
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 $C^2$.
Let $X$ denote a compact set which is laminated by Riemann surfaces. We assume that $X$ carries a positive CR line bundle $ L\rightarrow X$. The main result of the paper is that there exists a positive integer $s$ so that if $v$ is any continuous $(0,1)$ form with coefficients in $L^{\otimes s}$ there exists a continuous section $u$ of $L^{\otimes s}$ solving the equation $\bar\partial_b u=v$.
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.
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 $C^{n-1,α}(Ω)$.
Assume that $Ω_{1}$ and $Ω_{2}$ are two smooth bounded pseudoconvex domains in $\mathbb{C}^{2}$ that intersect (real) transversely, and that $Ω_{1} \cap Ω_{2}$ is a domain (i.e. is connected). If the $\overline{\partial}$-Neumann operators on $Ω_{1}$ and on $Ω_{2}$ are compact, then so is the $\overline{\partial}$-Neumann operator on $Ω_{1} \cap Ω_{2}$. The corresponding result holds for the $\overline{\partial}$-Neumann operators on $(0,n-1)$-forms on domains in $\mathbb{C}^{n}$.
In this paper, we prove weighted $L^p$ estimates for the canonical solutions on product domains. As an application, we show that if $p\in [4, \infty)$, the $\bar\partial$ equation on the Hartogs triangle with $L^p$ data admits $L^p$ solutions with the desired estimates. For any $ε>0$, by constructing an example with $L^p$ data but having no $L^{p+ε}$ solutions, we verify the sharpness of the $L^p$ regularity on the Hartogs triangle.
We construct a metric on a vector bundle $E \rightarrow Z$ restricted to a conic neighbourhood of a relatively compact $1$-convex section of a submersion $Z \rightarrow X$ with at most polynomial poles at the boundary and positive Nakano curvature tensor in bidegree $(p,q).$
