Related papers: A berndtsson-Andersson operator solving \bar\parti…

In this paper we obtain sharp weighted estimates for solutions of the $\partial$-equation in a lineally convex domains of finite type. Precisely we obtain estimates in spaces of the form L p ({\Omega},$\delta$ $\gamma$), $\delta$ being the…

In the late ten years, the resolution of the equation $\bar\partial u=f$ with sharp estimates has been intensively studied for convex domains of finite type by many authors. In this paper, we consider the case of lineally convex domains. As…

We reduce the problem of constructing a linear solution operator to the $\bar{\partial}$-equation on smoothly bounded weakly pseudoconvex domains, $\Omega$, in $\mathbb{C}^2$ to the problem of the boundary $\bar{\partial}_b$-equation. We…

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…

In this paper first we define generalized Carleson mea- sure. Then we consider a special case of it, named conditional Carleson measure on the Bergman spaces. After that we give a characterization of conditional Carleson measures on Bergman…

We show there is a solution operator to $\bar{\partial}$ which is bounded as a map $W^{s}_{(0,1)}(\Omega)\cap\mbox{ker }\bar{\partial}\rightarrow W^{s}(\Omega)$ for all $s\ge 0$.

We prove regularity of solutions of the $\bar\partial$-problem in the H\"older-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…

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…

We prove estimates for solutions of the $\bar \partial u=\omega $ equation in a strictly pseudo convex domain $ \Omega $ in ${\mathbb{C}}^{n}.$ For instance if the $ (p,q)$ current $\omega $ has its coefficients in $L^{r}(\Omega )$ with…

We use weighted $L^2$-methods to obtain sharp pointwise estimates for the canonical solution to the equation $\bar\partial u=f$ on smoothly bounded strictly convex domains and the Cartan classical domain domains when $f$ is bounded in the…

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 $\Lambda_{\alpha},$ where $\alpha$ is an…

We study an elliptic operator $L:=\mathrm{div}(A\nabla \cdot)$ on the upper half plane $\mathbb{R}^2_+$. There are several conditions on the behavior of the matrix $A$ in the transversal $t$-direction that yield $\omega\in…

Let $\Omega\subset\mathbb{C}^m$ be a bounded pseudoconvex domain with smooth boundary. For each $k\in\mathbb{N}$, we give a sufficient condition to estimate the $\bar\partial$-Neumann operator in the Sobolev space $W^k(\Omega)$. The key…

In this paper we derive Carleman estimates for the fractional relativistic operator. We consider changing-sign solutions to the heat equation for such operators. We prove monotonicity inequalities and convexity of certain energy functionals…

We obtain sharp weighted estimates for solutions of the equation $\partial$ u = f in a lineally convex domain of finite type. Precisely we obtain estimates in the spaces L p ($\Omega$,$\delta$ $\gamma$), $\delta$ being the distance to the…

We solve the $\bar{\partial}$-problem for differential forms in the sens of Whitney.

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 present in concise form recent results, with illustrative proofs, on solvability of the $L^p$ Dirichlet, Regularity and Neumann problems for scalar elliptic equations on Lipschitz domains with coefficients satisfying a…

In the paper we give a lower estimate for the essential norm of the $\bar{\partial}$-Neumann operator on convex domains and worm domains of Diederich and Fornaess.

Let $\Omega$ be an unbounded, pseudoconvex domain in $\Bbb C^n$ and let $\varphi$ be a $\mathcal C^2$-weight function plurisubharmonic on $\Omega$. We show both necessary and sufficient conditions for existence and compactness of a weighted…