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Register a new userExact sequences and estimates for the $overline{partial}$-problem
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We study Sobolev estimates for solutions of the inhomogenous Cauchy-Riemann equations on annuli in $cx^n$, by constructing exact sequences relating the Dolbeault cohomology of the annulus with respect to Sobolev spaces of forms with those of the envelope and the hole. We also obtain solutions with prescibed support and estimates in Sobolev spaces using our method.
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We develop a method for proving sup-norm and Holder estimates for $overline{partial}$ on wide class of finite type pseudoconvex domains in $mathbb{C}^n$. A fundamental obstruction to proving sup-norm estimates is the possibility of singular complex curves with exceptionally high order of contact with the boundary. Our method handles this problem, and in $mathbb{C}^3$, we prove sup-norm and Holder estimates for all bounded, pseudoconvex domains with real-analytic boundary.
We investigate regularity properties of the $overline{partial}$-equation on domains in a complex euclidean space that depend on a parameter. Both the interior regularity and the regularity in the parameter are obtained for a continuous family of pseudoconvex domains. The boundary regularity and the regularity in the parameter are also obtained for smoothly bounded strongly pseudoconvex domains.
In the present investigation, we introduce a new class k-US_{s}^{{eta}}({lambda},{mu},{gamma},t) of analytic functions in the open unit disc U with negative coefficients. The object of the present paper is to determine coefficient estimates, neighborhoods and partial sums for functions f(z) belonging to this class.
In this article, high frequency stability estimates for the determination of the potential in the Schrodinger equation are studied when the boundary measurements are made on slightly more than half the boundary. The estimates reflect the increasing stability property with growing frequency.
In the late ten years, the resolution of the equation $barpartial u=f$ 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 the method used to obtain global estimates for a support function cannot be carried out in this case, we use a kernel that does not gives directly a solution of the $barpartial$-equation but only a representation formula which allows us to end the resolution of the equation using Kohns $L^2$ theory. As an application we give the characterization of the zero sets of the functions of the Nevanlinna class for lineally convex domains of finite type.
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