... and Schrödinger operators.
Integral Formulas for the ... ....-Equation and Zeros of Bounded Holomorphic Functions in the Unit Ball.
A Formula for Interpolation and Division in Cn.
Integral Kernels and Approximation on Totally Real Submanifolds of Cn.
The extension theorem of Ohsawa-Takegoshi and the theorem of Donnelly-Fefferman
We give a short proof of the extension theorem of Ohsawa-Takegoshi. The same method also gives a generalization of the -theorem of Donnelly and Fefferman for the case of -forms.
Cauchy-Leray forms and vector bundles
Integral formulas on projective space and the radon transform of Gindikin-Henkin-Polyakov.
We construct a variant of Koppelman's formula for (0,q)-forms with values in a line bundle, O(l), on projective space. The formula is then applied to a study of a Radon transform for (0,q)-forms, introduced by Gindikin-Henkin-Polyakov. Our presentation follows along the basic lines of Henkin-Polyakov [3], with some simplifications.
-estimates for the -equation and Witten’s proof of the Morse inequalities
This is an introduction to Witten’s analytic proof of the Morse inequalities. The text is directed primarily to readers whose main interest is in complex analysis, and the similarities to Hörmander’s -estimates for the -equation is used as motivation. We also use the method to prove -estimates for the -equation with a weight where is a nondegenerate Morse function.
Convexity on the space of Kähler metrics
These are the lecture notes of a minicourse given at a winter school in Marseille 2011. The aim of the course was to give an introduction to recent work on the geometry of the space of Kähler metrics associated to an ample line bundle. The emphasis of the course was the role of convexity, both as a motivating example and as a tool.
Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains
Let be a pseudoconvex domain in and let be a plurisubharmonic function in . For each we consider the -dimensional slice of , , let be the restriction of to and denote by the Bergman kernel of with the weight function . Generalizing a recent result of Maitani and Yamaguchi (corresponding to and ) we prove that is a plurisubharmonic function in . We also generalize an earlier results of Yamaguchi concerning the Robin function and discuss similar results in the setting...
Non-holomorphic functional calculus for commuting operators with real spectrum
We consider -tuples of commuting operators on a Banach space with real spectra. The holomorphic functional calculus for is extended to algebras of ultra-differentiable functions on , depending on the growth of , , when . In the non-quasi-analytic case we use the usual Fourier transform, whereas for the quasi-analytic case we introduce a variant of the FBI transform, adapted to ultradifferentiable classes.
On interpolation and sampling in Hilbert spaces of analytic functions.
Quasi-isometric vector bundles and bounded factorization of holomorphic matrices
We give a sufficient condition for a hermitian holomorphic vector bundle over the disk to be quasi-isometric to the trivial bundle. One consequence is a version of Cartan's lemma on the factorization of matrices with uniform bounds.
Real Monge-Ampère equations and Kähler-Ricci solitons on toric log Fano varieties
We show, using a direct variational approach, that the second boundary value problem for the Monge-Ampère equation in with exponential non-linearity and target a convex body is solvable iff is the barycenter of Combined with some toric geometry this confirms, in particular, the (generalized) Yau-Tian-Donaldson conjecture for toric log Fano varieties saying that admits a (singular) Kähler-Einstein metric iff it is K-stable in the algebro-geometric sense. We thus obtain a new proof and...
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