Holomorphic Functions and Integral Representations in Several Complex Variables

Static Vacuum Solutions from Convergent Null Data Expansions at Space-Like Infinity

“…so that the f j n converge uniformly to z j on K. Standard results on compactly converging sequences of holomorphic functions ( [22]) then imply that the z j = z j (s,u 0 , v 0 ,ẇ 0 ) are holomorphic function of all four variables on P 1/t (0) × U . For later use we note that (7.7), (7.9) imply as an immediate consequence that…”

Uniformisation of Foliations by Curves

“…The theorem above is a generalization of, and was motivated by, a classical result of Hartogs [Ran,II.5], asserting (in modern language) that a domain U in D n × C of the form (Hartogs tube) U = { (z, w) | |w| < e −f (z) }, where f : D n → [−∞, +∞) is an upper semicontinuous function, is Stein if and only if f is plurisubharmonic. Indeed, in this special case the Poincaré metric is easily computed, and one checks that the plurisubharmonicity of f is equivalent to the plurisubharmonic variation of the fiberwise Poincaré metric.…”

“…By pseudoconvexity, the Levi form of g at (z 0 , w 0 ) is therefore nonpositive on the complex tangent space T C (z 0 ,w 0 ) (∂U 0 ), i.e. on the Kernel of ∂g at (z 0 , w 0 ) [Ran,II.2]. By developing, and using also the fact that g is w-harmonic, we obtain ∂ 2 g ∂z∂z (z 0 , w 0 ) ≤ 2Re .…”

OPTIMAL NON-ISOTROPIC Lp ESTIMATES WITH WEIGHTS FOR ∂ IN STRICTLY PSEUDOCONVEX DOMAINS

“…(for details see [8]). Since the kernel K r (z, ζ ) has singularities at ζ = z, it is enough to prove the estimate…”