An elementary theory of hyperfunctions and microfunctions
An embedding of C in C² with hyperbolic complement.
An embedding relation for bounded mean oscillation on rectangles
In the two-parameter setting, we say a function belongs to the mean little BMO if its mean over any interval and with respect to any of the two variables has uniformly bounded mean oscillation. This space has been recently introduced by S. Pott and the present author in relation to the multiplier algebra of the product BMO of Chang-Fefferman. We prove that the Cotlar-Sadosky space of functions of bounded mean oscillation is a strict subspace of the mean little BMO.
An embedding theorem for real analytic spaces
An energy estimate for the complex Monge-Ampère operator
We prove an energy estimate for the complex Monge-Ampère operator, and a comparison theorem for the corresponding capacity and energy. The results are pluricomplex counterparts to results in classical potential theory.
An Envelope of Holomorphy for Certain Normal Complex Spaces.
An equivariant version of Grauert's Oka principle.
An estimate of the essential norm of a composition operator from to in the unit ball.
An example concerning a question of Zariski
An example for the holomorphic sectional curvature of the Bergman metric
We study the behaviour of the holomorphic sectional curvature (or Gaussian curvature) of the Bergman metric of planar annuli. The results are then utilized to construct a domain for which the curvature is divergent at one of its boundary points and moreover the upper limit of the curvature at that point is maximal possible, equal to 2, whereas the lower limit is -∞.
An example of a Banach space V such that all the degree ≥ 2 hypersurfaces of P(V) are singular.
Here we give an example of a Banach space V such that all the degree ≥ 2 hypersurfaces of P(V) are singular.
An example of a non-projective smooth compactification of C3 with second Betti number equal to one.
An example of a pseudoconvex domain whose holomorphic sectional curvature of the Bergman metric is unbounded
Let a and m be positive integers such that 2a < m. We show that in the domain the holomorphic sectional curvature of the Bergman metric at z in direction X tends to -∞ when z tends to 0 non-tangentially, and the direction X is suitably chosen. It seems that an example with this feature has not been known so far.
An example of the Heisenberg group
An Example on the Heisenberg Group Related to the Levy Operator.
An expansion theorem for nonanalytic functions in several complex variables.
An explicit formula for period determinant
We consider a generic complex polynomial in two variables and a basis in the first homology group of a nonsingular level curve. We take an arbitrary tuple of homogeneous polynomial 1-forms of appropriate degrees so that their integrals over the basic cycles form a square matrix (of multivalued analytic functions of the level value). We give an explicit formula for the determinant of this matrix.
An explicit holomorphic map of bounded domains in Cn with C2-Boundary onto the poydisc.
An exponential transform and regularity of free boundaries in two dimensions
An expression for the superposition of general algebraic functions in terms of hypergeometric series.