Mathematics

We investigate the quaternionic extension of the fractional Fourier transform on the real half-line leading to fractional Hankel transform. This will be handled à la Bargmann by means of hyperholomorphic second Bargmann transform for the slice Bergman space of second kind. Basic properties are derived including inversion formula and Plancherel identity.

We introduce the concept of Bergman bundle attached to a hermitian manifold X, assuming the manifold X to be compact - although the results are local for a large part. The Bergman bundle is some sort of infinite dimensional very ample Hilbert bundle whose fibers are isomorphic to the standard L 2 Hardy space on the complex unit ball; however the bundle is locally trivial only in the real analytic category, and its complex structure is strongly twisted. We compute the Chern curvature of the Bergman bundle, and show that it is strictly positive. As a potential application, we investigate a long standing and still unsolved conjecture of Siu on the invariance of plurigenera in the general situation of polarized families of compact K{ä}hler manifolds.

We give a general inequality for Bergman kernels of Bergman spaces defined by convex weights in $\C^n$. We also discuss how this can be used in Nazarov's proof of the Bourgain-Milman theorem, as a substitute for Hörmander's estimates for the $\dbar$-equation.

We give quantitative estimates of the Bergman distance through positivity of capacity densities.

We apply the Bekollé-Bonami estimate for the (positive) Bergman projection on the weighted L p spaces on the unit disk. We then obtain the boundedness of the Bergman projection on the weighted Sobolev space on the symmetrized bidisk, by the reduction to the (positive) Bergman projection on the weighted L p space on the unit disk,. We also improve the boundedness result of the Bergman projection on the unweighted L p space on the symmetrized bidisk in \cite{CKY}.

For appropriate domains Ω 1 , Ω 2 we consider mappings Φ A : Ω 1 → Ω 2 of monomial type. We obtain an orthogonal decomposition of the Bergman space A 2 ( Ω 1 ) into finitely many closed subspaces indexed by characters of a finite Abelian group associated to the mapping Φ A . We then show that each subspace is isomorphic to a weighted Bergman space on Ω 2 . This leads to a formula for the Bergman kernel on Ω 1 as a sum of weighted Bergman kernels on Ω 2

Let Ω be a Stein space with a compact smooth strongly pseudoconvex boundary. We prove that the boundary is spherical if its Bergman metric over Reg(Ω) is Kähler-Einstein.

We construct a pointwise Boutet de Monvel-Sjöstrand parametrix for the Szegő kernel of a weakly pseudoconvex three dimensional CR manifold of finite type assuming the range of its tangential CR operator to be closed; thereby extending the earlier analysis of Christ. This particularly extends Fefferman's boundary asymptotics of the Bergman kernel to weakly pseudoconvex domains in C 2 , in agreement with D'Angelo's example. Finally our results generalize a three dimensional CR embedding theorem of Lempert.

We construct embeddings of Bers slices of ideal polygon reflection groups into the classical family of univalent functions Σ . This embedding is such that the conformal mating of the reflection group with the anti-holomorphic polynomial z↦ z ¯ ¯ ¯ d is the Schwarz reflection map arising from the corresponding map in Σ . We characterize the image of this embedding in Σ as a family of univalent rational maps. Moreover, we show that the limit set of every Kleinian reflection group in the closure of the Bers slice is naturally homeomorphic to the Julia set of an anti-holomorphic polynomial.

We investigate a variant of the Beurling-Ahlfors extension of quasisymmetric homeomorphisms of the real line that is given by the convolution of the heat kernel, and prove that the complex dilatation of such a quasiconformal extension of a strongly symmetric homeomorphism (i.e. its derivative is an A ∞ -weight whose logarithm is in VMO) induces a vanishing Carleson measure on the upper half-plane.