The Bergman–Milton bounds provide limits on the effective permittivity of a composite material comprising two isotropic dielectric materials. These provide tight bounds for composites arising from many conventional materials. We reconsider the Bergman–Milton bounds in light of the recent emergence of metamaterials, in which unconventional parameter ranges for relative permittivities are encount...

In the paper, we discuss on the egg domains: Ω a = ξ = (z, w) ∈ C n+m : z ∈ C n , w ∈ C m , |z| 2 + |w| 2/a < 1 , 0 < a ≤ 2. We show that Gleason's problem can be solved in the weight Bergman space on the egg domains. The proof will need the help of the recent work of the second named author on the weighted Bergman projections on this kind of domain. As an application, we obtain a multiplier th...

We relate the classical approximations SN (f)(x) of O.Szasz to the Bergman kernel of the Bargmann-Fock space H(C, e |z| 2 dm(z)). This relation is the analogue for compact toric varieties of the relation between Bernstein polynomials and Bergman kernels on compact toric Kähler varieties of S. Zelditch. The relation is then used to generalize the Szasz analytic functions to any infinite volume t...

This paper gives embedding theorems for a very general class of weighted Bergman spaces: the results include a number of classical Carleson embedding theorems as special cases. The little Hankel operators on these Bergman spaces are also considered. Next, a study is made of Carleson embeddings in the right half-plane induced by taking the Laplace transform of functions defined on the positive h...

2 Tau-function on spaces of Abelian differentials over Riemann surfaces 7 2.1 Spaces of holomorphic differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Variational formulas on Hg(k1, . . . , kM ) . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Basic Beltrami differentials for Hg(k1, . . . , kM ) . . . . . . . . . . . . . . . . . . . . . . 11 2.4 Definition of...

We characterize, using the Bergman kernel, Carleson measures of Bergman spaces in strongly pseudoconvex bounded domains in C, generalizing to this setting theorems proved by Duren and Weir for the unit ball. We also show that uniformly discrete (with respect to the Kobayashi distance) sequences give examples of Carleson measures, and we compute the speed of escape to the boundary of uniformly d...

Let φ and ψ be holomorphic maps on such that φ( ) ⊂ . Let Cφ,Mψ and D be the composition, multiplication and differentiation operators, respectively. In this paper, we consider linear operators induced by products of these operators from Bergman-Nevanlinna spaces AβN to Bloch-type spaces. In fact, we prove that these operators map AβN compactly into Bloch-type spaces if and only if they map A β...

Let D be the open unit disk in the complex plane. For ε > 0 we consider the sector Σε = {z ∈ C : | arg z| < ε}. We prove that for every α ≥ 0 and for each ε > 0 there is a constant K > 0 depending only on α and ε such that for any function f in the weighted Bergman space Aα univalent on D, and f(0) = 0, then ∫ f−1(Σε) |f(z)|dAα(z) > K‖f‖1,α. This result extends a theorem of Marshall and Smith i...

For a smoothly bounded strictly pseudoconvex domain, we describe the boundary singularity of weighted Bergman kernels with respect to weights behaving like a power (possibly fractional) of a defining function, and, more generally, of the reproducing kernels of Sobolev spaces of holomorphic functions of any real order. This generalizes the classical result of Fefferman for the unweighted Bergman...

Noncommutative algebras, defined by the generators and relations, are considered. The definition and main results connected with the Gröbner basis, Hilbert series and Anick’s resolution are formulated. Most attention is paid to universal enveloping algebras. Four main examples illustrate the main concepts and ideas. Algorithmic problems arising in the calculation of the Hilbert series are inves...