Alexander Brudnyi

Methods of geometric analysis in extension and trace problems

Part 3. Lipschitz Extensions from Subsets of Metric Spaces.- Chapter 6. Extensions of Lipschitz Maps.- Chapter 7. Simultaneous Lipschitz Extensions.- Chapter 8. Linearity and Nonlinearity.- Part 4. Smooth Extension and Trace Problems for Functions on Subsets of Rn.- Chapter 9. Traces to Closed Subsets: Criteria, Applications.- Chapter 10. Whitney Problems.- Bibliography.- Index.

Metric Spaces with Linear Extensions Preserving Lipschitz Condition

We study a new bi-Lipschitz invariant λ(M) of a metric space M; its finiteness means that Lipschitz functions on an arbitrary subset of M can be linearly extended to functions on M whose Lipschitz constants are expanded by a factor controlled by λ(M). We prove that λ(M) is finite for several important classes of metric spaces. These include metric trees of arbitrary cardinality, groups of polynomial growth, Gromov-hyperbolic groups, certain classes of Riemannian manifolds of bounded geometry and the finite direct sums of arbitrary combinations of these objects. On the other hand we construct an example of a two-dimensional Riemannian manifold M of bounded geometry for which λ(M) = ∞.

Local inequalities for plurisubharmonic functions

The main objective of this paper is to prove a new inequality for plurisubharmonic functions estimating their supremum over a ball by their supremum over a measurable subset of the ball. We apply this result to study local properties of polynomial, algebraic and analytic functions. The paper has much in common with an earlier paper [Br] of the author.

On local behavior of holomorphic functions along complex submanifolds of ℂ N

In this paper we establish some general results on local behavior of holomorphic functions along complex submanifolds of ℂN. As a corollary, we present multi-dimensional generalizations of an important result of Coman and Poletsky on Bernstein type inequalities on transcendental curves in ℂ2.

Holomorphic functions of slow growth on coverings of pseudoconvex domains in Stein manifolds

We apply the methods developed in [Br1] to study holomorphic functions of slow growth on coverings of pseudoconvex domains in Stein manifolds. In particular, we extend and strengthen certain results of Gromov, Henkin and Shubin [GHS] on holomorphic L^{2} functions on coverings of pseudoconvex manifolds in the case of coverings of Stein manifolds.

Grauert- and Lax–Halmos-type theorems and extension of matrices with entries in H∞☆

Abstract In the paper we prove an extension theorem for matrices with entries in H ∞ ( U ) for U a Riemann surface of a special type. One of the main components of the proof is a Grauert-type theorem for “holomorphic” vector bundles defined on maximal ideal spaces of certain Banach algebras.

On covering numbers of sublevel sets of analytic functions

In this paper we estimate covering numbers of sublevel sets of families of analytic functions depending analytically on a parameter. We use these estimates to study the local behavior of these families restricted to certain fractal subsets of R^N.

Hartogs Type Theorems on Coverings of Stein Manifolds

We prove an analog of the classical Hartogs extension theorem for certain (possibly unbounded) domains on coverings of Stein manifolds.

LINEAR AND NONLINEAR EXTENSIONS OF LIPSCHITZ FUNCTIONS FROM SUBSETS OF METRIC SPACES

A relationship is established between the linear and nonlinear extension constants for Lipschitz functions defined on subsets of metric spaces. Proofs of several results announced in our earlier paper are presented.

Projections in the spaceH∞ and the corona theorem for subdomains of coverings of finite bordered Riemann surfaces

AbstractLetM be a non-compact connected Riemann surface of a finite type, andR⋐M be a relatively compact domain such thatH1(M,Z)=H1(R,Z). Let