ABSTRACTS OF PUBLICATIONS

MARIA M. GORDINA

Heat kernel analysis and Cameron-Martin subgroup for infinite dimensional groups, to appear in the Journal of Functional Analysis, 1-32 (2000).

The heat kernel measure $\mu_t$ is constructed on GL(H), the group of invertible operators on a complex Hilbert space H. This measure is determined by an infinite dimensional Lie algebra $\mathfrak g$ and a Hermitian inner product on it. The Cameron-Martin subgroup G_CM is defined and its properties are discussed. In particular, there is an isometry from the $L^2_{\mu_t}$-closure of holomorphic polynomials into a space $\mathcal{H}^t(G_{CM})$ of functions holomorphic on G_CM. This means that any element from this $L^2_{\mu_t}$-closure of holomorphic polynomials has a version holomorphic on G_CM. In addition, there is an isometry from $\mathcal{H}^t(G_{CM})$ into a Hilbert space associated with the tensor algebra over $\mathfrak g$. The latter isometry is an infinite dimensional analog of the Taylor expansion. As examples I discuss a complex orthogonal group and a complex symplectic group.

Holomorphic functions and the heat kernel measure on an infinite dimensional complex orthogonal group, to appear in Potential Analysis, 1-33 (1999).

We use a diffusion in a Hilbert space to construct the heat kernel measure $\mu_t$ on an infinite dimensional complex group. Then it is proved that holomorphic polynomials on the group are square integrable with respect to the heat kernel measure. The closure of these polynomials, $\mathcal{H} L^2(SO_{HS}, \mu _t)$, is one of two spaces of holomorphic functions we consider. The second space, $\mathcal{H} L^2(SO(\infty))$, consists of functions which are holomorphic on a subset of the group. It is proved that there is an isometry from the first space to the second one. The main theorem is that an infinite dimensional nonlinear analog of the Taylor expansion defines an isometry from $\mathcal{H} L^2(SO(\infty))$ into the Hilbert space associated with a Lie algebra of the infinite dimensional group. This is an extension to infinite dimensions of an isometry of B. Driver and L. Gross for complex Lie groups. All the results of this paper are formulated for one concrete group, the Hilbert-Schmidt complex orthogonal group, though my methods can be applied in more general situations.

The Denseness of Fractions in L_p for 0<p<1, 25 (1992), no. 4, 11-16; English translation: Vestnik, St. Petersburg University, Mathematics, vol. 25(4), 1992, 11-16.

This paper deals with the study of non locally convex function spaces over manifolds. I proved the denseness of rational functions in L_p for 0 < p < 1. The fact that these L_p-spaces are non locally convex is the principal difficulty of this problem, since methods used in Banach spaces are not applicable in this case. Instead one has to use the theory of singular functions on manifolds. My main result is the following theorem : Theorem. Suppose M is an n-dimensional Riemannian manifold embedded in $\Bbb R^d$, $\mu$ is a measure on M, which is absolutely continuous with respect to the n-dimensional volume on M, $\ 0 < p < 1$. Then Span $\{ \frac1{\Vert x-a\Vert^{\alpha}_d}, a \in M \}$ is dense in $L^p(M, d\mu )$ if $\ \alpha >n$, $\ \alpha p < n$, $\ p<\frac{2n}{\alpha} - 1$. This problem is interesting in conjunction with the results by T. Wolff, A. B. Aleksandrov and others on the theory of multidimensional Hardy spaces, uniqueness theorems for gradients of harmonic functions and some other questions.