Abstract. We find the logarithmic L 2 -small ball asymptotics for a large class of zero mean Gaussian fields with covariances having the structure of "tensor product". The main condition imposed on marginal covariances is the regular behavior of their eigenvalues at infinity that is valid for a multitude of Gaussian random functions including the fractional Brownian sheet, Ornstein -Uhlenbeck sheet, etc. So we get the far-reaching generalizations of well-known results by Csáki (1982) and by Li (1992). Another class of Gaussian fields considered is the class of additive fields studied under the supremum-norm by Chen and Li (2003). Our theorems are based on new results on spectral asymptotics for the tensor products of compact self-adjoint operators in Hilbert space which are of independent interest.

Key words Small deviations, slowly varying functions, smooth covariances, tensor product of operators, spectral asymptotics MSC (2010) 60G15, 60G60, 47A80We find the logarithmic L 2 -small ball asymptotics for a class of zero mean Gaussian fields with covariances having the structure of "tensor product". The main condition imposed on marginal covariances is slow growth at the origin of counting functions of their eigenvalues. That is valid for Gaussian functions with smooth covariances. Another type of marginal functions considered as well are classical Wiener process, Brownian bridge, Ornstein-Uhlenbeck process, etc., in the case of special self-similar measure of integration. Our results are based on a new theorem on spectral asymptotics for the tensor products of compact self-adjoint operators in Hilbert space which is of independent interest. Thus, we continue to develop the approach proposed in the paper [6], where the regular behavior at infinity of marginal eigenvalues was assumed.

ABSTRACT. We construct the asymptotics as t ---* 0 of the trace of the operator exp(-tP) for an elliptic operator P on a manifold with conical points.KEY WORDS: elliptic operator,-heat kernel, trace asymptotics, manifold with conical points, Green's function. w IntroductionLet P be a self-adjoint positive definite elliptic operator of order 21 in the space L2 on a closed compact manifold M of dimension m. For the case in which the coefficients of the operator are smooth, the following statement is well known (e.g., see [1] H(t, x, x) ..~ ~ ck(z)t (-'~+2k)/(2') , t --* +0.( 1) k=0This azymptotics provides important information on the operator P and the geometry of the manifold (see ill).If the coefficients of the operator are discontinuous at some point, then the expansion becomes nonuniform and the coefficients ck(x) axe singular at the point of discontinuity. We consider an elliptic operator on a manifold M with a conical point x0. This means (e.g., see [21)that M\{x0} is an m-dimensional C ~ manifold and there exists a neighborhood U of the point x0 and a mapping k of this neighborhood (a diffeomorphism preserving the conical structure) onto a neighborhood of the vertex 0 of a conical manifold X. We denote by S a fixed directing submartifold of X (S is a compact manifold of dimension m -1 intersecting each ray from X at one point). For the points of X and hence for the points in the neighborhood U we can introduce the polar coordinates x = r, w, where r E R+, w E S and r-ix E S.To simplify the notation, we make no distinction between a function u on U and the function u o k -1 on X. We also choose a positive density dx on M smooth on M \ {x0 } and assume that dx = r m-1 dr dw in the neighborhood U, where dw is a smooth positive density on S. Of course, the manifold can have any number of conical points; in what follows, we consider only one such point x0.An important example of an operator on a manifold with a conical point is an operator on a closed smooth manifold with symbol having an isolated singularity at the point. At this point, the principal symbol must have the limit that can depend on the direction of approach, and lower terms can have power-law singularities. In this case, X = R "~ and S = S '~-] .The operator is defined in the space L2(M) of Cd-valued functions via the quadratic form (Pu, u)= fM ~ (aat,a2(x)D~tu, D'~u) dx, aat,a2(x)=a*2,at(x), D~j = -iO~j ,

No abstract

UDC 517.95We consider compact selfadjoint pseudodifferential operators under the assumption that the decay order of symbols with respect to ξ depends on a point x. We show that the asymptotic Weyl formula is valid for such operators. Bibliography: 12 titles. Dedicated with thanks to Nina Nikolaevna Uraltseva