Let M be an n-dimensional Hadamard manifold, that is, a complete simply connected C∞ Riemannian manifold with nonpositive sectional curvatures. Making use of geodesic rays, Eberlein and O’Neill [11] constructed a compactification = MS(∞) of M which gives a homeomorphism of (M, S(∞)) with the Euclidean pair (Bn, Sn-1). In this paper we shall study the asymptotic Dirichlet problem for the Laplace-Beltrami operator, which is stated as follows:

Let be a curve in R2 defined by y = A(x). The Cauchy transform on is defined by the kernel

In our previous work [8], we picked up the elliptic equation

(1)

with the nonlinearity f(u) ⊇ 0 in C1. We studied the asymptotics of the family {(λ, u(x))} of classical solutions satisfying

(2)

We will define local times of self-intersection for multidimensional Brownian motion as generalized Wiener functionals under the framework of white noise analysis as in H. Watanabe ([6]). By making use of the chaotic representation of -function and precise computation we get a deep insight into the problem. In the section 1 multiple Wiener integrals with respect to multidimensional Brownian motion and chaotic representations for square-integrable Wiener functionals are given. They are indispensable, but seem not to be formulated clearly and correctly before. The useful concepts and results of white noise analysis are illustrated in the section 2. Section 3 is the main part of the paper. The applications to local times are introduced in the section 4 briefly.

In [Mo.l], S. Momm studied the boundary behaviour of extremal plurisubharmonic functions by using the pluricomplex Green function gΩ of a bounded convex domain Ω in Cn to exhaust the domain by a family of sublevel sets. Let Ω be a bounded convex domain in Cn containing the origin in its interior. The pluricomplex green function of Ω with a pole is defined by

(1.1)

Let k be an infinite field and A a standard G-algebra. This means that there exists a positive integer n such that A = R/I where R is the polynomial ring R := k[Xv …, Xn] and I is an homogeneous ideal of R. Thus the additive group of A has a direct sum decomposition A = ⊕ At where AiAj ⊆ Ai+j. Hence, for every t ≥ 0, At is a finite-dimensional vector space over k. The Hilbert Function of A is defined by

The primary purpose of this paper is to give an affirmative answer to a problem posed by Ohtsuka [13] whether there exists a p-harmonic measure on the unit disc in the 2-dimensional Euclidean space R2 with an infinite p-Dirichlet integral for the exponent 1 < p < 2.

For any set S and any entire function f let

where each zero of f — a with multiplicity m is repeated m times in Ef(S) (cf. [1]). It is assumed that the reader is familiar with the notations of the Nevanlinna Theory (see, for example, [2]). It will be convenient to let E denote any set of finite linear measure on 0 < r < ∞, not necessarily the same at each occurrence. We denote by S(r, f) any quantity satisfying .

In this paper we compute dimension formulas for rings of Siegel modular forms of genus g = 3. Let denote the main congruence subgroup of level two, the Hecke subgroup of level two and the full modular group. We give the dimension formulas for genus g = 3 for the above mentioned groups and determine the graded ring of modular forms with respect to .

Let M be any number field. We let DM, dM, hu, , AM and RegM be the discriminant, the absolute value of the discriminant, the class-number, the Dedekind zeta-function, the ring of algebraic integers and the regulator of M, respectively.

we set If q is any odd prime we let (⋅/q) denote the Legendre’s symbol.