It is proved that the C2-smoothly bounded pseudoconvex domains in Pn admit bounded plurisubharmonic exhaustion functions. Further arguments are given concerning the question of existence of strictly plurisubharmonic functions on neighbourhoods of real hypersurfaces in Pn.

Representation of a Gaussian process in terms of a Brownian motion is a powerful tool in the investigation of its structure. Among various representations is the canonical representation which is viewed as the best one from the viewpoint of the prediction theory. We have discovered some significance of non-canonical representations and discuss their optimality in an information theoretical approach.

For Hunt processes on Rd, strong and weak transience is defined by finiteness and infiniteness, respectively, of the expected last exit times from closed balls. Under some condition, which is satisfied by Lévy processes and Ornstein-Uhlenbeck type processes, this definition is expressed in terms of the transition probabilities. A criterion is given for strong and weak transience of Ornstein-Uhlenbeck type processes on Rd, using their Lévy measures and coefficient matrices of linear drift terms. An example is discussed.

In this paper, we consider a kind of multilinear operators related to oscillatory singular integrals with rough kernels and give a criterion of certain boundedness for this kind of operators.

An Lp operator norm estimate of the difference between the Kac operator and the Schrödinger semigroup is proved and used to give a variant of the Trotter product formula for Schrödinger operators in the Lp operator norm. The method of the proof is probabilistic based on the Feynman-Kac formula. The problem is discussed in the relativistic as well as nonrelativistic case.

We will give certain asymptotic relations between p-eigenvalues and certain Fourier coefficients of Siegel cusp forms of genus g. In particular, it will turn out that potential strong bounds for the Fourier coefficients will imply potential strong bounds for the eigenvalues.

We systematically develop an algebraic technique of free products of operators and their Green’s functions. We apply this framework to obtain, in a simple and uniform fashion, several results on the spectra of graph Laplaceans and random walks.

The author continues his previous work with the purpose of establishing a theory of newforms in the case of half-integral weight. In this paper, the author formulates and proves a complete theory of newforms for Kohnen space. Kohnen space is an important canonical subspace in the space of cusp forms of half-integral weight k + 1/2 (k > 0). Every Hecke common eigenform in Kohnen space corresponds to a primitive form of integral weight 2k and of odd level via Shimura Correspondence.

These newforms for Kohnen space satisfy the Strong multiplicity One theorem. Moreover, we explicitly determine the corresponding primitive form (of weight 2k) to each newform for Kohnen space. The space of oldforms is also explicitly described.

In order to find all newforms for Kohnen space, the author needs a certain non-vanishing property of Fourier coefficients of cusp forms. Such property proves by using representation theory of finite linear groups. The method of proof of newform theory is mainly based on trace formulae and trace relations.

A generalized number operator and a generalized Gross Laplacian are introduced on the basis of white noise distribution theory. The equicontinuity is examined and associated one-parameter transformation groups are constructed. An infinite dimensional analogue of ax + b group and Cauchy problems on white noise space are discussed.

Studies for A.C. Stark Hamiltonian are closely related to that for the self-adjoint operator on torus. In this paper we use Mourre’s commutator method, which makes great progress for the study of time-independent Hamiltonian. By use of it we show the asymptotic behavior of the unitary propagator as σ → ± ∞.