As generalizations of zonal spherical functions and Whittaker functions, certain special functions on $p$-adic orthogonal groups closely related to automorphic forms are introduced. Their multiplicity one property and explicit formula are established.

The theory of stochastic differential equation on the field of $p$-adics is initiated by Kochubei. In this article, we will focus on a class of random walks with a certain integrability to develop the theory of stochastic analysis in a way similar to the existing theory of stochastic analysis based on the Brownian motion. In fact, for any random walk in the class, we can introduce the notion of stochastic integral with respect to the random walk and justify the existence of the solution of the stochastic differential equations based on the random walk, where the stochastic differential equations admit not only Lipschitz continuous path dependent coefficients but also continuous coefficients growing at most linearly with respect to $p$-adic norm. Finally, we will see an example of stochastic process which can be covered by Dirichlet space theory and obtained also by solving stochastic differential equation.

An explicit upper bound is given for (i) the number of $n$-dimensional monomial mod $p$ Galois representations of $G_{\Q} $ with bounded conductor and (ii) the order of the image of such a representation in terms of $n$, $p$, and the conductor.

We study a Riccati differential equation whose coefficient is expressible in terms of a special Weierstrass pe-function. We show that all the solutions are meromorphic, and examine the periodicity of them.

We prove sharp weighted norm inequalities for vector-valued singular integral operators and commutators. We first consider the strong $(p,p)$ case with $p>1$ and then the weak-type $(1,1)$ estimate. Our results do not assume any condition on the weight function and involve iterations of the classical Hardy-Littlewood maximal function.

We introduce a new class of selfadjoint compact pseudodifferential operators, which is analogous to a class of elliptic unbounded pseudodifferential operators and is, therefore, suitable for obtaining upper and lower estimates on the eigenvalues of operators in this class. We prove such estimates and, as an application, we show that any operator from this class belongs to the Schatten-von Neuman class if and only if its symbol belongs to the Lorentz space.

Based on two equalities for power series which are equivalent to the Tedone formulas, the elementary solution to the wave operator on the product of $k$ Riemannian manifolds is represented as a composition, with respect to the time variable, of $k$ elementary solutions to wave operators on factor manifolds. As a consequence, one has an infinite number of non-trivial momentary Huygens operators. For example, wave operators on the product of an odd numer of odd dimensional manifolds with constant curvature are revealed to be momentary Huygens operators for an appropriate choice of coefficients of the 0-th order terms.