Espinosa and Moll [2], [3] studied “the Hurwitz transform” meaning an integral over [0, 1] of a Fourier series multiplied by the Hurwitz zeta function $\zeta (z,u)$, and obtained numerous results for those which arise from the Hurwitz formula. Ito's recent result [4] turns out to be one of the special cases of Espinosa and Moll's theorem. However, they did not give rigorous treatment of the relevant improper integrals.

In this note we shall appeal to a deeper result of Mikolás [9] concerning the integral of the product of two Hurwitz zeta functions and derive all important results of Espinosa and Moll. More importantly, we shall record the hidden and often overlooked fact that some novel-looking results are often the result of “duplicate use of the functional equation”, which ends up with a disguised form of the original, as in the case of Johnson's formula [5]. Typically, Example 9.1 ((1.12) below) is the result of a triplicate use because it depends not only on our Theorem 1, which is the result of a duplicate use, but also on (1.3), the functional equation itself.

We explicitly describe the $({\mathfrak{g}}_{\mathit{C}},K)$-module structures of the principal series representations of $\mathit{SU}(2,2)$ associated with a maximal parabolic subgroup.

In this paper we deduce weak type extension theorems for the groups of measure-preserving homeomorphisms of noncompact manifolds. As an application, we show that the group of measure-preserving homeomorphisms with compact support of a noncompact connected manifold, endowed with the Whitney topology, is locally contractible.

The weak Harnack inequality for $L^p$-viscosity solutions is shown for fully nonlinear, second order uniformly elliptic partial differential equations with unbounded coefficients and inhomogeneous terms. This result extends those of Trudinger for strong solutions [21] and Fok for $L^p$-viscosity solutions [13]. The proof is a modification of that of Caffarelli [5], [6]. We apply the weak Harnack inequality to obtain the strong maximum principle, boundary weak Harnack inequality, global $C^{\alpha }$ estimates for solutions of fully nonlinear equations, strong solvability of extremal equations with unbounded coefficients, and Aleksandrov-Bakelman-Pucci maximum principle in unbounded domains.

Limit theorems for the normalized laws with respect to two kinds of weight functionals are studied for any symmetric stable Lévy process of index . The first kind is a function of the local time at the origin, and the second kind is the exponential of an occupation time integral. Special emphasis is put on the role played by a stable Lévy counterpart of the universal $\sigma$-finite measure, found in [9] and [10], which unifies the corresponding limit theorems in the Brownian setup for which $\alpha =2$.

In this paper, we investigate the asymptotic behavior of regular ends of flat surfaces in the hyperbolic $3$-space $H^3$. Gálvez, Martínez and Milán showed that when the singular set does not accumulate at an end, the end is asymptotic to a rotationally symmetric flat surface. As a refinement of their result, we show that the asymptotic order (called pitch $p$) of the end determines the limiting shape, even when the singular set does accumulate at the end. If the singular set is bounded away from the end, we have . If the singular set accumulates at the end, the pitch $p$ is a positive rational number not equal to $1$. Choosing appropriate positive integers $n$ and $m$ so that $p=n/m$, suitable slices of the end by horospheres are asymptotic to $d$-coverings ($d$-times wrapped coverings) of epicycloids or $d$-coverings of hypocycloids with $2n_0$ cusps and whose normal directions have winding number $m_0$, where $n=n_{0}d$, $m=m_{0}d$ ($n_0$, $m_0$ are integers or half-integers) and $d$ is the greatest common divisor of $m-n$ and $m+n$. Furthermore, it is known that the caustics of flat surfaces are also flat. So, as an application, we give a useful explicit formula for the pitch of ends of caustics of complete flat fronts.

We obtain a new construction of compact complex surfaces with trivial canonical bundle. In our construction we glue together two compact complex surfaces with an anticanonical divisor under suitable conditions. Then we show that the resulting compact manifold admits a complex structure with trivial canonical bundle by solving an elliptic partial differential equation. We generalize this result to cases where we have other than two components to glue together. With this generalization, we construct examples of complex tori, Kodaira surfaces and K3 surfaces. Lastly we deal with the smoothing problem of a normal crossing complex surface $X$ with at most double curves. We prove that we still have a family of smoothings of $X$ in a weak sense even when $X$ is not Kählerian or $H^1(X,{\mathit{O}}_X)\ne 0$, in which cases the smoothability result of Friedman [Fr] is not applicable.

On the harmonic Bergman space of the half space in ${\mathit{R}}^n$, we show that if the product of two or more Toeplitz operators with harmonic symbols that have certain boundary smoothness has finite rank, then one of the symbols must be identically 0. Our methods require the number of factors in the product to depend on the dimension $n$.

We consider asymptotic behavior of the dimension of the invariant subspace in a tensor product of several irreducible representations of a compact Lie group $G$. It is equivalent to studying the symplectic volume of the symplectic quotient for a direct product of several coadjoint orbits of $G$. We obtain two formulas for the asymptotic dimension. The first formula takes the form of a finite sum over tuples of elements in the Weyl group of $G$. Each term is given as a multiple integral of a certain polynomial function. The second formula is expressed as an infinite series over dominant weights of $G$. This could be regarded as an analogue of Witten's volume formula in 2-dimensional gauge theory. Each term includes data such as special values of the characters of the irreducible representations of $G$ associated to the dominant weights.