We develop a scattering theory for perturbations of powers of the Laplacian on asymptotically Euclidean manifolds. The (absolute) scattering matrix is shown to be a Fourier integral operator associated to the geodesic flow at time $\pi $ on the boundary. Furthermore, it is shown that on ${{\mathbb{R}}^{n}}$ the asymptotics of certain short-range perturbations of ${{\Delta }^{k}}$ can be recovered from the scattering matrix at a finite number of energies.

We present “reiteration theorems” with limiting values $\theta =0$ and $\theta =1$ for a real interpolation method involving broken-logarithmic functors. The resulting spaces lie outside of the original scale of spaces and to describe them new interpolation functors are introduced. For an ordered couple of (quasi-) Banach spaces similar results were presented without proofs by Doktorskii in $[\text{D}]$.

Explicit evaluations of the symmetric Euler integral $\int _{0}^{1}\,{{u}^{\alpha }}{{(1-u)}^{\alpha }}f(u)\,du$ are obtained for some particular functions $f$. These evaluations are related to duplication formulae for Appell’s hypergeometric function ${{F}_{1}}$ which give reductions of ${{F}_{1}}(\alpha ,\beta ,\beta ,2\alpha ,y,z)$ in terms of more elementary functions for arbitrary $\beta $ with $z=y/(y-1)$ and for $\beta =\alpha +\frac{1}{2}$ with arbitrary $y,z$. These duplication formulae generalize the evaluations of some symmetric Euler integrals implied by the following result: if a standard Brownian bridge is sampled at time 0, time 1, and at $n$ independent randomtimes with uniformdistribution on $[0,1]$, then the broken line approximation to the bridge obtained from these $n+2$ values has a total variation whose mean square is $n(n+1)/(2n+1)$.

Let $Y$ be an infinite covering space of a projective manifold $M$ in ${{\mathbb{P}}^{N}}$ of dimension $n\ge 2$. Let $C$ be the intersection with $M$ of at most $n-1$ generic hypersurfaces of degree $d$ in ${{\mathbb{P}}^{N}}$. The preimage $X$ of $C$ in $Y$ is a connected submanifold. Let $\phi$ be the smoothed distance from a fixed point in $Y$ in a metric pulled up from $M$. Let ${{\mathcal{O}}_{\phi }}(X)$ be the Hilbert space of holomorphic functions $f$ on $X$ such that ${{f}^{2}}{{e}^{-\phi }}$ is integrable on $X$, and define ${{\mathcal{O}}_{\phi }}(X)$ similarly. Our main result is that (under more general hypotheses than described here) the restriction ${{\mathcal{O}}_{\phi }}(Y)\to {{\mathcal{O}}_{\phi }}(X)$ is an isomorphism for $d$ large enough.

This yields new examples of Riemann surfaces and domains of holomorphy in ${{\mathbb{C}}^{n}}$ with corona. We consider the important special case when $Y$ is the unit ball $\mathbb{B}$ in ${{\mathbb{C}}^{n}}$, and show that for $d$ large enough, every bounded holomorphic function on $X$ extends to a unique function in the intersection of all the nontrivial weighted Bergman spaces on $\mathbb{B}$. Finally, assuming that the covering group is arithmetic, we establish three dichotomies concerning the extension of bounded holomorphic and harmonic functions from $X$ to $\mathbb{B}$.

It is proved that a “typical” $n$-dimensional quotient ${{X}_{n}}$ of $l_{1}^{m}$ with $n={{m}^{\sigma }},0<\sigma <1$, has the property

$$\text{Average}\int_{G}{||Tx|{{|}_{{{X}_{n}}}}d{{h}_{G}}(T)\ge \frac{c}{\sqrt{n{{\log }^{3}}n}}\left( n-\int_{G}{|trT|d{{h}_{G}}(T)} \right)},$$

for every compact group $G$ of operators acting on ${{X}_{n}}$, where ${{d}_{G}}(T)$ stands for the normalized Haar measure on $G$ and the average is taken over all extreme points of the unit ball of ${{X}_{n}}$. Several consequences of this estimate are presented.

Let $G$ be an algebraic group and let $X$ be a generically free $G$-variety. We show that $X$ can be transformed, by a sequence of blowups with smooth $G$-equivariant centers, into a $G$-variety ${{X}^{'}}$ with the following property: the stabilizer of every point of ${{X}^{'}}$ is isomorphic to a semidirect product $U\rtimes A$ of a unipotent group $U$ and a diagonalizable group $A$.

As an application of this result, we prove new lower bounds on essential dimensions of some algebraic groups. We also show that certain polynomials in one variable cannot be simplified by a Tschirnhaus transformation.

In this paper, we consider the (essential) spectrum of the discrete Laplacian of an infinite graph. We introduce a new quantity for an infinite graph, in terms of which we give new lower bound estimates of the (essential) spectrum and give also upper bound estimates when the infinite graph is bipartite. We give sharp estimates of the (essential) spectrum for several examples of infinite graphs.

We give a characterization of bounded plurisubharmonic functions by using their complex Monge-Ampère measures. This implies a both necessary and sufficient condition for a positive measure to be complex Monge-Ampère measure of some bounded plurisubharmonic function.

Let ${{G}_{n}}$ be the split classical groups $\text{Sp}(\text{2}n\text{),}\,\text{SO(2}n\text{+1})$ and $\text{SO(2}n\text{)}$ defined over a $p$-adic field F or the quasi-split classical groups $U(n,n)$ and $U(n+1,n)$ with respect to a quadratic extension $E/F$. We prove the self-duality of unitary supercuspidal data of standard Levi subgroups of ${{G}_{n}}(F)$ which give discrete series representations of ${{G}_{n}}(F)$.