Let $\Omega$ be a proper subdomain of $\mathbb{R}^{n}$, $n \ge 2$, and let $x_{0} \in \Omega$ be fixed. By $G_{\Omega}$ and $K_{\Omega}$ we denote the Green function and the Martin kernel for $\Omega$, respectively. Under a certain assumption on $\Omega$ near a boundary point $\xi$, we show that the product $G_{\Omega}(x, x_{0})K_{\Omega}(x, \xi)$ is comparable to $|x-\xi|^{2-n}$ for $x$ in a nontangential cone with vertex at $\xi$. We also give an estimate for the product $K_{\Omega}(x, \xi)K_{\Omega}(x, \eta)$ in a uniform domain, where $\eta$ is another boundary point.

We use a method of Berndtsson to obtain a simplification of Ohsawa's result concerning extension of $L^{2}$-holomorphic functions. We also study versions of the Ohsawa-Takegoshi theorem for some unbounded pseudoconvex domains, with an application to the theory of Bergman spaces. Using these methods we improve some constants, that arise in related inequalities.

Nakajima introduced a certain set of monomials realizing the irreducible highest weight crystals $\mathcal{B}(\lambda)$. The monomial set can be extended so that it contains crystal $\mathcal{B}(\infty)$ in addition to $\mathcal{B}(\lambda)$. We present explicit descriptions of the crystals $\mathcal{B}(\infty)$ and $\mathcal{B}(\lambda)$ over special linear Lie algebras in the language of extended Nakajima monomials. There is a natural correspondence between the monomial description and Young tableau realization, which is another realization of crystals $\mathcal{B}(\infty)$ and $\mathcal{B}(\lambda)$.

We show that if $\{ \varphi \}_{Q \in \mathcal{Q}} \in \bigcap \mathcal{M}_{\alpha}(\mathbb{R}^{n})$ is a frame and $\{ \psi_{Q} \}_{Q \in \mathcal{Q}} \in \bigcap \mathcal{M}_{\alpha}(\mathbb{R}^{n})$ is its dual frame (for the definition of $\mathcal{M}_{\alpha}(\mathbb{R}^{n})$, see Definition 2.1), where $\mathcal{Q}$ is the collection of dyadic cubes, then for any $f \in \mathcal{S}'(\mathbb{R}^{n})$, there exists a sequence of polynomials, $P_{L, L', L''}$, such that

\lim_{L, L', L'' \to \infty} \biggl\{ \sum_{-L' \le i \le L} \sum_{|k| \le \delta(i)2^{L''}} \langle f, \psi_{Q_{i, k}} \rangle \varphi_{Q_{i, k}}-P_{L, L', L''} \biggr\} = f

in the topology of $\mathcal{S}'(\mathbb{R}^{n})$, where $\delta(i) = \max(2^{i}, 1)$. We prove this result by explicitly constructing the polynomials $P_{L, L', L''}$. Furthermore, using the above result, we assert that the linear span of the one-dimensional wavelet system is dense in a function space if and only if the dual space of this function space has an trivial intersection with the set of polynomials. This is proved by using the annihilator of the one-dimensional wavelet system.

This paper is one step toward infinite energy gauge theory and the geometry of infinite dimensional moduli spaces. We generalize a gluing construction in the usual Yang-Mills gauge theory to an "infinite energy" situation. We show that we can glue an infinite number of instantons, and that the resulting ASD connections have infinite energy in general. Moreover they have an infinite dimensional parameter space. Our construction is a generalization of Donaldson's "alternating method".

This paper is concerned with the problem of the determination of unknown scalars involved in the algorithm of computing the generalized Green functions of reductive groups $G$ over a finite field. In the previous paper, we have treated the case where $G = SL_{n}$. In this paper, we determine the scalars in the case where $G$ is a classical group $Sp_{2n}$ or $SO_{N}$ for arbitrary characteristic.