In this article, via the classical complex interpolation method and some interpolation methods traced to Gagliardo, the authors obtain an interpolation theorem for Morrey spaces on quasi-metric measure spaces, which generalizes some known results on ${\mathbb R}^n$.

Dimension groups (not countable) that are also real ordered vector spaces can be obtained as direct limits (over directed sets) of simplicial real vector spaces (finite dimensional vector spaces with the coordinatewise ordering), but the directed set is not as interesting as one would like, i.e., it is not true that a countable-dimensional real vector space that has interpolation can be represented as such a direct limit over the a countable directed set. It turns out this is the case when the group is additionally simple, and it is shown that the latter have an ordered tensor product decomposition. In the Appendix, we provide a huge class of polynomial rings that, with a pointwise ordering, are shown to satisfy interpolation, extending a result outlined by Fuchs.

For the $n$-th order nonlinear differential equation, $y^{(n)} = f(x, y, y', \dots, y^{(n-1)})$, we consider uniqueness implies uniqueness and existence results for solutions satisfying certain $(k+j)$-point boundary conditions for $1\le j \le n-1$ and $1\leq k \leq n-j$. We define $(k;j)$-point unique solvability in analogy to $k$-point disconjugacy and we show that $(n-j_{0};j_{0})$-point unique solvability implies $(k;j)$-point unique solvability for $1\le j \le j_{0}$, and $1\leq k \leq n-j$. This result is analogous to $n$-point disconjugacy implies $k$-point disconjugacy for $2\le k\le n-1$.

The classical approach to studying operator ideals using tensor norms mainly focuses on those tensor norms and operator ideals defined by means of $\ell_p$ spaces. In a previous paper, an interpolation space, defined via the real method and using $\ell_p$ spaces, was used to define a tensor norm, and the associated minimal operator ideals were characterized. In this paper, the next natural step is taken, that is, the corresponding maximal operator ideals are characterized. As an application, necessary and sufficient conditions for the coincidence of the maximal and minimal ideals are given. Finally, the previous results are used in order to find some new metric properties of the mentioned tensor norm.

Let $f\colon \mathbb R^n\to \mathbb R$ be $C^\infty$ and let $h\colon \mathbb R^n\to\mathbb R$ be positive and continuous. For any unbounded nondecreasing sequence $\{c_k\}$ of nonnegative real numbers and for any sequence without accumulation points $\{x_m\}$ in $\mathbb R^n$, there exists an entire function $g\colon\mathbb C^n\to\mathbb C$ taking real values on $\mathbb R^n$ such that \begin{align*} &|g^{(\alpha)}(x)-f^{(\alpha)}(x)|\lt h(x), \quad |x|\ge c_k, |\alpha|\le k, k=0,1,2,\dots, \\ &g^{(\alpha)}(x_m)=f^{(\alpha)}(x_m), \quad |x_m|\ge c_k, |\alpha|\le k, m,k=0,1,2,\dots. \end{align*} This is a version for functions of several variables of the case $n=1$ due to L. Hoischen.

We show that if the general real method $(\cdot ,\cdot )_\Gamma$ preserves the Banach-algebra structure, then a bilinear interpolation theorem holds for $(\cdot ,\cdot )_\Gamma$.