On the Structure of a Limit Class of Logarithmic Interpolation Spaces

Abstract

We investigate the interpolation spaces $\left( A_{0}, A_{1}\right) _{1,\infty , (0, \alpha _{\infty })}$ formed by all $a \in A_{0}+A_{1}$, having a finite norm:

$$\begin{aligned} {}\left\Vert a \right\Vert _{\left( A_{0}, A_{1}\right) _{1,\infty , (0, \alpha _{\infty })}} = \max \left\{ \sup _{0< t< 1} \frac{K(t,a)}{t}, \sup _{1< t < \infty } \frac{(1+\log t)^{\alpha _{\infty }}K(t,a)}{t}\right\} , \end{aligned}$$

where K(t, a) is the K-functional. We show that they have a description in terms of the J-functional which is of another nature than the description of the other logarithmic interpolation spaces. We also determine the associate space of $\left( E_{0}, E_{1}\right) _{1,\infty , (0, \alpha _{\infty })}$ when $E_{0}$ and $E_{1}$ are Banach function spaces, and the dual space of $(A_0,A_1)_{1,\infty , (0, \alpha _\infty )}^{\circ }$.