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 \((A_0,A_1)_{1,\infty , (0, \alpha _\infty )}^{\circ }\).

中文翻译：

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关于对数内插空间的极限类的结构

我们研究由所有\（a \组成的内插空间\（\ left（A_ {0}，A_ {1} \ right）_ {1，\ infty，（0，\ alpha _ {\ infty}}} \）在A_ {0} + A_ {1} \）中，具有有限范数：$$ \ begin {aligned} {} \ left \ Vert a \ right \ Vert _ {\ left（A_ {0}，A_ {1} \ right）_ {1，\ infty，（0，\ alpha _ {\\ infty}}}} = \ max \ left \ {\ sup _ {0 <t <1} \ frac {K（t，a）} {t}，\ sup _ {1 <t <\ infty} \ frac {（1+ \ log t）^ {\ alpha _ {\ infty}} K（t，a）} {t} \ right \}， \ end {aligned} $$其中K（t，  a）是K函数。我们表明，它们具有J功能的描述，这与其他对数插值空间的描述不同。我们还确定了\（\左（E_ {0}，E_ {1} \右）_ {1，\ infty，（0，\阿尔法_ {\ infty}）} \）时\（E_ {0} \）和\（ E_ {1} \）是Banach函数空间，是\（（A_0，A_1）_ {1，\ infty，（0，\ alpha _ \ infty）} ^ {\ circ} \）的对偶空间。