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On the Structure of a Limit Class of Logarithmic Interpolation Spaces
Mediterranean Journal of Mathematics ( IF 1.1 ) Pub Date : 2020-09-10 , DOI: 10.1007/s00009-020-01602-7
Blanca F. Besoy , Fernando Cobos , Luz M. Fernández-Cabrera

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(ta) 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:

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Kt,  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\(\左(E_ {0},E_ {1} \右)_ {1,\ infty,(0,\阿尔法_ {\ infty})} \)\(E_ {0} \)\( E_ {1} \)是Banach函数空间,是\((A_0,A_1)_ {1,\ infty,(0,\ alpha _ \ infty)} ^ {\ circ} \)的对偶空间。
更新日期:2020-09-10
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