Interpolation theorems for the pairs of spaces $(L^{p}, L^{\infty })$ and $(L^{1}, L^{q})$

Abstract:

A Banach space $Z$ has the interpolation property with respect to the pair $(X,Y)$ if each $T$, which is a bounded linear operator from $X$ to $X$ and from $Y$ to $Y$, can be extended to a bounded linear operator from $Z$ to $Z$. If $X = {L^p},Y = {L^\infty }$ we give a necessary and sufficient condition for a Banach function space $Z$ on $(0,l),0 < l \leqq + \infty$, to have this property. The condition is that $g \prec {}^pf$ and $f \in Z$ should imply $g \in Z$; here $g \prec {}^pf$ means that ${g^{ \ast p}} \prec {f^{ \ast p}}$ in the Hardy-Littlewood-Pólya sense, while ${h^ \ast }$ denotes the decreasing rearrangement of the function $|h|$. If the norms $||T|{|_X},||T|{|_Y}$ are given, we can estimate $||T|{|_Z}$. However, there is a gap between the necessary and the sufficient conditions, consisting of an unknown factor not exceeding ${\lambda _p},{\lambda _p} \leqq {2^{1/q}},1/p + 1/q = 1$. Similar results hold if $X = {L^1},Y = {L^q}$. For all these theorems, the complete continuity of $T$ on $Z$ is assured if $T$ has this property on $X$ or on $Y$, and if $Z$ satisfies a certain additional necessary and sufficient condition, expressed in terms of $||{\sigma _a}|{|_Z},a > 0$, where ${\sigma _a}$ is the compression operator ${\sigma _a}f(t) = f(at),0 \leqq t < l$.