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

Interpolation theorems for the pairs of spaces (𝐿^{𝑝},𝐿^{∞}) and (𝐿¹,𝐿^{𝑞})