A CARLSON TYPE INEQUALITY WITH BLOCKS AND INTERPOLATION

An inequality, which generalizes and unifies some recently proved Carl son type inequalities, is proved. The inequality contains a certain nu mber of ''blocks'' and it is shown that these blocks are, in a sense, optimal and cannot be removed or essentially changed. The proof is bas ed on a special equivalent representation of a concave function (see [ 6, pp. 320-325]). Our Carlson type inequality is used to characterize Peetre's interpolation functor [ ]phi (see [26]) and its Gagliardo clo sure on couples of functional Banach lattices in terms of the Calderon -Lozanovskii construction. Our interest in this functor is inspired by the fact that if phi = t(theta) (0 < theta < 1), then, on couples of Banach lattices and their retracts, it coincides with the complex meth od (see [20], [271) and, thus, it may be regarded as a ''real version' ' of the complex method.