On the theory of spaces Λ
Abstract
l Introduction. In this paper we discuss properties of the spaces A(,p), which were defined for the special case (x) -QLx a ~, 0 < & < 1, in our previous paper [] . A function f(x), measurable on the interval (0,/), I < + 00 belongs to the class A(,p) provided the norm j/j , defined by .i) ii/ii = is finite. Here (x) is a given nonnegative integrable function on (0,/), not identically 0, and / *() is the decreasing rearrangement of \f(x) | , that is, the decreasing function on (0,1), equimeasurable with | /Gc)| . (For the properties of decreasing rearrangements see 1.5, 12, 7, and 8] .) We write also ((X,p) instead of A(, p) with {x) = (Xx (x ~, and A() instead of (,l). We shall also consider spaces A(,p) for the infinite interval (0,+ u0 ). In 2 we give some simple properties of the spaces , and show in particular that A(,p) has the triangle property if and only if (x) is decreasing. In 3 we discuss the conjugate spaces *(,p), and show that the spaces A(,p) are reflexive. In 4 we give a generalization of the spaces A(,p), and characterize the conjugate spaces in case p = 1. In 5 we give applications; we prove that the Ilardy-Littlewood majorants {x 9 f) of a function f A(,p) or/ C (,p) also belong to the same class. We give sufficient conditions for an integral transformation to be a linear operation from one of these spaces into itself, and apply them to solve the moment problem for the spaces A{ 9 p) and (,p).