On a negative result concerning interpolation with change of measures for Lorentz spaces

Abstract

In this note we show that the Stein-Weiss theorem on LP interpolation with change of measures cannot be extended to Lorentz spaces LP,s. By the well-known Marcinkiewicz interpolation theorem, the LP boundedness of a sublinear operator for all p in some open interval (P1,P2) follows from some weak type inequalities for the extreme values P1 and P2. These inequalities can be viewed as boundedness properties on some appropiate Lorentz spaces, and, in fact, the theorem admits an extension to such spaces (see Theorem 1 below). In 1958, Stein and Weiss proved an interpolation result on LP spaces which allows one to change measures simultaneously with changing exponents. (A particular case is stated below as Theorem 2). It is natural to ask whether it is possible to obtain an extension of their result in the context of Lorentz spaces. In this note we prove that the answer is negative. In what follows we first give some definitions; next we state some known interpolation results and present the conjecture about interpolation with change of measures on Lorentz spaces. Finally we state and prove two lemmas establishing that the conjecture is false. The fact that this conjecture is false is a stumbling stone in the study of oneweight norm inequalities for the fractional maximal operator on Lorentz spaces. In fact, about the one-weight weak inequalities of Muckenhoupt and Wheeden [M-W] for such an operator it is only known that they extend to Lorentz spaces LP,s when the range of s is restricted. More precisely, the author obtained in [F] the following result. Let Mf(x) =sup Ql-oVn If (y)Idy, where 0 < a < n and the sup is taken over all cubes Q with center at x. Assume p, q and s are such that l/q = l/pa/n and that either 1 < p < n/a, 1 < s < q or p= s = 1. Then there exists a constant C, independent of f, such that fIMcef flq,oo,wq < C||f |P S, I Received by the editors May 25, 1995 and, in revised form, November 16, 1995. 1991 Mathematics Subject Classification. Primary 46B70; Secondary 46E30.