By J. Wolka: pp. 518. £50.00/£17.50. (Cambridge University Press, 1987)

Abstract. We present "reiteration theorems" with limiting values θ = 0 and θ = 1 for a real interpolation method involving broken-logarithmic functors. The resulting spaces lie outside of the original scale of spaces and to describe them new interpolation functors are introduced. For an ordered couple of (quasi-) Banach spaces similar results were presented without proofs by Doktorskii in [D].

No abstract

Sharp reiteration theorems for the K-interpolation method in limiting cases are proved using two-sided estimates of the K-functional. As an application, sharp mapping properties of the Riesz potential are derived in a limiting case.

We define generalized Lorentz -Zygmund spaces and obtain interpolation theorems for quasilinear operators on such spaces, using weighted Hardy inequalities. In the limiting cases of interpolation, we discover certain scaling property of these spaces and use it to obtain fine interpolation theorems in which the source is a sum of spaces and the target is an intersection of spaces. This yields a considerable improvement of the known results which we demonstrate with examples. We prove sharpness of the interpolation theorems by showing that the constraints on parameters are necessary for the interpolation theorems.RUDNICK [BR]: they defined the so-called Lorentz -Zygmund spaces, which include the spaces of Lebesgue, Lorentz, and Zygmund, and proved interpolation theorems for operators T of joint weak typethese satisfy a certain rearrangement inequality, see 1991 Mathematics Subject Classification. Primary 46 E 30; Secondary 46 B 70, 47 B 38, 47 G 10, 26 D 10. Keywords and phrases. Generalized Lorentz -Zygmund spaces, operators of joint weak type, scaling property, Hardy inequality, exact interpolation theorems, embedding theorems. 42 Q1 pl < pz 5 oo, -= -m ( k -i).