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].

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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).

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In [4] we proved that for the symmetric second-order differential expression My=-(py'y+qy(1.1) on [0, oo) with p and q real-valued and p > 0, the well-known Levinson criterion for M to be in the limit-point case also implies that any power of M (and hence any real polynomial expression in M) is also in the limit-point case. In this paper we consider the analogous problem for the general 2/ith-order symmetric expressionMy= Z (-\yXp n -j/ J) ) u) (1.2) ;=o on [0, oo) with the p n -. } real-valued for./ = 0, 1, ..,, n and p 0 > 0.The deficiency index d{M) of the real symmetric expression M can be defined as the number of linearly independent L 2 (0, oo) solutions of My = Ayfor any non-real number L This integer d(M) is independent of X (as long as im A # 0) and it was shown by Glazman in [7] that n < d(M) ^ 2/i and also that every value in this range is possible. When d(M) = n we say that M is in the limit-point case or that M is limit-point.If the coefficients of M are sufficiently differentiate we can form powers M k in the natural way: M l =M, M k y = M k~l (My), fc = 2, 3, .... These powers M 2 , M 3 , ... are again real symmetric differential expressions of type (1.2)-(see [3; p. 1289]). It is known that powers of Jimit-point expressions are not necessarily limit-point-(see [11] and [13]). Our aim in this paper is to find conditions on the coefficients pj which are sufficient for M k , if it exists, to be in the limitpoint case. We have two main results. The first shows that Hinton's condition in [9] for (1.2) to be limit-point (which almost reduces to Levinson's condition in the second-order case (1.1)) is sufficient not only for M to be limit-point as Hinton proves but for M k to be limit-point for any k = 1, 2, 3, .... Our second main result shows that Hartman's " interval type " sufficient condition in [8] for (1.1) to be limit-point, along with its extension by Brown and Evans [1] to the higher order case (1.2), is sufficient not only for M to be limit-point but also for M k to be limitpoint for any k = 1, 2, 3,