PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 84, Number 4, April 1982 INTERPOLATIONOF UNIFORMLYCONVEXBANACH SPACES MICHAEL CWIKEL AND SHLOMO REISNER Abstract. If A0 and At are a compatible couple of Banach spaces, one of which is uniformly convex, then the complex interpolation spaces [/lo'^ilo ale ^so um" formly convex for 0 < 0 < 1. Estimates are given for the moduli of convexity and smoothness of [A0, At]e in terms of these moduli for A0 and At. In general, up to equivalence of moduli these estimates are best possible. A result of Beauzamy [1, p. 71] states that if A0 and /I, are compatible Banach spaces, one of which is uniformly convex, then the spaces S(p, £0, A0, ¿,, Ax) obtained from A0 and Ax by the method of real interpolation, are also uniformly convex for 1 <p < oo. In this note we present an analogue of this result for the complex interpolation method and give an estimate for the modulus of convexity of [A0, A\]e in terms of the moduli of A0 and Av This estimate is best possible in general, up to equivalence of moduli. We remark that for the power type of the moduli, such an estimate is implicit in [10, section 1.9]. We use the notation and definitions of [7] and of [3,2] for concepts from the geometry of Banach spaces and from interpolation space theory respectively. The letters A, A0, Ax will always denote Banach spaces over the complex field. (A0, /!,) is a compatible couple (interpolation pair). The complex interpolation space [A0, Ax]e is denoted by Ae. We say that a positive function / on R+ is dominated by another function g (notation: / < g orf(t) < g(t)) if there are positive constants a, b such that f(t) < ag(bt) for all t G R+ . /is equivalent to g(f~ g)iif< g and g <f. We denote by/"1 the inverse function of / (if it exists). We denote the moduli of uniform convexity and uniform smoothness of A by 8A(e) and pA{r) respectively. It is known [4; 7, part II] that pA is an Orlicz function and that 8A is equivalent to an Orlicz function 8A(8A(e) is defined only for 0 < e «£ 2 but 8A is defined for all e > 0 and equivalent to 8A near 0). Also, pa(t) > 0 for t > 0 (this can be concluded from the duality formula pa(t) = \ supe3»0{Te — 28A.{e)} and the fact that 8A.(e) < e2) and being an Orlicz function, it turns out that pA is strictly increasing. 8A is also strictly increasing if A is uniformly convex (8A itself is strictly increasing when A is uniformly convex since 8A(e)/e is nondecreasing). If L is a Banach lattice of measurable functions on a measure space (Í2, 2, ¡i) we denote by L(A) the space of _ Received by the editors April 13, 1981 and, in revised form, June 29, 1981. 1980 Mathematics Subject Classification. Primary 46B20. ©1982 American Mathematical Society 0O02-9939/81/00O0-0343/$02.25 555 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use