Peetre, Jaak

English

Peetre Jaak

University of Szeged

On interpolation functions. II By J. PEETRE in Lund (Sweden) The purpose of this note is to indicate a certain simplification of the results of [2]. (We use throughout the terminology of that note.) If h = h(u) is a positive Borel measurable function on (0, we agree to say that h is pseudo-concave if it is equivalent to a concave function, i.e. if there exist a concave function h0 and a constant C such that C~1h0(ju)^h(u)SCh0(u) for all u. Lemma, h is pseudo-concave if and only if for some C holds (1) h(v) ^ C max(l,vju) h(u) for all u,v. Proof (necessity). It suffices to show that (1) holds for h0. But from the concav-ity it follows that h0(u) is increasing [but ¡ s decreasing. Therefore (1) follows with C= 1. Proof (sufficiency). Let a ; s 0 , 2ai = 1> u~ 2 a>Mi (finite sums). Using (1) we obtain Corollary. If h(u) is pseudo-concave so is k(u) = (h(ur))r where r M any real number ¿¿0. Proof . Obvious. — This corollary was obtained in [2] as a by-product of the main result there which we now reformulate as follows. Theorem. A function H=H(z0, ¿j) homogeneous of degree ¡1 and satisfying condition (2) in [2] is an interpolation function {of any power p) if and. only if it is of the form H(z0,z^) = z0h(zjz0) with h pseudo-concave. This establishes the pseudo-concavity because we may now take /z0(u) = sup 2 «Mud, which is by the way the least concave majorant of h. i 92 J. Peetre: On interpolation functions. II Proof (necessity). Consider the space X={x,y) provided with the measure H such that each of the two points x and y carries the mass 1. Take Co(*) = l . C o O O ^ l ; i i ( * ) = " . ti(y) = v and define a linear mapping n by na(x) = 0, na(y) = a(x). The corresponding operator norms of n (considered as a mapping from Lg0 into Lpn etc.) are M0= 1, Ml=—, M=——-. u h(u) By the inequality (cf. inequality (1) in [2]) M^Cmax (M0,Mj) we now get M « ) 5 C m a X ( 1 ' « ) which establishes (1). Proof (sufficiency). By the lemma H is equivalent to a function of the form (4) in [2]. Therefore we may use the corresponding part of the proof of theorem 1 in [2] (cf. in particular pp. 168—169). We conclude by illustrating our new result in a concrete case (cf. [1]). Let X=(0, ;u = Haar measure, (0(x) = l, (1(x) = ex" (a>0). With • h(u) = A = ( « > / ? > 0) we then have ex' — h(ex*) = CQIi • We claim that h is pseudo-concave. It is clear that h(u) is increasing. It suffices there-fore to show that is decreasing if u is large, since the values u < 1 do not interfere. To this end we consider the corresponding logarithmic derivative; we have log (h(u)/u) = (A(log u f - 1 -1)/« which, since A<1, eventually becomes < 0. Applying now our theorem we thus get the following interpolation theorem \ If n is a continuous linear mapping from Lg into Lg and from into it is also a continuous linear mapping from [Ljjj into LI (at. > P > 0). Here we have set L* = LP ^ with y = 0, a, p. References [1] C. GOULAOUIC, Interpolation entre les espaces Lp avec poids, C. R. Acad. Sci. Paris, 262 (1966), 3 3 3 — 3 3 6 . [2] J. PEETRE, On interpolation functions, Acta Sci. Math., 27 (1966), 167—171. (Received April 5,1967) 

On interpolation functions : II.

http://acta.bibl.u-szeged.hu/14179/1/math_029_fasc_001_002_091-092.pdf

On interpolation functions : II.

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