Proceedings of the International Symposium on Banach and Function Spaces Kitakyushu, Japan, October 2-4, 2003, pp. 83-120 TYPE, COTYPE AND CONVEXITY PROPERTIES OF QUASI–BANACH SPACES LECH MALIGRANDA Abstract. Results on quasi-Banach spaces, their type and cotype together with the convexity and concavity of quasi- Banach lattices are collected. Several proofs are included. Then the Lebesgue Lp, the Lorentz Lp,q and the Marcinkiewicz Lp,∞ spaces are the special examples. We review also several results of Kami´nska and the author on convexity, concavity, type and cotype of general Lorentz spaces Λp,w. Introduction The notions of type and cotype of Banach spaces emerged from the work of J. Hoffman-J¨orgensen, S. Kwapie´n, B. Maurey, B. Maurey and G. Pisier in early 1970’s. Since then, these notions have found frequent use in geometry of Banach spaces as well as in probability theory in Banach spaces. For the Banach lattices related notions of p-convexity, q-concavity and upper p-estimate, lower q-estimate appeared in Krivine paper and then in classical Lindenstrauss-Tzafriri book [30]. These concepts are natural even in quasi-Banach spaces where we have triangle inequality with a certain constant C ≥ 1 instead of C =1 or in p-Banach spaces (0 <p<1). Many definitions and classical results in Banach spaces can be translated in a natural way to definitions and results in quasi-Banach or p-Banach spaces. These were done initially in eighties by N. J. Kalton ([17, 18]). 2000 subject classification: Primary 46E30, 46B20; Secondary 46B42, 46B25. Key words and phrases: quasi-Banach spaces, type and cotype of quasi-Banach spaces, normability, quasi-Banach lattices, convexity and concavity of quasi-Banach lattices, rearrangement invariant spaces, Lorentz spaces, Marcinkiewicz spaces, Hardy operator, weights, weighted inequalities. 83 84 L. Maligranda We study properties and results on type, cotype, convexity and con- cavity of quasi-Banach spaces. Moreover, we will show how to use some p,q techniques for the classical Lorentz spaces L and Lorentz spaces Λp,w with 0 <p<∞ and arbitrary weight w, which were obtained in the papers [20] and [21]. 1. Quasi-Banach spaces All vector spaces will be real, although most arguments may be mod- ified to the complex case. A quasi-norm on a real vector space X is a map ·: X → [0, ∞) such that (1) x = 0 if and only if x =0, (2) ax = |a|x,a∈ R,x∈ X, (3) x + y≤C(x + y),x,y∈ X, where C ≥ 1 is a constant independent of x, y ∈ X. The smallest possible constant C = CX ≥ 1 is called the quasi-triangle constant of X = (X, ·). A quasi-norm induces a locally bounded topology on X and con- versely any locally bounded topology is given by a quasi-norm. If, in addition, we have for some 0 <p≤ 1, (4) x + yp ≤xp + yp for all x, y ∈ X, then a functional ·is called a p-norm. A complete quasi-normed (or p-normed) space is called a quasi-Banach space (or p-Banach space). Theorem 1.1 (completeness of quasi-normed spaces). A quasi- normed space X =(X, ·) with a quasi-triangle constant C ≥ 1 is complete (quasi-Banach space) if and only if for every series such that ∞ k ∞ k=1 C xk < ∞ we have k=1 xk ∈ X and ∞ ∞ k xk ≤ C C xk. k=1 k=1 Proof. Note that in a quasi-normed space X for natural numbers Type, cotype and convexity properties of quasi-Banach spaces 85 n>mwe have n n xk ≤ C xm + xk k=m k=m+1 n ≤ C xm + C xm+1 + xk k=m+2 n k−m+1 ≤ ...≤ C xk. k=m ∞ k Let X be a quasi-Banach space and assume that k=1 C xk = S< n ∞. For each n ∈ N let Sn = k=1 xk be the sequence of partial sums. Then n+m n+m k−n Sn+m − Sn = xk ≤ C xk k=n+1 k=n+1 n+m n+m n k k k ≤ C xk = C xk− C xk k=n+1 k=1 k=1 → S − S =0, as n, m →∞, imply that {Sn} is a Cauchy sequence, and so, convergent ∞ in X. Therefore, k=1 xk = x ∈ X. Moreover, n n ∞ k k xk ≤ C xk≤ C xk, k=1 k=1 k=1 ∞ k and so lim supn→∞ Sn≤ k=1 C xk from which we obtain ∞ k x≤C lim sup (x − Sn + Sn) ≤ C C xk. n→∞ k=1 Conversely, assume that (xn) is a Cauchy sequence in X. Then there is a −1 natural number n1 such that for all n>n1 we have xn −xn1 < (2C) . −2 We take n2 ∈ N,n2 >n1 such that for all n>n2 xn − xn2 < (2C) . Inductively we find an increasing sequence n1 <n2 <n3 <... such that − −k for all n>nk we have xn xnk < (2C) . Then (xnk ) is a subsequence of (xn) which satisfies 1 xn − xn < for k =1, 2,..., k+1 k (2C)k 86 L. Maligranda and ∞ ∞ k − ≤ −k ∞ xn1 + C xnk+1 xnk xn1 + 2 = xn1 +1< , k=1 k=1 which by the assumption is convergent in X, that is, N−1 − → ∈ SN−1 = xn1 + (xnk+1 xnk )=xnN y X, k=1 as N →∞. Since − ≤ − − → y xn C( y xnN + xnN xn ) 0, as n, N →∞it follows that (xn) converges to y ∈ X and so X is complete. 2 Two quasi-norms or p-norms ·and · are equivalent if there is a constant A ≥ 1 such that A−1x≤x ≤ Ax for all x ∈ X. We say that a quasi-Banach space X is p-normable (0 <p≤ 1) if there exists in X a p-norm equivalent to the quasi-norm in X or ≥ n ≤ equivalently if there is a constant B 1 such that k=1 xk n p 1/p B ( k=1 xk ) for all x1,x2,...,xn ∈ X. In the case when p =1we simply say that the space is normable. Clearly, any p-norm is a quasi-norm with C =21/p−1. Up to equiva- lence, the converse is also true by the following Aoki-Rolewicz theorem (Aoki [1] in 1942, Rolewicz [42] in 1957). Theorem 1.2 (Aoki-Rolewicz theorem). If (X, ·) is a quasi- normed space, then there is 0 <p≤ 1 and a p-norm · on X equivalent to ·. More precisely, if C is the quasi-triangle constant, then p is determined by C =21/p−1 and x ≤x≤21/px for all x ∈ X. Proof (cf. Gustavsson [11], Kalton-Peck-Roberts [19], Pietsch [37]). If (X, ·) is a quasi-normed space, then ⎧ ⎫ ⎨ n 1/p n ⎬ p ∈ ∈ N x = inf ⎩ xk : x = xk,xk X, n ⎭ k=1 k=1 Type, cotype and convexity properties of quasi-Banach spaces 87 1/p−1 is a p-norm equivalent to ·X, where 2 = C and C is a quasi- triangle constant of X. 2 The important class of quasi-Banach spaces which are not Banach p spaces is the class of L (µ) spaces for 0 <p<1 with the usual quasi- p 1/p norm fp =( Ω |f(t)| dµ) . In this case 1/p−1 f + gp ≤ 2 (fp + gp), i.e., the quasi-triangle constant of Lp(µ)is21/p−1. Also the Lorentz spaces Lp,q (0 <p,q≤∞) and Orlicz spaces Lϕ with not necessary convex Orlicz function ϕ are classical examples of quasi-Banach spaces. Several examples of quasi-norms we can produce already in R2 fol- lowing the special two examples given by Aoki [1] in 1942. Let X = R2 2 and for x =(x1,x2) ∈ R , 1 ≤ p ≤∞and a = 1 consider functionals p p 1/p xp =(|x1| + |x2| ) if x2 =0 , x = a|x1| if x2 =0. Then min(1,a)xp ≤x≤max(1,a)xp 1 and so · is a quasi-norm with the quasi-triangle constant C = max(a, a ) > 1. A linear operator T between quasi-Banach spaces X and Y is contin- uous if and only if it is bounded, and we put, as usual, T = sup{Tx : x≤1}. The standard results depending on Baire category, like the open mapping theorem, are valid also in the context of quasi-Banach spaces (with the same proof). On the other hand, quasi-normed spaces are not necessarily locally convex, and the Hahn-Banach theorem and results depending on it are in general false in this context. For exam- ple, the dual space to a quasi-Banach space can be trivial one, that is, X ∗ = {0}. 2. Type and cotype of quasi-Banach spaces Let rn :[0, 1] → R,n ∈ N, be Rademacher functions, that is, rn(t) = sign (sin 2nπt). The classical Khintchine inequality yields: for any 88 L. Maligranda 0 <p<∞ there are positive constants Ap,Bp such that 1/2 1/p 1/2 n 1 n p n 2 2 Ap |ak| ≤ akrk(t) dt ≤ Bp |ak| k=1 0 k=1 k=1 n for every choice of scalars {ak}k=1 and every n ∈ N. A quasi-Banach space X has type p (0 <p≤ 2), respectively cotype q (q ≥ 2) if there is a constant K>0 such that, for any choice of finitely many vectors x1,...,xn from X, 1 n n 1/p p rk(t)xk dt ≤ K xk , 0 k=1 k=1 respectively n n 1/q 1 q xk ≤ K rk(t)xk dt .