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
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 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 (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 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 Theorem 1.2 (Aoki-Rolewicz theorem). If (X, ·) is a quasi- normed space, then there is 0 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 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 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 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 n n 1/q 1 q xk ≤ K rk(t)xk dt . k=1 0 k=1 The best possible constant K is called the type p constant of X and denoted by Tp(X), and the corresponding cotype q constant of X is denoted by Cq(X). Let us collect properties of the type and cotype: • By Khintchine’s inequality the spaces R and C are both of type and cotype 2. • (Kwapie´n 1972). A Banach space X is isomorphic to a Hilbert space if and only if it has type 2 and cotype 2. • No quasi-Banach space X = {0} can be of type p>2 or of cotype q<2. 1 In fact, for x ∈ X with x = 1, let x1 = x2 = ... = xn = n x. Then 1 n 1 1 n xkrk(t) dt | rk(t)| dt 0 k=1 0 n k=1 ≈ 1/2−1/r n r 1/r = 1/r−1 n . ( k=1 xk ) n Now, estimation from above is possible for any n ∈ N only if r ≤ 2 and estimation from below only if r ≥ 2. Type, cotype and convexity properties of quasi-Banach spaces 89 • Any Banach space X is of type 1 and cotype ∞, that is, has trivial type and trivial cotype. In fact, by the triangle inequality we have 1 n 1 n n xkrk(t) dt ≤ xkrk(t) dt ≤ xk, 0 k=1 0 k=1 k=1 and 1 n xi = ri(t) xkrk(t) dt 0 k=1 1 n ≤ xkrk(t) dt, i =1, 2,...,n, 0 k=1 from which we obtain 1 n max xi≤ xkrk(t) dt. i=1,2,... ,n 0 k=1 2 • Any quasi-Banach space X is of cotype ∞. Let maxi=1,2,... ,n xi = x1. Then, for every t ∈ [0, 1], n n 2x1 = xkrk(t)+x1r1(t) − xkrk(t) k=1 k=2 n n ≤ C xkrk(t) + x1r1(t) − xkrk(t) . k=1 k=2 Integrating over t, 1 n 1 n 2x1≤C xkrk(t) dt + x1r1(t) − xkrk(t) dt 0 k=1 0 k=2 1 n =2C xkrk(t) dt. 0 k=1 2 90 L. Maligranda • If a quasi-Banach space X has type p [ cotype q], then it has type p1 ≤ p [cotype q1 ≥ q]. The type and cotype indices of a Banach space X defined in 1976 by Maurey-Pisier [35] we can defined for quasi-Banach space X by p(X) = sup{0 and if X has trivial cotype then q(X)=∞. • In the above definition of type or cotype we can replace an average 1/r 1 n 1 n r 0 k=1 rk(t)xk dt by the r-average 0 k=1 rk(t)xk dt for any 0 Theorem 2.1 (Kahane inequality). If X is a quasi-Banach space and 0 {f ∈ L0(X):|{t ∈ [0, 1] : f(t)≥a}| ≤ b},a,b≥ 0. It is clear that the topology induced by ·p is stronger than τ.In the proof of the Kahane’s inequality it was established that the opposite Type, cotype and convexity properties of quasi-Banach spaces 91 relation holds on Rad(X). In fact, it was showed that for every neighbor- hood of zero {f ∈ Rad(X):(f≥a)|≤b} of τ, and every 0 Theorem 2.3. (a) If 0 Proof. (a) Let 0 n p/2 2 = |xk(ξ)| dµ(ξ) Ω k=1 n p ≤ |xk(ξ)| dµ(ξ) Ω k=1 n p = xkp, k=1 p p p that is L has type p and Tp(L ) = 1. To show that L has cotype 2 we use H¨older-Rogers∗ inequality, Fubini theorem, classical Khintchine inequality for numbers and Minkowski inequality p/2 1 n 2 1 n p xkrk(t) dt ≥ xkrk(t) dt 0 k=1 p 0 k=1 p 1 n p = xk(ξ)rk(t) dµ(ξ) dt 0 Ω k=1 1 n p = xk(ξ)rk(t) dt dµ(ξ) Ω 0 k=1 n p/2 p 2 ≥ Ap |xk(ξ)| dµ(ξ) Ω k=1 n 2/p p/2 p p ≥ Ap |xk(ξ)| dµ(ξ) k=1 Ω n p/2 p 2 = Ap xkp , k=1 p p and L has cotype 2 with C2(L ) ≤ 1/Ap. Now, let 2 ∗The classical H¨older inequality should historically correctly be called the H¨older- Rogers inequality (cf. [34]). Type, cotype and convexity properties of quasi-Banach spaces 93 and subadditivity of the Lp/2-norm we obtain 2/p 1 n 2 1 n p xkrk(t) dt = xk(ξ)rk(t) dµ(ξ) dt 0 k=1 p 0 Ω k=1 2/p 1 n p ≤ xk(ξ)rk(t) dµ(ξ)dt 0 Ω k=1 2/p 1 n p = xk(ξ)rk(t) dtdµ(ξ) Ω 0 k=1 ⎡ ⎤2/p n p/2 ⎣ p 2 ⎦ ≤ Bp |xk(ξ)| dµ(ξ) Ω k=1 n 2/p n 2 p 2 2 ≤ Bp |xk(ξ)| dµ(ξ) = Bp xkp, k=1 Ω k=1 p p and L has type 2 with T2(L ) ≤ Bp. Similarly, using the Fubini theorem, the imbedding Lp[0, 1] → L2[0, 1] and the orthonormality of Rademachers we have 1 n 1 n p p xkrk(t)pdt = | xk(ξ)rk(t)| dµ(ξ) dt 0 k=1 0 Ω k=1 1 n p = | xk(ξ)rk(t)| dtdµ(ξ) Ω 0 Ω k=1 p/2 1 n 2 ≥ | xk(ξ)rk(t)| dt dµ(ξ) Ω 0 k=1 n p/2 2 = |xk(ξ)| dµ(ξ) Ω k=1 n n p p ≥ |xk(ξ)| dµ(ξ)= xkp, Ω k=1 k=1 p p and L has cotype p with Cp(L )=1. p If dim L (µ)=∞, then there are disjoint sets A1,A2,... such that 0 <µ(Ak) < ∞,k =1, 2,.... Put χAk (ξ) xk(ξ)= 1/p . (µ(Ak) 94 L. Maligranda Now, if 0 1/p 1 n p xkrk(t)pdt 0 k=1 1/p−1/r = n →∞, n r 1/r k=1 xkp as n →∞. Thus r ≤ p. Similarly with the other cases. 2 We can give a nice application of the type-cotype result for Lp spaces and the fact that “being of type-p or cotype-q” is invariant under iso- morphism. Two quasi-Banach spaces X, Y are called isomorphic if there is a linear continuous mapping T : X → Y which is onto Y, that is, T (X)=Y (then T −1 is also continuous). Theorem 2.4. If Lp(µ),Lq(ν) are infinite dimensional and isomor- phic spaces, then p = q. Proof. Assume that p = q, for example p (i) 0 Remark 2.5. Note that the question when the spaces Lp(µ) and Lp(ν) are isomorphic is not so easy problem. It is well-known that: (a) L2[0, 1] is isometric to l2; (b) L∞[0, 1] is isomorphic to l∞ by a Pelczy´nski result (1958); (c) If p =2 , 1 ≤ p<∞, then Lp[0, 1] is not isomorphic to lp. In particular, we have that the spaces L1[0, 1] and l1 are not isomorphic but their duals L∞[0, 1] and l∞ are isomorphic. Type, cotype and convexity properties of quasi-Banach spaces 95 Remark 2.6. If X0 is a closed subspace of X and X has type p (or cotype q), then X0 has type p (or cotype q) and the quotient space X/X0 has type p but not necessarily cotype q. Namely, every Banach 1 1 1 space X is isomorphic to quotient of L , that is, X L /X0 and L has cotype 2. ∞ ∞ Spaces c0,l ,L [0, 1] or any quasi-Banach space containing an iso- morphic copy of one of these spaces have type 1 and cotype ∞. Of course, p-normable space has type p, but also conversely for 0 < p<1. Theorem 2.7 (Kalton theorem). (a) If a quasi-Banach space X has type p for some 0 Remark 2.8. The important numbers in the proof of the above theorem for a quasi-Banach space X and n ∈ N are (cf. [15] and [17]) bn(X) = sup inf σ1x1 + ···+ σnxn. σ =±1 xk≤1 k For the Banach space X the numbers bn(X) are called the measure of B-convexity of a Banach space X since X is a B-convex space if and only if bn(X) b2(X) = sup {min(x + y, x − y): x≤1, y≤1} is called the James constant J(X) of a Banach space X. This constant was investigated in details by Kato-Maligranda-Takahashi [27] and Kato- Maligranda [26]. The next observation is a theorem of the Orlicz type. Orlicz proved it in 1933 for Lp spaces (1 ≤ p<∞) which are of cotype max(p, 2). Theorem 2.9. If a quasi-Banach space X has cotype q (2 ≤ q< ∞ ∞ ) and n=1 xn is an unconditionally convergent series in X, then ∞ q n=1 xn < ∞. 96 L. Maligranda Proof. We have, for every n ∈ N, n 1 n q 1/q ( xk ) ≤ Cq(X) rk(t)xk dt k=1 0 k=1 n ≤ Cq(X) sup rk(t)xk t∈[0,1] k=1 n ≈ Cq(X) sup εkxk ≤ C. 2 ε =±1 k k=1 3. Type and cotype of Banach spaces We collect here properties of type and cotype which are mainly true in the class of Banach spaces. We say that the space has trivial type or trivial cotype respectively, if it does not have any type bigger than one or any finite cotype, respec- tively. There are well known criteria due to Pisier describing non-trivial type or finite cotype. The following definition is needed. p A Banach space X contains ln uniformly (1 ≤ p ≤∞) if there is A>0 such that, for all n ∈ N, there are points x1,x2,...,xn ∈ X with n 1/p n n 1/p 1 p p |ak| ≤ akxk ≤ A |ak| A k=1 k=1 k=1 n for all scalars {ak}k=1. Theorem 3.1 (Pisier theorem, 1982). A Banach space has non- 1 trivial type p>1(finite cotype) if and only if it does not contain ln s ∞ (respectively ln s) uniformly. Equivalently, a Banach space X contains 1 ∞ ln s (respectively ln s) uniformly if and only if p(X)=1(respectively q(X)=∞). For the proof see [3, pp. 313-317] or [9, pp. 260-267] or [4, pp. 443-444]. 1 It is known that a Banach space contains no isomorphic copy of ln ∗ 1 uniformly if and only if its dual X contains no isomorphic copy of ln uniformly. Thus p(X) > 1 ⇐⇒ p(X ∗) > 1. Type and cotype are dual notions under certain assumption. Type, cotype and convexity properties of quasi-Banach spaces 97 Theorem 3.2 (duality). If a Banach space X has type p, then its ∗ ∗ 1 1 dual space X has cotype p and Cp (X ) ≤ Tp(X), where p + p =1. Remark 3.3. The converse of theorem 3.2 is false, since l1 has cotype 2 but the dual [l1]∗ = l∞ has type 1. Moreover, this theorem is not true for the cotype since l1 has cotype 2 but its dual l∞ has type 1 not type 2. The last implications are true if space X has nontrivial type p>1. Theorem 3.4 (Pisier, 1982). Let a Banach space X has non-trivial type, i.e., p(X) > 1. Then, for 1 1 1 1 1 + = + =1. p(X) q(X ∗) p(X ∗) q(X) Observe that p(X) > 1 implies q(X) < ∞. Another words, if a Banach space X has a nontrivial type then it has also a nontrivial cotype. Theorem 3.5 (Maurey-Pisier 1976, Krivine 1976). Let X be an p infinite dimensinal Banach space. Then X contains ln’s uniformly for p both p = p(X) and p = q(X) and X does not contain uniformly ln for q p In 1986 Bastero-Uriz [2] proved that any r-Banach space (0 4. Convexity and concavity of quasi-Banach lattices A quasi-Banach space (X, ·) which in addition is a vector lattice and x≤y whenever |x|≤|y| is called a quasi-Banach lattice. The H¨older-Rogers inequality in quasi-Banach lattices X is of im- portance (in Banach lattices see [28, pp. 42–43], [30, pp. 43–44] and in p-Banach lattices see [41, Lemma 15]). In the case of X = L1 we have classical H¨older-Rogers inequality. Theorem 4.1. Let X be a quasi-Banach lattice. Then, for every 0 <θ<1 and every x, y ∈ X, we have that |x|1−θ|y|θ ∈ X and |x|1−θ|y|θ≤Cx1−θyθ, 98 L. Maligranda where C ≥ 1 is a quasi-triangle constant of X. If we known, in addition, that X is a p-Banach lattice (0 |x|1−θ|y|θ≤max(1, 21/p−1)x1−θyθ. Proof. For any ε>0 we have |x|1−θ|y|θ =(ε1/(1−θ)|x|)1−θ(ε−1/θ|y|)θ ≤ (1 − θ)ε1/(1−θ)|x| + θε−1/θ|y|, and so |x|1−θ|y|θ ∈ X. Assume now that x = 0 (otherwise is nothing to y θ(1−θ) prove). For ε =(x ) we have |x|1−θ|y|θ = (ε1/(1−θ)|x|)1−θ(ε−1/θ|y|)θ ≤(1 − θ)ε1/(1−θ)|x| + θε−1/θ|y| ≤ C (1 − θ)ε1/(1−θ)x + θε−1/θy = Cx1−θyθ. In the case of p-Banach lattice with 0 |x|1−θ|y|θ = (ε1/(1−θ)|x|)1−θ(ε−1/θ|y|)θ ≤(1 − θ)ε1/(1−θ)|x| + θε−1/θ|y| 1/p ≤ (1 − θ)pεp/(1−θ)xp + θpε−p/θyp p p1/p = (1 − θ)p x1−θyθ + θp x1−θyθ 1/p =[(1− θ)p + θp] x1−θyθ ≤ 21/p−1x1−θyθ. 2 A quasi-Banach lattice X or its quasi-norm is said to be p-convex (0 Theorem 4.2. Let X be a quasi-Banach lattice. Given 0 n 1/r n 1/p n 1/q r r p q |xk| |yk| ≤ |xk| |yk| k=1 k=1 k=1 n n for any {xk}k=1, {yk}k=1 ⊂ X. Hence 1/r n r |xk| k=1 1/r n (1−r/p)r (r/p−1)r r = xk xk |xk| k=1 1/q 1/p n n (1−r/p)q (r/p−1)p p ≤ xk xk |xk| k=1 k=1 100 L. Maligranda n 1/q n 1/p r (r/p−1) p ≤ K(p)(X) xk xk |xk| k=1 k=1 n 1/q n 1/p r r = K(p)(X) xk xk k=1 k=1 n 1/r r = K(p)(X) xk , k=1 and so X is r-convex with K(r)(X) ≤ K(p)(X). 1 1 1 Recall that for r>p, there exists q<0 such that r = p + q and the reverse H¨older-Rogers inequality holds n 1/r n 1/p n 1/q r r p q |xk| |yk| ≥ |xk| |yk| k=1 k=1 k=1 n n for any {xk}k=1, {yk}k=1 ⊂ X. Therefore 1/r n r |xk| k=1 1/r n (1−r/q)r (r/q−1)r r = xk xk |xk| k=1 1/p 1/q n n (1−r/q)p (r/q−1)q q ≥ xk xk |xk| k=1 k=1 n 1/p n 1/q ≥ r 1 r/q−1| |q xk (q) xk xk k=1 K (X) k=1 n 1/p n 1/q 1 r r = (q) xk xk K (X) k=1 k=1 n 1/r 1 r = (q) xk , K (X) k=1 and so X is r-concave and K(r)(X) ≤ K(q)(X). Suitable relations among upper and lower estimates can be shown in a similar way. 2 Type, cotype and convexity properties of quasi-Banach spaces 101 Theorem 4.3. If 0 n n 1/r 1/r r r (1) |xk| ≤ xkp and k=1 p k=1 n n 1/s 1/s s s xkp ≤ |xk| . k=1 k=1 p p p p Moreover, if dim L (µ)=∞ then pc(L (µ)) = qc(L (µ)) = p. s Proof. If 0 n n 1/q p s |xk(ξ)| |ak|dµ ≤{ak}q |xk(ξ)| dµ Ω k=1 Ω k=1 n 1/s p s = {ak}q |xk| . k=1 p Taking the supremum over all {ak} such that {ak}q≤1 we obtain by the Landau theorem n p sup |xk(ξ)| |ak|dµ : {ak}q ≤1 Ω k=1 n n p = sup |ak| |xk(ξ)| dµ : {ak}q ≤ 1 k=1 Ω k=1 n q 1/q p p = |xk(ξ)| dµ = |xk(ξ)| dµ Ω q k=1 Ω n 1/q n p/s s s = xkp = xkp . k=1 k=1 n s 1/s n s 1/s Thus ( k=1 xkp) ≤( k=1 |xk| ) p. 102 L. Maligranda r If 0 If a quasi-Banach lattice (X, ·X) is normable, then n n x = inf xkX : x = xk,xk ∈ X, n ∈ N k=1 k=1 is a lattice norm in X equivalent to ·X . It is clear that · is a norm and x≤xX ≤ Cx for all x ∈ X. The norm ·is also monotone, that is, if x, y ∈ X and |x|≤|y|, then x≤y which follows from the Riesz decomposition property. If a quasi-norm ·X is 1-convex, then the above formula on · defines in X an equivalent norm to ·X. Thus a quasi-Banach space (X, ·X) is normable if and only if it is 1-convex. We also observe that for 0 Theorem 4.4. Let X be L-convex quasi-Banach lattice. (a) If X satisfies an upper p-estimate, then X is r-convex for every 0 < n | |r 1/r≤ r 0 so that ( k=1 xk ) n p 1/p C( k=1 xk ) . (b) If X satisfies a lower p-estimate, then X is r-concave for every p For a given quasi-Banach lattice X we define two types of convexity and concavity indices as follows (cf. [25]): pc(X) = sup{p>0:X is p-convex}, qc(X) = inf{q>0:X is q-concave}, pd(X) = sup{p>0:X satisfies an upper p-estimate}, qd(X) = inf{q>0:X satisfies a lower q-estimate}. The indices pd(X) and qd(X) were introduced by T. Shimogaki in 1965 for order complete Banach lattices, by J. Grobler in 1975 for Banach function spaces and in 1977 by P. Dodds for general Banach lattices (cf. [47] for suitable references). Obviously, pc(X) ≤ pd(X) ≤ qd(X) ≤ qc(X), and by the Aoki-Rolewicz theorem, pd(X) > 0. Also pc(X)=pd(X)if and only if X is L-convex. Moreover, if X is L-convex then qc(X)= 104 L. Maligranda qd(X). Kalton [18] example of a quasi-Banach lattice which is not L- convex shows that 0 = pc(X) Theorem 4.5. Let 1 ∗ p p ∗ p ∗ p ∗ X [ln ]=(X[ln]) → (ln[X]) = ln [X ], which again is equivalent to p-concavity of X ∗. 2 There are close connections between convexity, upper estimate and type, and respectively between concavity, lower estimate and cotype of Banach lattices. The diagram in Lindenstrauss-Tzafriri book [30] on page 100 presents in details all these relations for Banach lattices. Theorem 4.6. Let X be L-convex quasi-Banach lattice. Then (i) X is p-convex, 1 The results of (ii) and (v) follow from the extension of the Maurey (1974) result to a quasi-Banach lattices: if X is a quasi-Banach lattice which is p-convex for some p>0 and q-concave for some q<∞, then n n 1/2 1 Ap 2 |xk| ≤ xkrk(t) dt K(p)(X) k=1 0 k=1 n 1/2 (q) 2 ≤ BqK (X) |xk| k=1 for all x1,x2,...,xn ∈ X. 5. Convexity, concavity, type and cotype of Lorentz Lp,q-spaces Let I =[0, ∞)or[0, 1] and 0 ∞ 1/q 1/p ∗ q dt xp,q = [t x (t)] , 0 t where x∗ is a decreasing rearrangement of |x| defined by ∗ x (t) = inf{λ>0:df (λ) ≤ t},df (λ)=|{s ∈ I : |x(s)| >λ}|. Lorentz spaces Lp,q have the following properties: • Lp,p = Lp with equality of the quasi-norms, • 1 ≤ q ≤ p =⇒·p,q is a norm, ∗∗ • 1 Theorem 5.1. (a) If 0 (b) If 0 Proof. Consider on I =(0, ∞) a sequence of functions n χ[i−1,i) xk = 1/p ,k=1, 2,...,n. i=1 [(i + k)modn] n p 1/p n 1 1/p Then ( k=1 |xk| ) =( k=1 k ) χ[0,n) and so n n 1/p 1/p p 1 1/p 1/p 1/p |xk| = n ≈ n (lnn +1) . k=1 p,q k=1 k ∗ n χ[i−1,i) On the other hand, xk = i=1 i1/p and so 1/p 1/p n n q/p − − q/p p p i (i 1) ≈ 1/p 1/q xk p,q = n q/p n (lnn) . k=1 q i=1 i Thus n p 1/p 1/q ( k=1 xk p,q) ≈ (lnn) → ∞ n p 1/p 1/p 0or ( k=1 |xk| ) p,q (lnn +1) as n →∞(when p The Marcinkiewicz spaces Lp,∞ or the weak Lp-spaces (for I =[0, ∞) or [0, 1] and 0 • 1 Theorem 5.2 (normability). The Lorentz spaces Lp,q for 0 Then xkp,q = 1 and since yn are decreasing functions we also have for p = q, ∞ q q 1/p ∗ dt ynp,q = t yn(t) 0 t −np n q q 2 1 = tq/p−1 2k dt p 0 n k=1 n −(n−m)p n−m q q 2 1 + tq/p−1 2k dt −(n−m+1)p p m=1 2 n k=1 q q 2n+1 − 2 2n−m+1 − 2 = + 1 − 2−q n2n n2n−m Type, cotype and convexity properties of quasi-Banach spaces 109 q n q 2 − 21−n 2 − 21−(n−m) = + 1 − 2−q n m=1 n q n q 2 − 21−n 2 − 21−(n−m) ≥ + (1 − 2−q) n m=1 n q − 1−n q q n 2 2 2 − −q − 1 = + q (1 2 ) 1 n−m n n m=1 2 q q 2 − 21−n 2q − 1 n(n +1) = + − 2+21−n →∞ n nq 2 as n →∞. Therefore, n k=1 xkp,q n = ynp,q →∞as n →∞. k=1 xkp,q p,p For p = q we immediately have that yn ∈ L . (b) For 0 n n xk(t)= χ[ i−1 , i )(t) 1/p n n i=1 [1 + (i + k)modn] for all t ∈ [0, 1] and k =1, 2,...,n. Then n n n xk(t)= 1/p χ[0,1)(t), k=1 k=1 k and for 1 ≤ q<∞ and q = ∞ we obtain n 1/q n n n p n n xk = 1/p and xk = 1/p , k=1 p,q q k=1 k k=1 p,∞ k=1 k respectively. Moreover, for 1 ≤ q<∞, n i q q n n q/p−1 xkp,q = t dt i−1 i1/p i=1 n n q q/p − q/p p n i − i 1 = 1/p q i=1 i n n 110 L. Maligranda n q/p − − q/p n p q−q/p i (i 1) ≤ q−q/p 1 = n q/p n , q i=1 i i=1 i and so 1/q 1/q n p n n p n 1 xk q k=1 1/p q k=1 1/p k=1 p,q k ≥ k n = n 1/q k=1 xkp,q nxkp,q 1−1/p 1 n i=1 i 1/q n 1/p−1 ≥ p 1 n →∞ →∞ 1/p 1/q as n . q k=1 k (1 + ln n) If q = ∞, then n 1/p ∗ n xk p,∞ = sup t x (t) = sup 1/p tχ[ i−1 , i )(t) t>0 t>0 i n n ⎧ i=1 ⎫ ⎨ ⎬ n = max sup t1/p = n1−1/p, i=1,... ,n ⎩ 1/p ⎭ i−1 1/q n 1−1/q 1 1 = →∞as n →∞, q k=1 k and for q = ∞, n n n n k=1 xk 1,∞ k=1 k 1 n = = →∞as n →∞, k1,∞ k=1 x n k=1 k Type, cotype and convexity properties of quasi-Banach spaces 111 and this completes the proof of the theorem. 2 In the further investigations of weak Lp-spaces important is so called Kolmogorov-Cotlar equivalence of the weak Lp quasi-norms. Theorem 5.3. If 0 1/p−1/q xp,∞ ≈ sup xχAq m(A) . A⊂I,0 Proof. First, it is easy to see that 1/p xp,∞ = sup λdx(λ) , λ>0 where dx(λ)=m({t ∈ I : |x(t)| >λ}). Denote 1/p−1/q C = sup xχAqm(A) A⊂I,0 Aλ = {t ∈ I : |x(t)| >λ},λ>0. Then ≥ 1/p−1/q C xχAλ qm(Aλ) ∞ 1/q ∗ q 1/p−1/q = (xχAλ ) (s) ds m(Aλ) 0 1/q m(Aλ) q 1/p−1/q ≥ λ ds m(Aλ) 0 1/p 1/p = λm(Aλ) = λdx(λ) . Since λ>0 was arbitrary it follows that xp,∞ ≤ C. On the other hand, let A ⊂ I with 0 −1/p xp,∞m(A) , then ∞ q q−1 xχAq = qλ m({t ∈ I : |xχA(t)| >λ}) dλ 0 a ∞ q−1 = + qλ m({t ∈ I : |xχA(t)| >λ}) dλ 0 a a ∞ ≤ q−1 q−1 x p,∞ m(A) qλ dλ + qλ p dλ 0 a λ q q p−q p = m(A)a + a xp,∞ q − p 1−q/p q q 1−q/p q = m(A) xp,∞ + m(A) xp,∞ p − q p 1−q/p q = m(A) xp,∞. p − q Hence, 1/q 1/p−1/q p xχAqm(A) ≤ xp,∞, p − q and so 1/q q xp,∞ ≥ 1 − C. p We proved the estimates 1/q q 1/p−1/q 1 − sup xχAqm(A) ≤xp,∞ p A⊂I,0 1/p−1/q ≤ sup xχAqm(A) . A⊂I,0 As a corollary from the above theorems and earlier theorems we obtain equalities for the indices: p,q p,q pc(L ) = min(p, q),qc(L ) = max(p, q), Type, cotype and convexity properties of quasi-Banach spaces 113 and p(Lp,q) = min(p, q, 2),q(Lp.q) = max(p, q, 2). 6. Convexity, concavity, type and cotype of Lorentz spaces Λp,w Let I =[0, ∞). A weight function w will be a non-negative locally t integrable function on I and W (t)= 0 w(s)ds for all t ∈ I. The Lorentz space Λp,w for 0 ∞ 1/p ∗ p xΛp,w = xp,w = x (t) w(t) dt < ∞. 0 The properties of Lorentz set Λp,w rather than space (since sometimes it is not a linear space) started in papers by Lorentz (1951) and Haaker (1970), and were investigated in details in [20], [21] and [6]. Theorem 6.1. (a) The space Λp,w is linear if and only if either (i) W satisfies the ∆2-condition (∆2-condition for all arguments), that is there exists K>0 such that W (2t) ≤ KW(t), for all t>0. or (ii) W (t)=0on some interval (0,a) and W satisfies the ∆2-condition in some interval (b, ∞). (b) The functional ·p,w is a quasi-norm on Λp,w if and only if W satisfies condition ∆2. (c) The functional ·p,w is a norm on Λp,w if and only if w is decreasing and 1 ≤ p<∞. (d) If W satisfies condition ∆2, then (Λp,w, ·p,w) is a rearrange- ment invariant quasi-Banach function lattice with the Fatou property. Moreover, W (∞)=∞ if and only if ·p,w is order continuous. We assume later on that a weight w is a positive function on I such t ∞ that W (t)= 0 w(s)ds < ∞ for every t>0,W(∞)= 0 w(s)ds = ∞ and W satisfies ∆2-condition (we denote this simply by writing W ∈ ∆2). Otherwise, space is either {0} or contains an isomorphic copy of l∞. Note that two Lorentz spaces Λp,w1 and Λp,w2 coincide and the cor- responding quasi-norms are equivalent if and only if the functions W1 114 L. Maligranda and W2 are equivalent. This follows from the another description of the ∞ p−1 1/p quasi-norm x Λp,w = p 0 W (dx(λ)) λ dλ . The normability conditions are due to Boyd (1967), Sawyer (1990), Ari˜no-Muckenhoupt (1990) and Raynaud (1992) [cf. [20] and [21] for the proofs]: Theorem 6.2. Let 1 (i) Λp,w is normable. ∗∗ 1 t ∗ (ii) The Hardy operator x (t)= t 0 x (s)ds is bounded in Λp,w. (iii) The weight w satisfies condition Bp, that is, there exists B>0 such that ∞ s−pw(s) ds ≤ Bt−pW (t) for all t>0. t (iv) An upper index of W satisfies β(W ) Theorem 6.3 (Kami´nska-Maligranda theorem). If 0 Theorem 6.4. Let 0 p/r (a) Λp,w satisfies an upper r-estimate ⇐⇒ W (t)/t is pseudo- decreasing and r ≤ p. If in addition 0 Theorem 6.5. Let 0 (i) Λp,w is r-convex. (r) 1 t ∗ r 1/r (ii) The Hardy operator H x(t)=(t 0 x (s) ds) is bounded in (r) Λp,w, that is, the quasi-norms xp,w and H xp,w are equiva- lent. (iii) The weight w satisfies condition Bp/r, that is, for some B>0 ∞ s−p/rw(s) ds ≤ Bt−p/rW (t) for all t>0. t (iv) β(W ) < p/r, or equivalently, for some >0, W (t)/tp/r− is pseudo-decreasing. II.Ifr = p, then the following assertions are equivalent: (i) Λp,w is p-convex. (ii) Λp,w satisfies an upper p-estimate. (iii) W (t)/t is pseudo-decreasing. (iv) There exists a decreasing weight w0 such that Λp,w =Λp,w0 and ·p,w and ·p,w0 are equivalent. Moreover, for 0 (v) Λp,w is p-normable. III.Ifr>p, then Λp,w is not r-convex. Remark 6.6. (a) If 0 so β(W ) < p/r for some r>0. Hence in view of Theorem 2, Λp,w is r-convex and thus L-convex. Theorem 6.7. Let 0 (i) Λp,w is r-concave. 1 ∞ ∗ r 1/r (ii) The Hardy operator H(r)x(t)=(t t x (s) ds) is bounded in Λp,w. (iii) α(W ) > p/r, or equivalently, W (t)/tp/r+ is pseudo-decreasing for some >0. 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Wojtaszczyk, Banach Spaces for Analysts, Cambridge Univ. Press, Cambridge 1991. [47] A. C. Zaanen, Riesz Spaces II, North-Holland, Amsterdam 1983. Lech Maligranda Department of Mathematics Lule˚a University of Technology SE-971 87 Lule˚a, Sweden E-mail address: [email protected]