CC BY
9
Владикавказский математический журнал июль-сентябрь, 2007, Том 9, Выпуск 3
UDC 517.98
HOLDER TYPE INEQUALITIES FOR ORTHOSYMMETRIC BILINEAR OPERATORS1
A. G. Kusraev
Dedicated to the memory of G. Ya. Lozanovskii
An interplay between squares of vector lattice and homogeneous functional calculus is considered and
Holder type inequalities for orthosymmetric bilinear operators are obtained.
Mathematics Subject Classification (2000): 47A63, 47B65, 47A60.
Key words: orthosymmetric bilinear operator, lattice bimorphism, homogeneous functional calculus,
squares of vector lattices, Holder type inequalities.
1. Introduction
The homogeneous functional calculus on vector lattices is a useful tool in a variety of areas. One of the interesting application is the study of powers of Banach lattices initiated by G. Ya. Lozanovskii [18]. Recently G. Buskes and A. van Rooij [8] introduced the concept of squares of Archimedean vector lattices which allows to represent orthoregular bilinear operators as linear regular operators. In particular, it is proved in [9] that the square of a relatively uniformly complete vector lattice can be constructed by well known p-convexification procedure (with p = 1/2) which is also based on the homogeneous functional calculus, see [17, 22].
The aim of this paper is to consider some interplay between squares of vector lattices and homogeneous functional calculus and obtain Holder type inequalities for orthosymmetric bilinear operators. We also collect several useful facts concerning homogeneous functional calculus on relatively uniformly complete vector lattices some of which despite of their simplicity does not seem appeared in the literature.
There are different ways to introduce the homogeneous functional calculus on vector lattices, see [6, 13, 17, 19, 21, 22]. We follow the approach [6, 9] going back also to G. Ya. Lozanovskii [19]. For the theory of vector lattices and positive operators we refer to the books [2] and [14]. All vector lattices in this paper are real and Archimedean.
1.1. We start by recalling some definitions and results from [7]. Let E and G be vector lattices. A bilinear operator b : E x E ^ G is said to be orthosymmetric if |x| A |y| =0 implies b(x, y) = 0 for arbitrary x, y £ E, see [8]. If b(x, y) ^ 0 for all 0 ^ x £ E and 0 ^ y £ E, then b is named 'positive. The difference of two positive orthosymmetric bilinear operators is called orthoregular. Denote by BLor (E; G) the space of all orthoregular bilinear operators from E x E
© 2007 Kusraev A. G.
1The author is supported by a grant from Russian Foundation for Basic Research, project № 06-01-00622.
to G ordered by the cone of positive orthosymmetric operators. Aspects of orthosymmetric bilinear operators are presented in [7, 15, 16].
We say that a bilinear operator b is symmetric if b(x, y) = b(y, x) for all x, y G E. The following important property of orthosymmetric bilinear operators was established in [8, Corollary 2].
An orthosymmetric positive bilinear operator is symmetric.
A bilinear operator b is said to be lattice bimorphism if the mappings y ^ b(e, y) (y G F) and x ^ b(x, f) (x G E) are lattice homomorphisms for all 0 ^ e G E and 0 ^ f G F. For a lattice bimorphism the converse is also true, see [7, Proposition 1.7].
A lattice bimorphism is orthosymmetric if and only if it is symmetric.
1.2. For an arbitrary vector lattice E there exists a vector lattice E0 and an orthosymmetric lattice bimorphism 0 : (x, y) ^ x 0 y from E x E to E0 such that the following universal property holds: whenever b is a symmetric lattice bimorphism from E x E to some vector lattice F, there is a unique lattice homomorphism $5 : E0 ^ F with b = $60.
The pair (E0, 0) is essentially unique, i.e. if for some vector lattice E® and symmetric lattice bimorphism @ : E x E ^ E® the pair (E®, @) obeys the said universal property, then there exists a lattice isomorphism 1 from E0 onto E® such that 10 = @ (and, of course, l-1@ = 0).
The vector lattice E0 and the lattice bimorphism 0 are called the square of E and the canonical bimorphism, respectively. The following result claims that the mentioned universal property of squares can be essentially extended, see [7, Theorem 3.1] and [9, Theorem 9].
1.3. Let E and G be vector lattices with G relatively uniformly complete. Then for every bilinear orthoregular operator b : E x E ^ G there exists a unique linear regular operator $5 : E0 ^ G such that
b(x,y)=$6(x 0 y) (x,y G E).
The correspondence b ^ $5 is an isomorphism of the ordered vector spaces BLor (E, G) and Lr (E0, G).
2. Homogeneous functions on vector lattices
In this section we introduce homogeneous functional calculus on relatively uniformly complete vector lattices and state some useful facts.
2.1. Denote by H(RN) the vector lattice of all continuous positively homogeneous functions f : RN ^ R. In accordance with [6] we say that f (xi,... ,xn ) exists in E and write y = f (xi,..., xn) if there is an element y G E such that u(y) = f (u(xi),..., )) for every R-valued lattice homomorphism u on the sublattice of E generated by {xi,... ,xn, y}. The definition is correct in the sense that if L is any vector sublattice of E containing {x1,... ,xn , y} and u(y) = f (u(x1),...,u(xN)) (u G fi) for some point separating set fi of R-valued lattice homomorphisms on L, then y = f (xi,... ,xn). It is immediate from the definition that f (x,..., x) = xf (1,..., 1) whenever 0 ^ x G E. Define dtj G H(RN) by dtj (ti ,...,tN) = tj (j := 1,...,N).
2.2. Theorem. Let E be a relatively uniformly complete vector lattice and xi,..., xn G E. Then f (xi,... ,xn) exists for any f G H(RN) and the mapping
f ^ f(X1,...,XN) f G H))
is a unique lattice homomorphism from H(RN) into E with dtj(xi,...,xn) = xj (j : = 1,...,N).
In particular, if f,g £ H(RN) and f ^ g, then f(x1,...,xn) ^ g(x1,...,xN) for all (xi,... ,xn) £ EN. Moreover, the inequality holds:
N
|f (xi,...,xN )| < ||f H V |xj I, j=1
where ||f || := sup{f (ti,..., ¿n) : (ti,..., ¿n) £ RN, maxj 11j| = 1}.
2.3. Let K, M, N £ N and consider positively homogeneous functions fi,...,fM £ H (RN ) and gi,...,gK £ H (RM). Denote f := (fi,...,fM) and g := (gi,..., g*). Then gi o f,... ,gK o f £ H(RN) and, for any x = (xi;... ,xN) £ EN and y = (yi;..., yM) £ EM, the elements f (x) := (fi (x),..., fm(x)) £ EM and g(y) := (gi(y),... (y)) £ EK are well defined. Moreover,
(g o f)(x) = g(f (x)) (x = (xi,..., xn) £ EN). In particular, if N = M = K and g = f-i, then
f-i(f (x))= x, f (f-i(y)) = y (x,y £ EN).
We define also fi x gi : EN+M ^ E2 by (fi x gi)(x,y):= (fi(x),gi(y)).
2.4. Let E and F be relatively uniformly complete vector lattices, h : E ^ F be a lattice homomorphism, xi,...,xn £ E, and f £ H(RN). Then
h(f (xi,... ,xn)) = f (h(xi),..., h(xN))).
If E is a relatively uniformly complete vector sublattice of F containing xi,..., xn £ F and h is the inclusion map E ^ F, then f (xi,...,xn) relative to F is contained in E and its meaning relative to E is the same.
2.5. Assume that f £ H(RN) possesses the following property:
(Vti,...,tN £ R) tit2 ■ ... ■ ¿n = 0 ^ f (ti,t2,...,tN) =0.
Then for any u,xi,...,xn £ E and fixed integer 1 ^ k ^ N we have
xfc ± u ^ f (xi,..., xfc_i, xfc, xfc+i,..., xn) ^ u.
Moreover, for any band L C E there holds f (xi,...,x^^x^, x&+i,...,xn) £ L whenever xk £ L. If L admits a band projection n, then
nf (xi, . . . ,xfe-i,xfe ,xfe+i,... ,xn ) = f (xi,.. .,xk-i,nxk ,xfe+i . . . ,xn ). Now, we consider concrete examples of homogeneous functions.
2.6. Homogeneous functional calculus is used to introduce the so called p-convexification and p-concavification procedures for a Banach lattice, see [17, 22]. Consider three functions aa>N, a'a,N : RN ^ R, and J : R2 ^ R defined by
(ti, . . . , tN) := 0-i(0a(ti) + ... + 0a (¿N)),
^,N(ti, . . . , tN) := 0a(0-i(ti) + ... + 0-i(tN)),
J (r, s) := 0-i(rs) (r,s,ti,...,tN £ R),
where 0 < a G R and : t ^ sgn(t)ta is an order preserving bijection of R. Obviously, , N belong to H (RN) and J belongs to H (R2), so that (xi,...,xn ), aa N(xi,..., xn), and J(x, y) are well defined for all x, y, xi,..., xn in a relatively uniformly complete vector lattice E. From the above definitions the following implication is easily deduced
(Vx, y G E) |x| A |y| = 0 ^ aa ,2(x, y) = a^*, 2(x, y) = x + y,
since it is true in the real context. Denote for brevity 0 := $2, a = a 2 , 2, and a' = a2 2.
Given a relatively uniformly complete vector lattice E := (E, +, ■, and 0 < a G R, the a- convexxification E («) of E is defined as the same underlying set equipped with the same order and new vector operations
x+y := aa,2(x, y) := (x1/a + y1/a)a (x, y G E), A * x := (A) ■ x := Aa ■ x (A G R, x G E),
so that E(a) := (E, +, *, Then E(a) is also a relatively uniformly complete vector lattice. Moreover, E(i) = E, (E(a))(^ = E(a^), and (E(a))(1/a) = E, where identities meant in the sense of vector and lattice isomorphism, see [22, Proposition 4.8].
In particular, the square (E0, 0) can be defined as E0 := E(1/2) := (E, +, *, <) and 0 := J, where x+y := a(x, y), A * x := $-i(A)x, and ^ is the given ordering in E, see [9, Theorem 9].
2.7. We say that a function f G H(RN) is multiplicative or modulus 'preserving if respectively f (siti,..., Sn tN) = f (si,..., Sn )f (ti,...,tN) and f (1111,..., |tN |) = |f (ti,...,tN )| for all Si, ti,..., Sn , tN G R. The general form of a nonzero positively homogeneous multiplicative and modulus preserving function is given by
tit2 ■ ... ■ tN = 0 ^ f (ti ,t2,...,tN) = 0,
f (ti, . . . , tN) = f (|ti|, . . . , |tN |) Sgn f (ti, . . . , tN),
f (|ti|,..., |tN|) = exp(gi(ln |ti|)) ■ ... ■ exp(gN(ln |tN|)),
where gi,...,gN are some additive functions in R (i. e. solutions to Cauchy functional equation, see [3]) with ^ Ni gi = Jr. In the case of continuous gi,...,gN we get a Kobb-Duglas type function f and if, in addition, f is nonnegative, then f(ti,...,tN) = c|t1|pi ■... ■ |tN|PN with 0 ^ c G R, pi,... ,pn G R and N=1 Pi = 1. Therefore, the expression |x1|pi ■ ... ■ |xn|PN is well defined in E for p& ^ 0, p1 + ... + Pn = 1. Moreover,
|xi|pi ■ ... ■ |xn p < Pi|xi| + ... + PN |xn | by the inequality between the weighted arithmetic and geometric means.
3. Gauges and Holder type inequalities
Now we consider some interplay between the square of a vector lattice and homogeneous functional calculus and deduce some Holder type inequalities. In the sequel E denotes a relatively uniformly complete vector lattice.
3.1. A gauge is a nonnegative sublinear function defined on a convex cone contained in . The polar k° of a gauge k defined by
k°(t) := inf{A > 0: (Vs G )(s,t> < Ak(s)} (y G )
is also a gauge. Moreover, k°° := (k°)° = k if and only if k is lower semicontinuous (for more details see [20]).
A gauge k : RN ^ R is strictly positive provided that k(s) > 0 for every s = 0. Here we consider only strictly positive gauges defined everywhere on RN. The totality of such gauges on RN will be denoted by G(RN). Every gauge from G(RN) is continuous. The polar of a gauge k £ G(RN) is also contained in G(RN) and can be calculate by
k°(t) = sup = sup{(s, t> : k(s) < 1} (t £ RN).
s=0 k(s)
Since G(RN) C H(RN), there exist k(xi,... ,xN) £ E and k°(xi,... ,xN) £ E for any xi,... £ E. Moreover, the mapping (xi,... , xn) ^ k(xi,... , xn) is a sublinear operator from EN to E and
N
|k(xi,... ,XN) - k(yi,... ,yN)| ^ I|k|| \/ |xi - y»|.
i=i
3.2. If k £ G(Rn) and xi,..., xn £ E, then the representation holds
k°(xi ,...,xn ) =sup j : (ai,...an ) £ RN, k(Ai,...AN) < 1 j .
Moreover, k°(xi,...,XN) is a relatively uniform limit of an increasing sequence which is comprised of the finite suprema of linear combinations of the form E^ A»x» with
k(Ai,... An) < 1.
< Observe that the set U := | Ei=i A»x» : k(Ai,... An) ^ 1 j is norm totally bounded in the AM-space E«, u := |xi| V ... V |xn|, since it is the image of the compact set {A £ Rn : k(A) ^ 1} under the map A = (Ai,...,An ) ^ E fe=i A» x». Denote by Uv the subset of E consisting of the suprema of the finite subsets of U. Then by Krengel's Lemma (see [1, Lemma 3.13]) y : = sup U exists in E« and belongs to the norm closure Uv of Uv. Since Uv is upward directed, Uv is norm convergent to y. Therefore, for any R-valued homomorphism u on E« we have
u(y) = lim u(u) = sup{u(u) : u £ Uv} sup{u(u) : u £ U} = k°(u(xi),..., u(xN)).
«SU v
Thus, y = k°(xi,... ,xN) by [6, Corollary 3.4]. >
3.3. Take a gauge kp,N : (ti,...,tN) ^ (EN=i 11»|p)P with 1 < p < to. For the corresponding mapping from EN into E an expressive notation is used, see [17, 21, 22]:
i
ip f ._
jxi^ := (xi,...,xn) (xi£ E).
For p = to, we define kp,N (ti, ...,iN) = max {jtjj : i := 1,...,^} and, obviously, kp,N(xi,..., xn) = xi V ... Vxn• Of course, kp,N £ H(RN) and the mapping (xi,..., xn) ^ kp,N(xi,..., xn) £ E is well defined even if 0 < p < 1, but in this case kp,N £ G(RN) and the corresponding mapping is not sublinear.
3.3. If k G G(RN), Xi,... G E, and T is a positive operator from E to a F, then
k(Txi,...,Txw ) < T (k(xi )).
and equality holds if T is a lattice homomorphism. The inequality can be derived from the representation 3.2 using the same arguments as in 3.6 below. In particular, specializing k : = kp,N with 1 ^ p ^ œ yields
p
|
k=i 7 v k=i
£|Txfc|p < T £ |xfc|
and equality holds if T is a lattice homomorphism [17, 21]. On the contrary, if 0 < p < 1, then
i i / n v / n V
T E |xkf < E |Txk|p .
A more general fact will be proved below in 4.3.
3.4. For any k G G(RN) and x1,... , y1,..., yw £ E the inequality holds
N
E^i © y ^ k(Xi,...,XN ) 0 k°(yi,...,yN ).
i=1
< It is an easy exercise to check that the inequality (see 2.6 for definitions of CT2,n and J)
^2,N (J(si,tl), . . . , J(sn)) ^ J(k(si, . . . ,SN), k°(ti, . . . ,tw)) .
is equivalent to the well known property of gauges [20]:
(s,t) < k(s)k°(t) (s = (si,...,sn), t = (ti,...,tw) G Rn).
Combining this with 2.3 and 3.6 we obtain the desired inequality. >
In the special case of k := kp,N and k° = kq,N, 1 ^ p, q ^ ro, 1/p + 1/q = 1, we have
N / N \ p / N \ i
u. ^ y.i ^ I V^ ix.ip I ^ I V^ iy.iq
E|Xi © yi| < £ |xi|p J 0 £|y*|q
i
3.5. If b : E x E ^ G is a positive orthosymmetric bilinear operator and Xj,yj G E, i := 1,..., N, then
N
£ |b(xk,yk)| ^ b(k(xi,. . . ,XN), k°(yi,. . . ,yN)).
k=i
< Apply $5 to 3.4 and use 1.3. >
Again, if 1 ^ p, q ^ œ and 1/p + 1/q = 1, then
N / / N / N
X)|b(xfc)| < b( £ |xfc|p , PT |yk|q k=i \ Vk=i / Vk=i J
3.6. Let b : E x E ^ G be a positive orthosymmetric bilinear operator and b = $60 for a positive linear operator $5 from E0 to G. Then for xi, yi,..., xn, yw £ E and k £ we have
k(b(xi,yi),... ,b(xw,yw)) ^ $6(k(xi 0 yi,... ,xw 0 yw)). In particular, if 1 ^ p, q ^ to and 1/p + 1/q = 1, then
N \p / / N \p
\p\ if>, I hr |x• ^ 1».Ip I
ib(x^irj < $5 ^xi ixi 0 yf
< Taking into consideration 1.3, 3.2, and positivity of $5 we deduce N / N \
^ Aib(x,, y,) = $6 I ^ A^x, 0 yj ^ $b(k(xi 0 yi,... ,xn 0 yw)) j=i \i=i /
for any finite collection Ai,... an £ R with k°(Ai,... an) ^ 1. It remains to apply 3.2 again. > 3.7. If in 3.6 b is a symmetric lattice bimorphism, then
k(b(xi,yi),... ,b(xw,yw)) = $6(k(xi 0 yi,... ,xn 0 yw)).
In particular, if 1 ^ p, q ^ to and 1/p + 1/q = 1, then we have
N \p f N
P / N \p
J>(x,y/| =$6 I PT |xi 0 y/1
\j=i / \ \j=i /
< Since $6 is a lattice homomorphism by 1.2, we only need to apply 2.4 and 1.3. >
4. Inequalities with monomials
In this section we prove several inequalities containing homogeneous expressions of the form |xi|pi ■ ... ■ |xN|PN with pi + ... + pN = 1, see 2.7.
4.1. Assume that a homogeneous function f £ H(RN) is multiplicative and modulus preserving. Then for all xi, yi,..., xn, yw £ E we have
f(xi 0 yi,... ,xw 0 yw) = f(xi,... ,xw) 0 f(yi, . . . ,yw).
In particular, if 0 ^ pi,... ,pN £ R, pi + ... + pN = 1, then
N , N x , N x
nix, 0 y,iPt = (n wpi) ^rniyiipM.
< If f is multiplicative and modulus preserving, then 0(f (si,...,sw)) = f (0(si),..., 0(sn)) and the equality f o (J x ... x J) = J o (f x f) holds, see 2.6. Applying 2.3 and 2.6 we come to the desired inequalities. >
4.2. Theorem (The generalized Holder inequality). Assume that E and G be relatively uniformly complete vector lattices. If a mapping f : E+ ^ G is increasing and sublinear (f (x + y) < f (x) + f (y), f (Ax) = Af (x); x, y £ E, 0 < A £ R), then
N x N
f n |x*r <nf(N)p
for xi,..., xN G E and 0 ^ pi,...,pN G R with pi +... + pN = 1. Equality holds if f : E ^ G is lattice homomorphism.
< Without loss of generality we may assume that 0 ^ x» and 0 < p» < 1 for all i : = 1,...,N. Indeed, if (ii,...,ifc} = {j < N : pj = 0}, then |xi|p1 ■ ... ■ |xw|PN = K|Pii ■ ... ■ |xik |Pik. Now we observe that, for 0 ^ x, y G E and 0 < p < 1, the equality holds
xpyi-p = inf{pAi/px + (1 - p)A-i/(i-p)y : 0 < A G Q}.
Indeed, by 2.6 for an arbitrary 0 < A G R the inequality is valid:
xPyi-P = (Ai/Px)P(A-i/(i-p)y)i-p < pAi/px + (1 - p)A-i/(i-p)y.
Assume that v ^ ^>a := pAi/px + (1 — p)A-i/(i-p)y for all 0 < A G Q. Actually this inequality is true for all 0 < A G R, since |^>A — ^ C(e)|A — ^|(x + y) whenever 0 < e < A,^ < 1/e. By the Kreins-Kakutani Representation Theorem we can view the principal ideal Eu generated by u = x + y + |v| as C(S) for some compact space S. Then v, x, y, ^a, and xpyi-p lie in C(S) and for 0 < A G R the pointwise inequality v(s) ^ ^a(s) is true. If x(s) = 0, then trivially
v(s) < inf {(1 — p) A-i/(i-p)y(s) : 0 < A G q} = 0 = x(s)py(s)i-p.
If x (s) = 0, then for A := (y(s)/x(s))p(i-p) we have ^a(s) = x(s)py(s^ p ^ v(s). Thus, v ^ xpyi-p and the desired representation for xpyi-p follows.
Now, taking into consideration that f is sublinear and increasing, we deduce
f(xpyi-p) < inf {f(pAi/px + (1 — p)A-i/(i-p)y) : 0 < A G q} < inf {pAi/pf (x) + (1 — p)A-i/(i-p)f (y) : 0 < A G q} = f (x)pf (y)i-p.
The general case is handled by induction. Suppose f (x^ ■ ... ■ xN--1) ^ f (xi)qi ■ ... ■ f(xn-i)qN-1, whenever qi -f ... + qw-i = 1. Put p := pi + ... + pw-i, q» := p»/p (i := 1,..., N — 1), and u:= (xf ■ ... ■ xN-1 )i/p = xi1 ■ ... ■ xN-1. Then
f (xi1 ■ ... ■ xNN) = f (upxNN) < f (u)pf (xn)PN = f (xi1 ■ ... ■ xN-i1 )pf (xn)PN < f (xi)p1 ■ ... ■ f (xn)PN,
and the required inequality follows. The remaining part is obvious. >
4.3. Theorem (The generalized Minkowski inequality). Assume that E, G, f, and xi,...,xn are the same as in 4.2 and 0 < p < 1. Then the inequality holds:
N \ i/p\ / N \ i/p
|x.|p i i ^ I V^(f(|x.|))p
f(( £ |Xi^ ')<( it(f (|Xi|))P)
Equality holds if f : E ^ G is a lattice homomorphism.
< The same line of reasoning as in 4.4 works. We may assume without loss of generality that xi ^ 0 i := 1,...,N. First we prove that, for 0 ^ x, y G E and 0 < p < 1, the representation holds
(xp + yp)i/p = inf{A-i/px + (1 — A)-i/py : 0 < A < 1, A G Q}.
Next, applying f to this equality and taking into account that f is increasing and sublinear we deduce
f ((xp + yp)1/p) < (f (x)p + f (y)p)1/p. The general case is again settled by induction. Put u:= (xp +... + xN_i)1/P and observe that
i^vP + xn) — + ... + xn) . Moreover, by induction f (u) ^ (f (xi)p + ... + f (xn_i )P)1/p. Thus
f ((x? + ... + xN )1/p) — f (up + xN )1/p < (f(u)p + f(xn)p)1/p < (f(x1)p + ... + f(xn)p)1/p
and we are done. [>.
4.4. We can take in 4.2 an arbitrary increasing gauge k £ G(RM) instead of f and consider the corresponding sublinear operator from EM to E. Suppose that M £ N and for every j:— 1,..., M a finite collection of elements (x1j,..., xn?) £ EM is given. Replacing f, for example, by kp,M (1 ^ P ^ ro) we arrive at the following version of Holder inequality:
M , \ p\ 1/P / M \Pl/P / M \PN/p
]T(|x1j|P1 ■ ... I^l) ) <l£ j ■ ... ' E^f .
4.5. Let (fi, S, be a measure space with a a-finite positive measure ^ and F be a Banach lattice. Let L 1(fi, S,^,F) be the space of all Bochner integrable functions on fi with values in F and E :— L1 F) :— L(fi, S,^, F)/ ~ denotes the space of all equivalence classes (of almost everywhere equal) functions from L 1(fi, S, F). Then E — L1 F) is also a Banach lattice and hence f (x1,..., xn ) is well defined in E for f £ H (RM) and x1,..., xn £ E. Denote by x the equivalence class of x £ L 1(fi, S, F). Making use of the continuity of functional calculus (see [9, Theorem 7]) one can deduce that the equality f (x1,..., xn)(u) — f (x1(u),..., xn (u)) is true for almost all u £ fi (or more precisely f (x1,..., xn)(u) is the equivalence class of the measurable function u ^ f (x1 (u),..., xn (u))) for any finite collection x1,...,xn £ L 1(fi, S,^,F). Since the Bochner integral defines a linear and increasing operator from E to F, we can replace f in 4.2 and 4.3 by the Bochner integral. Thus, we get the following inequalities (0 ^ p1,..., pN £ R, p1 + ... + pN — 1, 0 < p < 1):
N \ N
nixi(u)r d^(u) ^ / |xi(u)| d^(u)
Pi
/■ / N \ Vp / N / r \p\Vp
I i £ |xi(W)|M d^) < ( £ i I |x,(W)| J
for xi(-),...,xw (■) £ £,^,F).
4.6. Let E, F, and G be relatively uniformly complete vector lattices, f, g : E+ ^ F be ingreasing sublinear operators, and b : F x F ^ G be a positive orthosymmetric bilinear operator. Then
b(f(n NPi) ,g(II Npi)) <rib(f(|x,i),g(|y,i))pt.
for all xi, yi,..., xN, yN G E and 0 ^ pi,..., pN G R with pi + ... + pN = 1. Equality holds if f, g : E ^ F are lattice homomorphisms and b is a symmetric lattice bimorphism.
< By applying 4.2 to f and g and using 4.1 we obtain , N \ , N \ N
f (n |x»|p0 © g( II |yi|pt ^U (f(|xi|) © g(|yi|))pt.
M=i 7 M=i 7 i=i
Now, apply $5 to the last inequality, use again 4.2 with f := $5, and take 1.3 into account. If f and g are lattice homomorphisms and b is a lattice bimorphism, then we apply 2.4 and 4.1 and observe that, according to 1.2, $5 is a lattice homomorphism if and only if b is a lattice bimorphism. >
4.7. Remarks 1. The generalized Holder inequalities 4.2 for increasing sublinear operator between relatively uniformly complete vector lattices was obtained in [16, Theorem 5.2]. Both, the generalized Holder and Minkowski inequalities 4.2 and 4.3 for operators between spaces of measurable functions were established in [12, Remark 1.2(5)] and [12, Remark 1.2(6)].
2. In [10] and [11] some interesting estimates for the Hadamard weighted geometric means of positive kernel operators on Banach function spaces were obtained. Holder type inequalities for operators can be useful in such studies. For example, the inequalities (1) of [11, Theorem 2.1] and (4) of [11, Theorem 2.2] are the easy consequences of 4.2 (f (k) := ||K||) and 4.6 (b(H, K) := H ■ K, f (h) := H, g(k) := K), respectively. More applications see in [12] and [16].
3. In [12, Proposition 1.1] a Jensen type convexity inequality for sublinear operator on spaces of measurable functions was proved. (The scalar case see in [5, § I.1, Proposition 1].) Certainly, this result can be generalized to operators on vector lattice so that 4.2 and 4.3 will become its special cases. The corresponding improved version of homogeneous functional calculus on vector lattice will be developed in a forthcoming paper.
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Received 25 April, 2007
ANATOLY G. KUSRAEV
Institute of Applied Mathematics and Informatics Vladikavkaz Science Center of the RAS Vladikavkaz, 362040, RUSSIA E-mail: kusraev@alanianet.ru
