CC BY
9
Владикавказский математический журнал 2011, Том 13, Выпуск 1, С. 38-43
УДК 512.555+517.982
A BECKENBACH-DRESHER TYPE INEQUALITY IN UNIFORMLY COMPLETE /-ALGEBRAS
A. G. Kusraev
To the memory of Gleb Akilov on the occasion of the 90th anniversary of his birth
A general form Beckenbach-Dresher inequality in uniformly complete /-algebras is given.
Mathematics Subject Classification (2000): 06F25, 46A40.
Key words: /-algebra, vector lattice, lattice homomorphism, positive operator.
An easy modification of the continuous functional calculus on unitary f-algebras as defined in [3] makes it possible to translate the Fenchel-Moreau duality to f-algebra setting and to produce some envelope representations results, see [8]. This machinery, often called quasi-linearization (see [2, 9]), yields the validity of some classical inequalities in every uniformly complete vector lattice [4, 5]. The aim of this note is to give general forms of Peetre-Persson and Beckenbach-Dresher inequalities in uniformly complete f-algebras.
The unexplained terms of use below can be found in [1] and [6].
1°. We need a slightly improved version of continuous functional calculus on uniformly complete f-algebras constructed in [3, Theorem 5.2].
Denote by B(RN) the f-algebra of continuous functions on with polynomial growth; i. e., p £ B(R+) if and only if p £ C(RN) and there are n £ N and M £ R+ satisfying |p(t)| < M (1 + w(t))n (t £ RN), where t := (t1,...,tN), w(t):= |ti| + ... + | tN | and 1 is the function identically equal to 1 on RN. Denote by B0(RN) the set of all functions in B(RN) vanishing at zero. Let A (RN) stands for the set of all p £ B(RN) such that lima^0 a-1p(at ) exists uniformly on bounded subsets of RN. Evidently, A (RN) C B0(RN). Finally, let H(RN) denotes the set of all continuous positively homogeneous functions on RN.
Lemma 1. The sets B(R N), B0(RN), and A (RN) are uniformly complete f -algebras with respect to pointwise operations and ordering. Any p £ A (RN) admits a unique decomposition p = p1 + wp2 with p1 £ H(RN) and p2 £ B0(RN), i■ e.
A (RN) = H (RN) e wB0(R N).
Moreover, p1(t) = p'(0)t := limaj0 a-1p(at) for all t £ RN.
< See [3, Lemma 4.8, Section 5]. >
© 2011 Kusraev A. G.
2°. Consider an f-algebra E. Denote by H(E) the the set of all nonzero R-valued lattice homomorphisms on E and by Hm(E) the subset of H(E) consisting of multiplicative functional. We say that u G H(E) is singular if u(xy) = 0 for all x,y G E. Let Hs(E) denotes the set of singular members of H(E). Given a finite tuple x = (xi,..., xN) G EN, denote by ((x)) := ((x1,... ,xN)) the f-subalgebra of E generated by {x1,... , xN}.
Definition. Let E be a uniformly complete f-algebra and x1,...,xn G E+. Take a continuous function p : R+ — R. Say that the element </?(x1,..., xN) exists or is well-defined in E provided that there is y G E satisfying
u(y) = p(u(X1),..., u(XN)) (u G Hm(((X1,...,XN ,y))), (^
u(y) = ^1(u(X1), . . . ,u(XN)) (u G HS(((X1, . . . ,xn,y))),
cp. [3, Remark 5.3 (ii)]. This is written down as y = (p(x1,... , xN).
Lemma 2. Assume that E is a uniformly complete f -algebra and x1,..., xN G E+, and x:= (x1,..., xN). Then x(p) := </?(x1,..., xN) exists for every p G A(RN), and the mapping x : p — x(p) = <x(x1 , ...,xN) is the unique multiplicative lattice homomorphism from N) to E such that dj(x1,..., xN) = x^ for all j := 1,..., N. Moreover, X(A(RN)) =
((X1, . . . ,XN)).
< Take p G A (RN). In view of Lemma 1 p = p1 + wp2 with p1 G H (R+), p2 G and w(t) = |t1| + ... + |tN|. For x G E denote by X G Orth(E) the multiplication operator y — xy (x G E). According to [5, Theorem 3.3] and [8, Theorem 2.10] we can define correctly p1 (x1;... ,xN) in E and p2(X 1;... ,XN) in Orth(E), respectively. Now, it remains to put p(x1;..., xN) := p1(x1,..., xN) + p2(X 1;..., XN)w(x1,..., xN) and check the soundness of this definition. Closer examination of the proof can be carried out as in the case of p G A(Rn), see [3]. >
Lemma 3. Assume that p G A (RN) is convex. Then for all x := (x1,...,xn ) G EN, y := (y1;..., yN) G En, and n, p G Orth(E)+ with n + p = we have p(nx + py) ^ np(x) + pp(y), where nx := (nx1;... ,nxN). The reverse inequality holds whenever p is concave.
< Let L be the order ideal generated by ((x1 ,...,yN)). Clearly, L is an f-subalgebra of E. If n0 := n|L and p0 := p|L then n0, p0 Orth(L). For any u G H(L) there exists a unique u G Hm(Orth(L)) such that u(nx) = u(n)u(x) for all x G L and n G Orth(L), [3, Proposition 2.2 (i)]. If u is nonsingular then au is multiplicative for some a > 0 [3, Corollary 2.5 (i)], and thus we may assume without loss of generality that u G Hm(L). By using (1), the convexity of p, and the relation u(n) + u(p) = 1 we deduce
u(x(nx + py)) = p(u(n0)u(x) + u(p0)u(y)) ^ u(n0)p(u(x)) + u(p0)p(u(y)) = u(n0)u(px(x)) + u(p0)u(px(y)) = u(np(x) + pp(y)),
where u(x) := (u(x1),..., u(xN)). If u is singular then by above definition we have u(p(x)) = u(p1(x)), u(p(y)) = u(p1(y)), and u(p(nx + py)) = u(pX1(nx + py)). At the same time p1 is sublinear, since it coincides with the directional derivative of the convex function p at zero, see Lemma 3. Thus, by replacing p by p1 in the above arguments we again obtain u(p(nx + py)) ^ u(np(x) + pp(y)). It remains to observe that every u0 G H(((x1,... ,xN))) admits an extension to u G H(L) and thus H(L) separates the points of ((x 1,... ,xN)). >
Lemma 4. If p G A (RN) is isotonic, then p is also isotonic, i. e. x ^ y implies X(x) ^ p(y) for all x, y G EnN. (The order in EN is defined componentwise.)
< Follows immediately from the above definition (1). >
3°. Everywhere below (G, +) is a commutative semigroup, while E is a uniformly complete f-algebra and f1;..., fN : G ^ E+. Let P(M) stands for the power set of M. Assume that some set-valued map F : G ^ P(Orth(E) +) meets the following three conditions:
(i) n-1 exists in Orth(E) for every n £ F(u),
(ii) F(u) + F(v) C F(u + v) - Orth(E) + for all u, v £ G, and
(iii) the infimum (the supremum) of {np(n-1f(u)) : n £ F(u)} exists in E for each u £ G, where f(u):= (f1(u),... ,fN(u)) £ Ef and n-1f(u):= (n-1 f1(u),..., n-1 fN(u)).
Lemma 5. Given a function p : A(RN) and a set-valued map F : G ^ P(Orth(E)+) satisfying 3 (i-iii), we have the operator g : G ^ E (h : G ^ E) well defined as
g(u):= inf W(n-1 f(u))}, (h(u) := sup (np(n-1f(u))} ). (2)
neF(u) v neF(u) 7
< By 3 (i) and Lemma 2 p (n 1f (u)) exists in E and by 3 (iii) g and h are well defined. > 4°. Now we are able to state and prove our main result. A function g : G ^ F is said
to be subadditive if g(u + v) ^ g(u) + g(v) for all u, v £ G and superadditive if the reversed inequality holds for all u, v £ G.
Theorem. Suppose that the operators g, h : G ^ E are defined as in (2). Then:
(1) g is subadditive whenever f1;... ,fN are subadditive and p £ A(R+) is increasing and convex;
(2) h is superadditive whenever f1;..., fN are superadditive and p £ A (R+) is increasing and concave.
< We restrict ourselves to the subadditivity of g. The superadditivity of h can be proved in a similar way. Take u, v £ G and let n £ F(u) and p £ F(v). By 3 (ii) we can choose a £ F(u + v) with a ^ n + p. In view of 3 (i) n, p, and a are invertible. Taking subadditivity of f : G ^ En and some elementary properties of orthomorphisms into account we have
a-1 f (u + v) ^ a-1 (f(u) + f (v)) ^ na-1 (n-1 f (u)) + pa-1 (p-1f(v)).
Putting t := a — n — p and making use of Lemmas 3, 4 and 5 we deduce
g(u + v) ^ ap(a-1f(u + v)) ^ ap(na-1 (n-1 f(u))) + pa-1 (p-1f(v) + Ta-10)
^ n<^(n-1 f(u)) + pp(p-1 f(v)) + a-1 tp(0) = n<^(n-1 f(u)) + p<^(p-1 f(v)).
By taking infimum over n £ F(u) and p £ F(v) we come to the required inequality. >
Remark 1. Suppose that the hypotheses of 3 (i-iii) are fulfilled for some fixed u, v £ G. Then the inequality g(u + v) ^ g(u) + g(v) (h(u + v) ^ h(u) + h(v)) holds.
Remark 2. An f-algebra E can be identified with Orth(E) if and only if E has a unit element. Thus, above theorem remains true if E is a uniformly complete unitary f-algebra and the map F : G ^ P(E+) satisfies the condition 3 (i-iii) with Orth(E) replaced by E.
5°. For a single-valued map F(x) = {f0(x)} (x £ G) with f0 : G ^ Orth(E)+ we have the following particular case of the above Theorem, see [8].
Corollary 1. Suppose that f1;..., fN are subadditive, f0 : G ^ Orth(E)+ is superadditive, and f0(u) is invertible in Orth(E) for every u £ G. Then, given an increasing continuous convex function p £ A(R+), the Peetre-Persson inequality holds:
The reverse inequality holds in (3) whenever f0,f1;...,fN are superadditive, and p is an increasing concave function.
Remark 3. The above theorem in the particular case of E = R was obtained by Persson [12, Theorems 1 and 2], while Corollary 2 covers the "single-valued case" by Peetre and Persson [11]. A short history of the Beckenbach-Dresher inequality is presented in [13]. Some instances of the inequality are also addressed in [9, 10].
6°. We need two more auxiliary facts. First of them is a generalized Minkowski inequality.
Lemma 6. Let E and F be uniformly complete vector lattices, f : E+ — F an increasing sublinear operator. If either and 0 < a ^ 1 or a < 0, then for all x1,...,xn £ E we have
' N \ 1/a \ ( N \ 1/a
ENaJ Ef(wrj . (4)
The reverse inequality holds if f : E+ — F is superlinear and a ^ 1.
< The function 0a(t) = (if + ... + if)1/a (t £ Rf) is superlinear if 0 < a < 1 and sublinear if a ^ 1. In case a < 0 we define 0a(t) = 0 whenever t1 ■ ... ■ tN = 0 and then
is superlinear on int(Rf). In all cases £ H (R+) and (4) follows from the generalized
Jensen inequality in vector lattices, see [4, Theorem 5.2] and [7, Theorem 4.2]. >
Let A and B be uniformly complete unitary f-algebras, while E C A is a vector sublattice.
For every x £ A+ and 0 < p £ R the p-power xp is well defined in A, see [3, Theorem 4.12].
If x £ A+ is invertible and p < 0, then we can also define xp := (x-1 )-p. It can be easily seen
that u(xp) = u(x)p for any u £ Hm(A0) with an f-subalgebra A0 C containing x. Assume
i
that R : E —> B is a positive operator. Given x £ A with xp £ E, we define Rp(x) := R(xp)p. This definition is sound provided that x is invertible in A and R(xp) is invertible in B.
Lemma 7. If p ^ 1 and x1;..., xN £ A+ are such that xp",..., xf £ E and (x1 + ... + xN)p £ E, then the inequality holds:
Rp(x1 + ... + xn) ^ Rp(x) + ... + Rp(XN). (5)
The reversed inequality is true whenever p ^ 1, p = 0. (In case p < 0 the positive elements x » and R(xP) are assumed to be invertible in A.)
< Denote Ui := x1-, a:= 1/p, and observe that (uf + ... + u%) a = (pa(ui,..., un) where 0a(u1;..., uN) is understood in the sense of homogeneous functional calculus. In particular,
{x\ + ... + xn)p = (uf + ... + ufj) a £ E for every p / 0. We need consider three cases. If p ^ 0 then by applying Lemma 6 to the right-hand side of the equality
Rp(x i + ...+xN) = + ... + = (R((pa(ui,...,uN)))a
with uf £ E replaced by x» and making use of Rp(xj) = R(u»)a (i := 1,..., N), we arrive immediately at the desired inequality (5). The same arguments involving the reversed version of (4) leads to the reversed inequality in (5) whenever 0 < p < 1. Finally, in the case p < 0,
again by Lemma 6, we have i?((u" + ... + u%)<*) ^ (R(u\)a + ... + R(uN)a)a and rising
i. . i
both sides of this inequality to the ath power we get the reversed inequality (5). >
7°. Now, we can deduce a generalization of one more Beckenbach-Dresher type inequality due to Peetre and Persson [11].
Corollary 2. Let S : E — F and T : E — Orth(F) be positive operators. Take x1,...,xn £ A+ such that xf,xf, (£N=1 xj)a, (£N=1 x»)^ £ E (i := 1,...,N). If p ^ 1,
ß ^ 1 ^ a, ß = 0, and T(xß) are invertible in Orth(F) whenever ß < 0, then
^ Z-, ^^(p-l)//?' W
xi
(T((Eil,x,)ß^(p-1)/ß fei (T(xft)(
< Put G = E, f (x) := f(x) := S(xa)1/a, fo(x) := T(xß)1/ß, N = 1, and p(t) = in Corollary 1. By Lemma 7 f is subadditive, f0 is superadditive, and f0(x») is invertible in Orth(F). Moreover, p e A(R+) is convex and increasing whenever p ^ 1. Now, the desired inequality is deduced by induction. >
Remark 4. If 0 < p < 1 then the concave function p(t) = is not in A(R+) and we cannot guarantee the reversed inequality in (6). Nevertheless, in the case that F is a unitary f-algebra one can take p e ) in Peetre-Persson's inequality (3) and thus the reversed
inequality is true in (6) whenever 0 < p ^ 1, a, ß ^ 1, and a, ß = 0.
References
1. Aliprantis C. D., Burkinshaw O. Positive operators.—London etc.: Acad. Press Inc., 1985.—xvi+367 p.
2. Beckenbach E. F., Bellman R. Inequalities.—Berlin: Springer, 1983.
3. Buskes G., de Pagter B., van Rooij A. Functional calculus on Riesz spaces // Indag. Math.—1991.— Vol. 4, № 2.—P. 423-436.
4. Kusraev A. G. Homogeneous functional calculus on vector lattices.—Vladikavkaz: Southern Math. Inst. VSC RAS, 2008.—34 p.—(Preprint № 1).
5. Kusraev A. G. Functional calculus and Minkowski duality on vector lattices // Vladikavkaz Math. J.—2009.—Vol. 11, № 2.—P. 31-42.
6. Kusraev A. G., Kutateladze S. S. Subdifferentials: Theory and applications.—Dordrecht: Kluwer Academic Publ., 1995.—ix+398 p.
7. Kusraev A. G. Inequalities in vector lattices // Studies on Mathematical Analysis, Differential Equation, and their Applications / Eds. Korobeinik Yu. F., Kusraev A. G.—Vladikavkaz: SMI VSC RAS, 2010.— P. 82-96.—(Review of Science: The South of Russia).
8. Kusraev A. G., Kutateladze S. S. Envelopes and inequalities in vector lattices.—(to appear).
9. Mitrinovic D. S., Pecaric J. E., Fink A. M. Classical and new inequalities in analysis.—Dordrecht: Kluwer, 1993.—xvii+740 p.
10. Pecaric J. E., Proschan F., Tong Y. L. Convex functions, partial orderings, and statistical application.— Boston a. o.: Academic Press, 1992.—xiii+469 p.
11. Peetre J., Persson L.-E. A general Beckenbach's inequality with applications // Function Spaces, Differential Operators and Nonlinear Analysis. Pitman Res. Notes Math. Ser. 211.—1989.—P. 125-139.
12. Persson L.-E. Generalizations of some classical inequalities and their applications // Nonlinear Analysis, Function Spaces and Applications / Eds. Krbec M., Kufner A., Opic B., Rakosnik J.— Leipzig: Teubner, 1990.—P. 127-148.
13. Varocanec S. A generalized Beckenbach-Dresher inequality // Banach J. Math. Anal.—2010.—Vol. 4, № 1.—13-20.
AnATOLY G. Kusraev Southern Mathematical Institute Vladikavkaz Science Center of the RAS, Director RUSSIA, 362027, Vladikavkaz, Markus street, 22 E-mail: kusraev@smath.ru
НЕРАВЕНСТВО ТИПА БЕККЕНБАХА - ДРЕШЕРА В РАВНОМЕРНО ПОЛНЫХ f-АЛГЕБРАХ
А. Г. Кусраев
Установлено общее неравенство типа Беккенбаха — Дрешера в равномерно полных /-алгебрах.
Ключевые слова: /-алгебра, векторная решетка, решеточный гомоморфизм, положительный оператор.
