ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXIII (1983)
Leszek Jankowski (Poznan)
The existence of the superadditive functions useful for the M usielak-Peetre type modular spaces
Abstract. This note shows the existence of superadditive functions used to define F-norms in the generalized modular spaces considered by J. Musielak and J. Peetre [4 ].
Introduction. We recall some basic notions and facts concerning F- modulars and F-modular spaces introduced by Musielak and Peetre [4] (see also [2] for further development). Let F = ® be a binary operation on R + such that {R + , ©) is a topological ordered commutative semigroup with 0 as a neutral element. Let I be a (real or complex) linear space. Then a functional q: X -*R + is called an F -modular if
(1) £>(x) = 0 if and only if x = 0;
(2) e(ax) = g(x) if |a| = 1 ;
(3) g(ax + by) < e(x)©e(y) if a, b ^ 0 and a + b = 1
for all x, y in X and scalars a, b. The associated F -modular spade is the linear subspace
Xe = { x e X : lim g(ax) = 0}
я~*0 of X. The sets
6 - { x e J f e : e ( x ) < e } , e > 0 ,
are easily seen to form a base of neighbourhoods of 0 for a metrizable linear topology t on XQ. Moreover, if (x„) c Xe, then x„ -> O(t) if and only if g{ax„) -> 0 for all a > 0. It is desirable to have an explicit formula for an F-norm defining the topology t. Assuming the existence of a superadditive function k: (R +, + )-►(# + , ®) continuous and vanishing at 0, Musielak and Peetre have provided such a formula, viz.,
I|x||g,k = inf { a > 0: g{x/a) < k(a)}.
In view of this it is natural to ask whether such a function к does exist
for every semigroup (R +, ©) ([5], Problem 263). The theorem below answers this question on the affirmative.
The structure of semigroups on R + and R +. Below we shall deal with topological ordered commutative semigroups (S, ©), briefly called semi
groups, for which S is an interval [a, b) or [a, b] (0 ^ a < b ^ oo) considered with its natural topology and order, and with a as a neutral element.
Precisely, the order-topological requirement is that the function x -> x©y is increasing (= non-decreasing) and continuous on S fox each ye S'. (It follows easily that the map (x, y) -► x 0 y is jointly continuous.) Two such semigroups Sx and S2 are called isomorphic, and we write ~ S 2, if there exists an isomorphism between Si and S2, by which we mean an increasing homeo- morphism of S^ onto S2 which is simultaneously an algebraic semigroup- isomorphism.
Given a semigroup S = ([я, b), ©), we denote by S the semigroup ([a, b), 0 ) obtained from S by extending © so that x©b = b©x
= b for all xg [a, b].
The following standard examples of semigroups will be needed below:
(1) A = {R+, -f), where R + = [0, oo) and + is the usual addition.
(2) В = (I, + 1), where I = [0,1] and x + xy = min (x + y, 1) for x, у e l . (3) C = (R+, v ) , where x v у = max (x, y) for x, y e R + .
It is obvious that every semigroup supported by [a, b) (resp., [a, b]) is isomorphic to a semigroup supported by R+ (resp., R + = [0, oo]). The detailed knowledge of the structure of the latter semigroups will be of crucial importance for our construction of superadditive functions. The following description of those semigroups is based on the works of Faucett [1] and Mostert and Shields [3].
Let T = (R + , 0 ) be a semigroup. Then the set E = E(T) of its idem- potents e (c©e = e) is a closed subset of R+ containing 0 and oo. If a, b e E , a ^ b , then ([a, b], 0 ) is clearly a semigroup; here © is written instead of 0| [a, b] x [a, b]. The structure of T can be now described as follows:
(i) If (a, b) is a (connected) component of R+\E, then either ([я, b], ©)
~ Â — {R+, + ) or ([a, b], 0 ) ~ B.
(ii) If x, y e R + , e e E and x ^ e ^ y, then x©y = y ( = x vy).
Thus in particular, if E = {0, oo}, then either T ~ A or T ~ B ; if E
= R + , then T = C.
Conversely, let £ be a closed subset of R + containing 0 and oo.
For each component (a, b) of R + \E choose arbitrarily an increas
ing homeomorphism h of [я, b] either onto R + or onto I, and define x©y
= h~1 (b(x) + b(y)) (resp., = h~1 (b(x)-f xb(y))) for x, y e [ a , b]. Finally define x©y = x v у for those x, y eR + which are separated by an element of E. It
is then easily seen that R + is a semigroup under the binary operation © thus obtained, with E as the set of its idempotents.
The semigroups T such that T = S for a semigroup S = (R+, ©) are distinguished by the property: If c = sup (E\{cc}), then c = go or otherwise ([c, oo ], ©) is not isomorphic to B. (This answers [5], Problem 264.)
The superadditive functions. Let T = (R +, ©) be a semigroup and E
= E(T) the set of its idempotents. A function к : (R +, + )-+ (R +, ©) is superadditive if
k(x)@k{y) ^ k{x + y) for all x , y e R +,
and strictly super additive if the inequality is strict whenever x, у > 0.
Evidently, every superadditive or strictly superadditive function is increasing or strictly increasing, respectively.
Let a function к : (R+, +)-> (R + , ©) be superadditive, and define a = k(0), b = sup k(x).
xeR +
Then:
1° a, b e E .
2° If b' = sup E n [0, b) < b and ([b', b], ©) then k(x) — b for large values of x.
It is trivial that a e E : k(0) ^ /с(0)©/с(0) ^ /c(0 + 0). Suppose ЬфЕ and let (c, d) be the component of R + \E containing b. Choose any x with c
< k{x) < b. Since ([с,Л], ©) is isomorphic to Л or B, it follows easily that for n large enough we have b < k(x)@ . . . ®k(x) (n times) ^ k(nx) ^ b; a contradiction. The argument showing 2° is similar and uses the fact that in this case there must exist we(b', b) such that u©u = b: If x0 is chosen so that и ^ k(x0), then x ^ x0 implies b ^ k(x)@ k(x) ^ k(2x) ^ b.
Theorem.Let a, b e E , a < b. Then there exists a continuous superadditive function к: (R +, + )-> (R +,® ) such that k{0) = a and sup/c(x) = b. The function к can be chosen strictly subadditive if for b' = sup E n [0, b) we have :
b' = b or otherwise ([b', b], ©) is not isomorphic to B.
Proof. Sipce the semigroup ([a, b], ©) is isomorphic to a semigroup supported by R + , we may (and will) asume a = 0, b = oo. In our construc
tion of к we shall distinguish several cases and subcases.
(1) E = R + : Then T = {R+, v ). Choose as к any continuous strictly increasing function mapping R+ onto R+.
(2) E = {0, oo}: Then Т ~ Л or T ~ B . Let h be an isomorphism between A or В and T, and define к = h o f where:
(2.1) / : {R + , +) -*(R + , +) is any continuous strictly superadditive function with f{ R + ) = R+ (e.g., any strictly convex function with /(0) = 0) when A, or
(2.2) /(x) = min (x, 1) for x e R + when T ~ B.
(3) E' = E\{0, oo} is non-empty and different from (0, oo): Define a' = inf E\ b' = sup E'
and choose c e E so that c = a' if a! > 0 and 0 < c < b' if a! = 0. We first construct a continuous function
к : [0, l ] - > * + such that k(0) = 0, k(l) = c and
( * ) /c(x)0/c(x) < k(x + y) for x, у > 0, x + y < 1.
Note that ( * ) implies that к is strictly increasing.
(3.1) a' > 0: Then ([0, c], ©) ~ A or ~ B. Let h be an isomorphism between Л or В and ([0, c], 0 ) , and define к = h o f, where
(3.1.1) / is an increasing strictly convex functions mapping [0, 1) onto R + and /(1) = oo when ([0, c], ©) ~ Â, or
(3.1.2) / is an increasing strictly convex function mapping [0, 1] onto [0, 1] when ([0, c], 0 ) ~ B.
(3.2) a' — 0: Choose a strictly decreasing sequence (c„)n$:1 in E with c t
— c and c„ -► 0. Let к be any strictly increasing continuous function on [0, 1] such that /c(21_") = cn for n — 1 , 2 , . . . We check that к verifies ( *):
Let x, ye(0, 1), x + y ^ 1 . If x ^ 2~" < у for some n ^ 1, then k(x)®k(y)
= k(y) < k(x + y). If x, уе(2~и, 21~n) for some n ^ 2, then k(x)0/c(y) < c„
= k(21~n) < k(x + y).
The function к defined so far on [0, 1] will now be suitably extended to R+ •
(3.3) b' = oc: Choose a strictly increasing sequence (dn)„^0 in E such that d0 = c and dn -> oo. Then extend к to a continuous strictly increasing function on R + so that k (2") = dn for n — 0, 1, 2, ...
(3.4) b ' < оо : Then ([/)', oo], 0 ) ~ Л or ~ B . Let h be an isomorphism between Я or В and ( [ b \ oo], 0 ) .
(3.4.1) Suppose ([b', oo], 0 ) ~ A. If b' = c, then set k\ [1, oo) = h of, where /(x) = x — 1 for x ^ 1. If c < b', then extend к to a continuous strictly increasing function on R + so that k (2) = b' and k\ [2, oo) = h o f where/(x)
= x —2 for x ^ 2.
(3.4.2) Suppose ([b', oo], 0 ) ~ B. Then proceed similarly as in (3.4.1), but with /(x) = min (x —1, 1) when b' = c, and /(x) = min (x —2, 1) when c < b'.
It is routine to verify that к .thus obtained is as required in the theorem, and that it is strictly superadditive except for the case when b' < oo and (O', oo], 0 ) ~ B.
A similar result holds for the semigroups (R +, ®). We omit the precise formulation, and note only the following
Corollary. For every semigroup (R +, ©) there exists a continuous strictly superadditive function к: (R +, 4-)->(jR+, ©) such that k(0) = 0 and sup k(x) = со.
I would like to express my gratitude to Dr Lech Drewnowski for many fruitful discussion and his help in- preparing this paper.
References
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