ERGODIC RESULTS FOR CERTAIN CONTRACTIONS ON ORLICZ SPACES WITH FIXED POINTS

ERGODIC RESULTS FOR CERTAIN CONTRACTIONS ON ORLICZ SPACES WITH FIXED POINTS


. Introduction and preliminaries
Let (X, M, lc) be a u-finite measure space and Lo -Lo(X, M, ju) and Orlicz space associated to an N-function 0 (L o may be a complex Banach space) .In this paper we will consider linear operators T such that 1) fx0(ITfI)dp<_ fx0(1fj)dM, f E L a ii) T has a fixed point h, h 7É 0 a.e.iii) T is either a 11 11 1 -contraction in Lo n Ll or a 11 11,,-contraction in Lo n L. .The main aim of this paper is to prove that, for a wide class of N-functions ~, ¡he ergodic maximal operator MT defined by is bounded in Lp (dominated ergodic theorem) .Moreover, we shall prove that if {bk} is a bounded Besicovitch sequence, then for every f E Lp there exists f* E Lo such that lim 1 bkTkf(x) = f*(x) a.e ., n-oo n k=0 A sequence of complex numbers {bk} is called a Besicovitch sequence if for every e > 0 there exisis a trigonometric polynomial cY E such that 1 n-1 limsup-1: 1bk -aE(k)¡ < e.As a special case we obtain the almost everywhere convergence (individual ergodic theorem) and the norm convergence (mean ergodic theorem) of the Césiaroaverages n-1 (f + Tf + ---+ Tn-1 f) .
In the real Lp-case, with 1 < p < oo, and (X, .M, p) a finite measure space the corresponding dominated ergodic theorem is proved by A. de la Torre in [10] .R. Sato proved in [9] that the de la Torres result may be extended to the case (X, ,M, p) v-finite and a complex Lp-space .The ergodic result for an operator which only satisfies conditions i) and iii) is an open problem even in the Lp-case, 1 < p < oo.
The bounded Besicovitch sequences as weights in the averages were used by J.H. Olsen in [8] .
In order to obtain the dominated ergodic theorem we first need some extrapoldion theorems which extend the ones given by M .A. Akcoglu and R.V. Chacon in [1] and R. Sato in [9], for Lp , 1 < p < oo .Now, we shall present the basic definitions and results concerning to Nfunctions and Orlicz spaces which will be used in this paper.The proofs of most of there results can be found in [5] or in II-13 of [7] .
The function .0 is an N-function if and only if it has the representation O(s) = fo co where ;o : [0, oo) -> R is continuous from the right, non decreasing such that cp(s) > 0, s > 0, W(O) = 0 and cp(s) )oe as s)oo .More precisely cp is the right derivate of 0 and will be called the density function of 0.
Associated to ep we have the function p : [0, oo) -+ R defined by p(t) _ sup{s : ~o(s) < t} which has the same aforementioned properties of cp .We will call p ¡he generalized inverse of cp .
An N-function 0 is said to satisfy the ¿nS2 -condition in [so , 00), so _> 0, if there exists a constant a such that ¢(2s) < ceo(s) for every s > so .
If cp is the density function of 0, then 0 satisfies 02 in [so, oo) if and only if there exists a constant a > 1 such that scp(s) < ao(s) , s > so .
The Z~2 -condition for ¢ does not transfer necessarily to the complementary N-function.
If (X, M, h) is a Q-finite measure space we denote by M = M(X, M, lí) the space of M-measurable and p-a .e. finite functions from X ro R or to C. If 0 is an N-function we consider the Orlicz spaces Lp -LO (X, M, ¡i) and L p* -L, 5* (X, M, y) defined by Lo = {f E M : fx O(I fj)dlc < oo} and L,p* = {f E M : fg E L1 for all g E L o } where 0 is the complementary N-function of 0. We have Lo C Lo * and if 0 satisfies 2~12 then Lo = L,5* .
We have that Lp* is a linear space with the usual operations on which we may define the norms Ilf ¡lo = sup{fx JfgI dh : g E Sp}, where S,p = {g E L,p : fx 0(1g1)dp 5 1}, and lif11(0) = inf{A > 0 : fx 0(A-1 jfj)dh < 1} which are called Orlicz norm and Luxemburg norm respectively.Both norms are equivalent .
Holder's inequality asserts that for every f E Lp* and every g E Lp* we have jjfgjjl < jif 11(0)lIgil~p where ¢ and 0 are complementary N-functions .
The convergente fn --> f in [L o*j 110) implies the mean convergente lima-.fjfn -f j)dp = 0 but, in general, mean convergente only implies norm convergente when ¢ satisfies ZN2 .Then the set S of simple functions (with support of finite measure) is dense in [Ls, 11 jj<p] if 0 satisfies 02 .
If 0 verifies 1n12, then for every continuous linear functional F over [Lk, 11 there exists an unique function g E L,p* such that F(f) = fx fgdp, f E LO, and moreover JIF11(,p) = ligijo, where 0 is the complementary N-function of 0, but if 0 does not satisfy ¿n12 then there exist linear functionals on LS* which are not represented by functions of L,p* .
In the following, we shall always assume that (X, M, y) is a a-finite measure space and 0, together with its complementary N-function 0, satisfy the ¿n12 - condition in [0, oo) .The '~12 -condition for 0 is a very important condition that plays fundamental roles in many questions and the best known Orlicz spaces are associated to functions which satisfy ¿n12 .The \ 2 -condition for 0 may seem to be a restrictiva assumption.Some know Orlicz spaces as, for example, the Zygmund Orlicz space L Log L and the L Log k L spaces, k > 0, are associated to N-functions which satisfy Inl2 but their complementary N-functions do not ; but the above spaces do not satisfy our dominated ergodic result .In fact ¡he A2-condition for the complementary N-function is necessary for such result .
Precisely, let ([0,11, B, A) be the Lebesgue-space and let r an invertible Ameasure preserving transformation from [0,1] into itself.In [2] B. Bru and H. Heinich characterize the Orlicz spaces, associated to Young's functions, for which the ergodic maximal operator associated to the operator T, defined by Tf = f o r-1 , is bounded in Lo (classical dominated ergodic theorem) (the Young's functions in [2] are our N-functions) .The characterizing condition given in [2] is the condition of comoderation on 0.
The function 0 is said to be comoderated if there exist so , a and b > 1 such that cp(as) >_ bep(s) for s >_ so , where cp is the density function of 0 or, equivalently, if there exist so , a and b > 1 such that ¢(as) >_ abO(s) for s >_ so (in [2] a function continuous from the left is taken as density function of whereas our density function is right continuous) .
The papar [21 does not establisch the equivalence between the comoderation of ¢ and the moderation (Ini2 -condition in some [t o, oo)) of the complementary N-function 0 unless cp be continuous .However, we observe that the comode- ration of 0 is equivalent to the moderation of 0. At the same time, we shall prove another characterization of the moderation of 0, which is usad in this paper, and which appear in [2], [5] and in the test of the literature with more restrictiva hypothesis .Exactly : Proposition 1.2 .Let 0 be an N-function and 0 the complementary Nfunction of 0. The following conditions are equivalent: c) There exist so and ,l > 1 such that QO(s) < scp(s) for s > so .
c) ==:> a).Condition c) implies that there exist so and fl > 1 such that the function s % s -Q«s) increases for s >_ so (or for s > so if so = 0) .Then, if a > 1 is such that aQ-1 >_ 2 we have O(as) > aOO(s) > 2ao(s) for s >_ so and thus we obtain the comoderation of 0 .
Note.Since cp(0) = p(0) = 0, if some of the conditions of Proposition 1.2 is satisfied for every s >_ 0, then the others two conditions are also valids for every s > 0.
In this way, the moderation of 0 is necessary for the classical dominated ergodic result and, therefore, for our dominated ergodic result since that the operator T, defined by Tf = fo-r-1 satisfies conditions i), ii) and iii), whatever the N-function 0 may be.On the other hand, the space ([0,1],13, A) is of finite measure and our spaces can be of infinite measure.For this reason we shall assume the InS2 -condition in [0, oo), but un the case p(X) < oo the argument which we shall use can be adapted if only we suppose the A2-condition in some [so, 00) .
Our results are valid, for example, for the known Lp Logk L spaces; with p > 1 and k >_ 0, since the N-functions of the form ¢(s) = sp logk (1 + s) satisfy that 1 < p < O(s)/scp(s) < p + b for every s > 0 and certain constant b.

Extrapolation Theorems
We first observe that the convexity theorem for positive operators given by M.A .Akcoglu and R.V. Chacon in [1] can be easily extended to Orlicz spaces, following the same type of arguments, as follows Proposition 2 .1 .Let ¢ be an N-function strictly convex in some interval and le¡ T be a conservative positive contraction in .L1 such that Then, 11Tf ¡l .< II f 1I,, for every f E L1 n L~.
Proof.The operator T is said to be conservative when p(D) = 0, where D is the dissipative part of X with respect to T.
First assume that M(X) < oo.It is enough to prove that Tc _< c almost everywhere for some constant c qÉ 0.
We have that co increases strictly in some interval I, where yo is the density function of 0. Let c E I with c qÉ 0.Then, we get that Since T is conservative we have fX Tfdp = fX f dp for every f E L1 .
The general case follows from the preceding by a method similar to the one given in [1] using the following result : The proof of Lemma 2.4 can be obtained easily following the argumenta of Remarks .
1.The conservative condition of T cannot be eliminated from the hypothesis of Proposition 2.1 since in R with Lebesgue-measure if Tf(x) = -,í2f (2x) then T is a positive contraction in Ll, an isometry in L2 but 11Tf Ij,>,> v"2-1If11 2. There exist N-function which are strictly convex over no interval .An example is the following .We consider the dyadic intervals In = [2n-1 , 2n) and Jn = [2-n , 2-n+1) where n is a positive integer and let c,o : [0, oo) -+ [0, oo) be defined by cp(0) = 0, W(t) = 2-n if t E Jn and cp(t) = 2n-1 if t E In.Then ¢ defined by ¢(s) = fo ep is an N-function .Since 0(2s) = 4«s) we have that 0, as well as its complementary N-function, satisfy the ¿n12 -condition .However 0 is not strictly convex over any interval.Furthemore there is no constant c :,~0 such that (2.3) holds .
However most of N-functions are strictly convex in some interval .
In the following results the operators are not necessarily positive but they have a fixed point h with h :~0 a.e.Theorem 2.5.Let 0 be an N-function, strictly convex in some inierval and let T: Lo -> Lo be a linear operator such that Then, IITf Ii .< Ilf II<>~f or every f E Ll n L,,. and consequently for every f E Lo n L,,.
Proof. .In this proof we follows the idea given by Sato in [9].Let k be such that O(s) < s for 0 < s < k.Given f E Ll n L,,,, let B = {x E X : I f(x)l >_ k} ; then M(B) < oo and therefore fx 0(I f I )dp <_ llfll1 -}-p(B)0(11f II) < oo .Consequently Ll n L,,.C Lo .
Let T : Ll -> L1 be the linear extension of T : [L, n Lo , II 11,1 --, Ll and P the linear modulus of T. (See Theorem 4.1.1 in [6]) .We shall prove that P satisfles the hypothesesof Proposition 2.1 and therefore 11Pf ll<, .< Ilf IIf E Ll n L,,. ; in this way, since ITfI <_ Pi f I , f E L1 , and Ll n L. C Ll n Lo we obtain that 11Tf II < lif l¡ ., f E Ll n L,,,, and consequently for every f E Lo n L,,, since Ll n L,,, is dense in Lo n L,,,, with the L,,-norm .Now, we show that P satisfles the conditions of Proposition 2.1.The A2condition implies that L 1 n Lo is dense in [Lo , II II(o)] .On the other hand, it follows from i) that 11Tf 11 (0) < llfll(0) , f E Lo, and consequently given e > 0 there is fE E Ll n Lo such that for every n > 1 ii> IITf1I1 < Ilf1I1, (f E Ll n LO) .iii) There exists h E LO , h :~0 a. e., such that Th = h.If T is a power bounded linear operator in a reflexive Banach space V, that is, the powers Tk , k > 0, are uniformly bounded in V, then the Césáro-averages .Ix 0(Ihfé * j)dp < e .
On the other hand, fE *(x) = 0 for a.e.x E D, where D is the dissipa,tive part of X with respect to P, since (Theorem 3 .1 .6 in [6]) Ek>0 Pk f(x) < 00 on D for all f E L1 + .Since 0(Ihj) > 0 a.e.(2.7) shows that p(D) = 0 and thus P is conservative .Now, in order to prove that P satisfies condition (2.2) we consider the Akcoglu and Brunel's theorem related with the structure of T on the conservative part C of X with respect to P (see Theorem 4.1.10in [6]) .Let .'Fbe the family of P-absorbing subsets of C; there exists a set F E .'F and a function s E L,, .(F),with ¡si = 1 on F, such that Tf = jP(s f) for any f E Ll(I'), where s is the complex conjugate of s, and if ¿ni = C-F then (I'-T)L,(Li) is dense in L, (o) .
We have that supp T(Xr h) C F and supp T(Xoh) C ¿ni ; therefore Tg = g where g = Xoh .Carryng out a similar reasoning to the used for h we have that for every e > 0 there exist fE E Lj(o) n Lp(A) and f, * E Lo(A) such that Ii9 -fE*jj(0) < E and lim n,,> IIRnf, -fE * 1I(0) = 0.
Let T : L< p -> Lp be a bounded linear operator; more precisely, we suppose that there is a constant C such that 1I T f 11(0) <_ Cl i f 11( 0), f E L,4 .Then, if g E L b *, where 0 is the complementary N-function of 0, the linear functional F over [LO , 11 jj( .p)]defined by F(f) = fx gT f dM is continuous since by Holder's inequality we have IF(f)j < ClIgilplifjj(m) and therefore, since 0 satisfies ~1z, there exists an unique function g* E L,p* such that fx gT f dp = fx f g*dp, f E Lo .Then, we can define the bounded linear operator T* : Lp* -~L,p*, g -T *g, where T*g is the function in L,p* such that f 9Tfdh = f fT *9dlí, f E Lo.
x X We shall call T* the adjoint operator of T. T* satisfies IIT*gllp <_ CIigil+p .In our case we have Lemma 2 .8.Let T : Lo -+ Lo be a linear operator such that Then, its adjoint operator T* satisfies f 0(ITfi)dh <-f 0(If¡)dh (f E Lm).
x x (2 .9)f 0(IT*gl)dh <-f 0(Ig1)dli (g E L~p) x x and moreover, if T admits an invariant function h with h :~0 a. e., then there exists g E L,y with g :~0 a. e., such that T*g = g .
Proof.We write sig z for z11x1 and by ú we denote the complex conjugate of u .For g E Lo+ we have Let cp be the density function of ~and p the generalized inverse of cp.Since satisfies O2 there exists a > 1 such that sp(s) < aO(s) and therefore O(p(s)) _ sp(s) -O(s) < (a -1)0(s).Therefore, for every g E Lp the function p(IT*g1) belongs to L p+ and so (2.9) follows from (2 .10)for f = p(IT*g1) .Now, let us assume that Th = h with h :~0 a.e.If cp is not continuous then there exists an at most countable set of positive reals sl , s2, . . ., s,, where cp is not continuous ; in this situation, Since h E L<p, it is easy to see that {c > 0 : a{x E X : ¡s i i h(x)I = c} > 0} is at most countable and therefore there exists A > 0 such that for every si we have (2.11) p{x E X: JA-lh(x)I = si} = 0.
In the case cp continuous (2.11) holds trivially with A = 1.
Since ITf1 <_ PIf1 for f E L1, (3 .4)shows that there exists a constant Cl > 0 such that fx O(MT f )dp < Cl f x O(If 1)dit , f E L1 n LO, which proves that IIMT f II(0) < CII .f1I(0) , f E L1 n LO, where C = max(1, Cl) .Since L1 n Lp is a dense linear subspace of [L O , II II(m)l it follows that IIMTf II(0) < CIIf II(0) for every f E Lp, which proves a).Now, let {bk} be a bounded Besicovitch sequence; then a) and the Banach principle show that for almost everywhere convergence it is enough to prove that the weighted averages Let m E N and S : Lo --+ Lo defined by Sf = e"°T f .Since L, 5 is reflexive and the powers Sk , k >_ 0, are uniformly bounded, exactly 11 S'f11(0) <-11f11(0) for every f E L< p and k >_ 0, then , the Césáro averages Rnf = n-1(f + Sf + . . .Sn -1 ) converge in norm for every f E Ls .Therefore L, 5 is the closure of the direct sum of the set of fixed points of S and the space (I -S)Lo (see 2.1 in [6]).
On the other hand, given f3 > 1 such that &(s) < scp(s), s >_ 0, the function s--)s-QO(s) increases for s > 0 and consequently O(st) < sO¢(t) for 0 <_ s <_ 1 and t > 0. Therefore, if g E Lo we have J ~0(In -1 Sngl)dp X Since the maximal operator MS is bounded in [Lo, II II(o)] we obtain that, for any f E LO, n -1 rk=01 e'm kT k f converges a.e. and therefore for every trigonometric polynomial a and f E Lo we have that lim -1: a(k)T n-oo n k=0 exists and is finite a.e.
Then, for every f E Lo fl L., Tn f converges a.e.since for every e > 0 there exists a trigonometric pólynomial af such that and consequently In this way, since Lo fl L. . is dense in Lo, we conclude that Tn f converges almost everywhere for every f E Lo.Finally, let f*(x) = lim, Tnf(x).It follows from a) that f * E Lo and 0(I Tnf -f* I) is dominated by O(MT f) E L1 ; thus, taking into account the Lebesgue's dominated theorem, we get that lima, fx 0(I Tnf -f * I )dp = 0 which proves that lim n -w IITn f -f*II(0) = 0.

[ 6 Lef 2
fE * be the limit in [LO , II II(0)] of Rn f£ .It follows from (2.6) that for 0 < e < 1 we have II h -fÉ II (m) < e and consequently (to a T-invariant limit for all f E V (See Theorem 2.1 .2 in

that 3
For f E Ll n Lo set fA = fXA(A) and f a = ffa where A(A) = {x E X I f (x)I > A/2} .We have fa E L1 , f A E L1 n L. and therefore ( e. for all f in a dense subset of [LO, II II(o)] .