Multiparameter pointwise ergodic theorems for Markov operators on L∞

Publicacions Matemàtiques, Vol 38 (1994), 395-410 . MULTIPARAMETER POINTWIS E ERGODIC THEOREM S FOR MARKOV OPERATORS ON L oo RYOTARO SAT O A bstract  Let P 1 , . . . , P d be commuting Markov operators on L 00 (X, .~ , where (X, .T, I .r,) is a probability measure space . Assuming tha t each P i is either conservative or invertible, we prove that for ever y f in Lp (X,  with 1 Ç p C oo the average s .A n f = (n + 1 ) —d E Pí n1 P2 2 . . .Pd d f (n O ) OÇn i Ç n converge almost everywhere if and only if there exists an invarian t and equivalent finite measure A for which the Radon-Nikody m derivativa v = da/dp, is in the dual space L p ~ (X, ,~, ~c} . Nex t we study the case in which there exists p1, with 1 C p1 C_ oo , such that for every f in Lp (X, .~, p,) the limit function belongs t o Lp l (X, .~, ~c} . We give necessary and sufficient conditions far thi s problem . 1 . Introductio n Let (X, ,F, II) be a probability measure space and let P i (i = 1, 2, . . . ,d ) be commuting Markov operators defined on L oo (X, II) . In this pape r we assume that each P i is conservative or invertible, and prove that th e averages Anf = (n- }-1) — d E P1 1 P2~ . . .Pd d f (n O ) OÇn i Ç n converge almost everywhere for every f in L p (X,, F , , .c), 1 < p < oo, i f and only if there exists an invariant and equivalent finite measure a fo r which the Radon - Nikodym derivative v = da/dp, satisfies v E Lp,(pt ) with 1/p + 1/p' = 1 . This is a new result, even if d = 1 . Next we 396  R . SAT O consider the question whether there exists a constant p 1 , 1 Ç p 1 ç oa , far which the limit functions belong to L p ~ (X, .f ' , p,) . We give necessar y and sufficient conditions for this problem . Let (X, Y', p,) be a probability measure space . By a Markov operato r P defined on L oo (p,) = Loe (X, F, ~ .c} we mean a linear operator fro m L,,(p) into itself such tha t (i) P is positive : P f ~ 0 whenever f E L + ,,(p,) , (u) P is a contraction : II P lloe ç 1 , (iii) limP f n -= o a . e . on X whenever f n E Lt(p) and f n ., 0 a .e . on X . n By virtue of (iii) and the Radon-Nikodym theorem we may define a n operator on L 1 (11) = L i (X, .~, 11) by the relatio n uPdµ = Lu1B(u E L(µ) and B E F) , 113 being the indicator function of B . Here we use the same symbol P for the L i (p,) operator, but we write it to the right of its variable . P is called invertible if P1 = 1 and P is a one-to-one onto mapping o f L, 3 (p,), and conservative if f E L+ ,,(p) and P f Ç f imply P f = f . I t 0 0 is known (see e . g . [4]) that P is conservative if and only if E uP l = = o 0 z - o a .e . oil X whenever u E L1 (p) and u > o a .e . on X . Since P is positive , we may extend the domain of P to the space M+ (p) = M + (X, .F', o f all nonnegative extended real valued measurable functions on X, by th e relations Pf = 1im P f n a .e . where f 7z E L 4- (p) and f n fi f on X n uP =1im u n P a .e . where un E Lt(t .c) and u n fi u on X . n It is easily seen that by this process Pf and uP are uniquely determine d a .e . on X . We then hav e (u P , f) = X (uP)fdji = x u ( P .f) d u = (u, P .Í ) In this paper we consider several commuting Markov operators P i (i = 1, 2, . . . , d) defined on L oo (p) . Throughout the paper we will assume tha t L and MULTIPARAMETER PoINTWISE ERGODIC THEOREMS  397 eac h P i is conserva t ive or invertible . Associated with these operators w e define the average s A n f= (n-}-1 ) --d E P1 i P2 2 .Pd d f (n~O ) o5 , n i c n and the maximal operator M f = sup A n ~ f I n> 0 Also we define the o--field of all invariant subsets of X : 1 ={ B EF :Pz1 B = 1 B foralli = We call the system {P 1 , P 2 , . . . , P d } ergodic if 1 is the trivial cr-field . It is known that if p, is invariant under the P i , i .e ., 1Pz = 1 for al l i = 1, 2, . . . , d then the sequence {Af} converges a .e . for every f i n Lp (p)„ 1 Ç p Ç oc . But, if ~ is not assumed to be invariant, we canno t expect the almost everywhere convergence of {Af} for every f in L p (ji) , 1 Ç p := oo . Therefore the author thinks that it would be of interest t o characterize those probability measures ~ for which the sequence {Af } converges a . e . for every f inL p 4 -(p) , 1 Ç p Ç oo . As is easily seen , such a characterization for p = oo is the existence of an invariant an d equivalent finite measure . Thus we will concern ourselves with the cas e 1 Ç p < oo below . It is interesting to note here that this problem fo r d -parameter groups of null preserving point transformations was recentl y examined by Martín-Reyes [51 . Hence our results may be regarded a s generalizations (and improvements) of those due to Martín-Reyes . W e will give a new characterization which has a connection with the invarian t measure problem . That is, we will prove in Section 3 that the sequenc e {Af} converges a . e . for every f in Lp (p,) if and only if there exists a n invariant and equivalent finite measure a for which the Radon-Nikody m derivative v = da / d p, satisfie s v E LP (X,  with 1/p + 1/p ' = 1 . Next, let us suppose that f* (x) = limA 7z f (x) exists a .e . for all f i n n L p + (p,), 1 Ç p Ç oo . Do the functions f* belong toLp (p,)? As is easil y seen, this is not true in general, unless p = oc . So the following questio n arises riaturally . Does there exist a constant p i , 1 Ç p i < oc, such tha t f* E L p + i (¡c) for all f E L p + (p,) ? We will prove in Section 3 that the limi t f* = 1 ~m An f exists a .e . and f* E L~ (p,) for all f in L p (~c) if and only 398  R . SAT O if for any u E L pi (II), 1/p i +  = 1, the limit uo = lim uA n exist s n a .e and in the norm topology of L 1 (and further we have u¿ E L p ► 1/p + 1/p' = 1 . In Section 2 we study the special case where ~ is invariant under th e Pi, and consider the measure a = V dp, where V is a positive measurabl e function on X . Regarding the P i as commuting Markov operators define d on L 0 , 0 ( X, , V d,u) , we obtain some preliminary results which may be o f independent interest by themselves . In what follows two functions f and g are nat distinguished provide d that f = g a .e . on X , and if 1 Ç p Ç oo then p ' will be its conjugat e exponent, Le . e . 1/p + 1/p' = 1 . 2 . The invariant measure cas e In this section we assume that the probability measure is invarian t under the or equivalently that 1P i = 1 = P i 1 for all i = 1, 2, . . . , d , and consider the measure a = Vd í u, where V is a positive measurabl e function on X . It may happen that aX = f X V d p, = oo . Theorem 2 .1 . Let P i (1 Ç i Ç d) be commutirig M arkov operator s on L~ (X, Y', l .c), where (X, F, ~u} is a probability measure space . Assum e that p, is invariant under the Pi . Let V be a positive measurable functio n on X, and let 1 Ç p < oo . Then the followingare equivalent . (a) Por every f in L p + (V d p,) , limA n f exists and is finite a . e . on X . n (b) Por every f in L p +(V d p,) , M f < cc a .e . on X . (c) There exists a positive measurable function U on X such tha t Ud,u Çt TP  f P Vd~c (t>O,feL(Vd)) . J ~vr f~t}  x (d) There exists a positive measurable function U on X such tha t  lim inf  U d,u t -P  f P V dp, (t > o, f E LP (V dtc)} . n  {An f >t}  X (e) ess sup{ W E M + (X, Z, µ) : W< V } > 0 a .e . on X (if p = 1) ; E{V l-P' 11} c oo a .e . onX (if 1 < p < oo) . Proof : It is clear that (a)  (b) and (c) ~ (d) . (b) ~ (c) . (b) implies that far each n ~ 0 and all f E L l v - (V dp,) , ~ A n f 1 C oo a . e . on X . Thus the operator A, z can be considered to be a MULTIPARAMETER POINTWISE ERG4DIC THEOREMS  399 continuous mapping from L p (V dp) to Lo (,u), where Lo (p) denotes th e space of all finite valued measurable functions on X, equipped with th e topology of the convergence in measure . By this and Banach's principi e (cf . e .g . [3, p . 2] ) we see that the sublinear operator f —> MI f ~ i s continuous from L p (V d,u) to Lo (p,) . Hence (c) follows from Nikishin ' s theorem (cf . [2, p . 536]) . (d) (a) . Let f E Lp (V d p,) , and choose f N E Lt(p,) , N = 1, 2, . . . , so that f N fi f on X . Since is invariant under the P i , it foliows tha t lI1iI1r = 1 for all 1 ç r Ç oo, and hence the classical pointwise ergodi c theorem far d -parameter semigroups of Dunford-Schwartz operators ca n be applied to infer tha t lim A n f N = E{ f N II} a . e . on X . n Using this we see that the pointwise limit f * (x) = limA n f (x) exists a .e . n on X (but may equal to infinity on some subset of X ) and tha t f* = E{ f l Z} a .e . on X . To prove that f* < oo a .e . on X, let us write B = {x : f * (x) = oo } . Then, since B c Hm inf {A n f ~ t} far any t ~ O, we have by Fatou ' s n lemma, and (d) , B U d p, ç lim inf  U 46 Ç t -p f P V d~c . n  {A n, ~~~}  ~ Letting t fi oo, it follows that f B U dp, = O, and consequently p,B = O . (a) -~ (e) . As in [S], we use the ergodic decomposition technique . W e first note that for the proof it may be assumed without ioss of generalit y that (X, .~, II) is a Lebesgue measure space in the sense of Rokhlin [6] . Then using Rokhiin's theory we can find a countable family {E} of set s in 1 such that if denotes the decomposition of X induced by {E}, i .e . , C E 1 has the form C = flE(e ) where Ei = +1, E i (l) = Ei and E i ( --1 ) = X \ E i, then : (i) The factor space (X/,  p l ) of (X, p,) with respect to is a Lebesgue measure space . (u) To a .e . C E X/e with respect to  there correspondsa Lebesgu e measure p,c on C such that if B E .~ then B ~1 C is measurable with 400  R . SAT O respect to p,c for a .e . C E X /1, and the function h(C) = p , c ($ n C) i s measurable with respect to /I I and satisfies, for all Z E F of the for m Z = c -1 (ZM , where t : X —> X /1 denotes the canonical mapping , (BnZ)=fh(C)d(C) = JZ/(B n C ) d ( C ) . (iii) To a .e . C E X /e there correspond commuting Markov operators P(C) 1 , P(C) 2 , . . . , P(C)d defined on L,,,(C, p , c ) such that th e system {P(C) 1 , P(C)2, . . . , P(C)d} is ergodic, and also such that i f u E L I (X, F, tt) and f E L,,(X, F, p) then, for a .e . C E X/ , ucP(C) i = (uP 2 )c and P(C)ifc = (Pjf)c for all i = 1, 2, . . . d , where uc, (uP i )c , fc and (Pf)c denote, respectively, the restrictio n functions of u, uPz 7 f and P i f to the set C . To prove the implication (a) ~ (e) for p = = 1 we defin e ~ = ess sup { W E M+ (X, Z, ~,c} : W V } , and assuming that 11{12 - = 0} > 0, we derive a contradiction as follows : Since {V = 0} E 1, it may be supposed without loss of generality tha t 02 - = 0} = X . Then we use the ergodic decomposition technique . Defin e the funciton h n on X /1 b y h n (C) = µc({V< 1/n} n c) (n > 1) . Since h n (C) > 0 for a .e . C E X /1' because ~ = 0 on X, if f n denote s the functionoil X defined b y then we hav e .fn( x ) = n—1 (hn(c)Í -11{V<1/n}(x) (x E C E X/0 , dp,c = 1/n (C E X/ ) f c 1 f nV dµ =  ( f nV dµc ) dµl (c ) L  L ,e c f X ~~( 1 / n2 ) d µ~( C ) = 1/n 2 . and MULTIPARAMETER POINTWISE ERGODIC THEOREMS  40 1 0 0 Therefore the function f = E f n satisfies f E L 1 (V 4) . But, sinc e n= x f c Li) for a . e . C E X/1, it follows from (iii) and the classical pointwise ergodic theorem for d -parameter semigroups of Dunford - Schwarz operators that for a .e . C E X /e , f * (x) = lim A n f (x) = oc a .e . on C n with respect to the measure pc . It follows that f* (x) =oo a .e . on X with respect to the measure ~c . This contradicts (a) . Next, let us consider the case 1 C p < oo, and suppose that the se t B = {x : E {V 1—P ' 1 Z} (x) = oa } is not a null set . Since B E 1, we then suppose without loss of generalit y that B = X, and from this we derive a contradiction as follows . Firs t we note that if E{ V 1 ^ p' jz} = oo a . e . on X the n fc V 1 —P dµc = oo for a .e . C E X/ 1 . Using this and doing as in the proof of the implication (a) ~ (b) o f Theorem 1 in [8] (see especially p . 75 in [81) it is possible to constructa function f in L - p F (V dp,) so that for a .e . C E F/1, where F is a set in 1 and satisfies p,F ~ o, we have f c f dp,c = oo . Then it follows from (iii ) that for a .e . C E F7 , f * (x) = lim AJ(x) = co a .e . on C n with respect to the measure pe . Hence f* (x) = oo a .e . on F wit h respect to the measure ~ . This contradicts (a), because f E L p (V dp,) . (e) (a) . (e) implies the existence of a sequence {X N } of sets in 1 , with XN fi X, such that if VN denotes the restriction function of V t o XN then V I ç7 1 E LP (X N , V dp,) . Here, since XN E 1, for the proof of (a) it may be supposed withou t loss of generality that X N = X . Then for any f E Lp (V d~c} the Holde r inequality yields V (L f V dµ) L~ (X,V dµ} C o c 1 Lf=Lf~ CL f pV dµ /  ,x ( v i )P'vdp)l'P' <0 0 (1 < p < oo), 402  R . SAT O so that f E Li Cu), and hence (a) follows from the classical pointwis e ergodic theorem for d -parameter semigroups of Dunford-Schwartz operators . The proof is complete . ■ The following corollary is immediate from the equivalente of (a) and (e) in Theorem 2 .1 . Corollary 2 .2 . Let P i (i = 1, 2, . . . , d) be as in Theorem 2 .1 . Suppos e in addition that the system {P 1 , P 2 , . . . , P d }is ergodic . Then 1imA n f n exists and is finite a . e . on X for every f in Lp (V dp,) , 1 Ç p < oo, i f and only if V —1 E L p, {X, .1 , V d,u} . 3 . The general cas e Theorem 3 .1 . Let (X, .I T , l .c) be a probability measure space and le t P i (i = 1, 2, . . . , d) be commuting Markov operators on L co ( X, Y", p,) . Assume that each P i is conservative or invertible . If 1 Ç p < oo, the n the f ollowing are equivalent . (a) Por every f in Lp (p,), 1 ñ mAn f exists and is finite a . e . on X . (b) Por every f in Lp (11), 11r1 f < oo a . e . on X . (c) There exists a positive measurable function U on X such tha t f  Udµ < t -p J f P dµ (t>0,fEL()) . M f >t}  X (d) There exist a positive measurable function U on X, a positiv e constant r and a subsequence {n(k)} of the sequence {n} suc h that r/ p U d á u Ç t -r  f P olc  (t ~ o, f E L - 1 ;(p)) . f An ( k ) f >t} X (e) Por every u in L 1 (p,), the sequence {uA n } converges in the nor m topology of L 1 (p,) and also a .e . on X ; further to each v E Lt(11 ) with vPz = v for all i = 1, 2, . . . , d there corresponds a sequenc e { X N } of sets in 1 such that XN fi X and the restriction functio n vN of v to XN is in L p ,(X N , ¡~} for each N > 1 . (f) There exists v E L -F ,(p) with v ~ 0 a . e . on X and v P i = v f or al l p 1 ,2 , . . . ,d . In order to prove Theorem 3 .1 we begin by proving the following MULTIPARAMETER POINTWISE ERGODIC THEOREMS  40 3 Lernma 3 .2 . Let P i (i = 1, 2, . . . , d) be commuting Markov operator s on L oc (X, Y', p,), where (X, p,) is a probability measure space . Assum e that each P i is conservative or invertible . Then the sequence {Af } converges a . e . on X for every f in L eo (X, Y', ,u) if and only if there exist s a function v in LRX, p,} such that v > 0 a . e . on X and vPz = v fo r all i =- 1,2, . . .,d . Proof : Suppose the first assertion of the lemma holds . Then far an y u E L 1 (,u) and f E L,,(p,) the sequenc e ( uA na f) =  (uAn) f d~c =  u(A n f) d~c = (u, A nf i  x  x converges to a finite limit as n — } oo . It follows from the Vitali-Hahn - Saks theorem that {uA} converges weakly in L 1 (,u) . Hence by a mea n ergodic theorem (c .f . e .g . [4, Theorem 2 .1 .5]), {uA} converges strongl y in L 1 () . Let v E LRp,} be the limit function of the sequence {1A} i n L 1 (p,) . Since v P i = v for each i, it follows that the set B = {x : v(x) = 0 } satisfies P i 1 B Ç 1 B for each i . Here if P i is conservative, then w e have P a B = 113 . On the other hand, if P i is invertible then, sinc e vPz = v= vPti — 1 , we have Pz 1 B = 1 B , too . Consequently B E 1, and thus p,B = ( 1,An1B) = (1A n ,1 B ) —>  v d~u = o . B Conversely, if the second assertion holds, then the P i may be regarde d as commuting Markov operators defined on L oo (X, .~, v d p,) such tha t = 1 ( 1 Ç i d) . Thus we may apply the classical pointwise ergodic theorem far d -parameter semigroups of Dunford-Schwartz operators to infer that the firs t assertion of the lemma holds . ■ P r oof of Theorern . 3 .1 : (a) ~ (b) and (c) ~ (d) are immediate . Th e proof of (b)  (c) is the same as that of the corresponding part o f Theorem 2 .1 . (f)  (a) follows from the classical d -parameter pointwis e ergodic theorem, since L P (14 c L 1 (v d p,) . (d) ~ (a) . We may suppose that 0 < U < 1 on X . Given an e > 0 , choose (5 > 0 so that 1 .1{U < 6} C E . Then, since A n 1 B Ç 1 on X for any B  (d) implie s (Ar i (/e)l B ) 41, < 6—1 J (A (k) 1 B )Ud + {U < 6 } eS~ l  Udp+ t Udp, + e {A,(k)a .B>t}  x < 6 -1 t —r (p,B) r/P + 8 —1 t + E (t 0) . L 410  R . SAT O 8. SATO R ., On pointwise ergodic theorems for positive operators, Stu - dia Math . 97 (1990), 71-84 . 9. SATO R ., Pointwise ergodic theorems in Lorentz spaces L(p, g) for null preserving transformations, submitted for publication . Department of Mathematic s School of Scienc e Okayama Universit y okayama 70 0 JAPA N Rebut el 4 de Març de 1994 .