On a pointwise ergodio theorem for multiparameter semigroups

Publicacions Matemátiques, Vol 38 (1994), 81—87 . ON A POINTWISE ERGODIO THEORE M FOR MULTIPARAMETER SEMIGROUP S RYOTARO SAT O A bstract  Let Ti (i = 1, 2, . . . ,d) be commuting null preserving transforma - tions on a finite measure space (X, F, p) and let 1 Ç p < oo . I n this paper we prove that far every f E L P (p,) the average s An .f (x) = (n+ 1)—d E f(T i nl T2 2 oÇni Ç7 a converge a .e . on X if and only if there exists a finite invarian t measure v (under the transformations Ti) absolutely continuou s with respect to and a sequence {X - N} of invariant sets wit h XN T X such that vB ~ Q for all nonnuil invariant sets B an d such that the Radon-Nikodym derivative v = dv/dp satisfies v E (xN, 11), l/p + 1/q = 1, for each N > 1 . 1 . Introductio n We refer to [2] for the basic notation in ergodic theory . Let (X, /2 ) be a finite measure space and let T i : X - -> X (i = 1,2, . . . , d) be commuting null preserving transformations, where d ~ 1 is a fixed integer . Associated with these transformations T i and far any measurable function f on X we have the average s A n f (x) = (n+ 1) —d E f (T1 x T2 2 . . . Td d x) (n ~ o ) QÇn i Çy t and the maximal operator M f — sup A n ~ f 1 . n ~ o 82  R . SAT O Further each T i defines, by the Radon-Nikodym Theorem, a unique positive linear contraction operator T i * on L 1 0C} by the relation fBfT'B (ú E L 1 (), B E ) . Under the additional hypothesis that all the transformations T i ar e invertible, Martín-Reyes [3] has recently proved the equivalence of th e following conditions, for 1 Ç p < oa . (a) The sequence {Af} converges ase . for all f in L p(p,) ; (b) There exists a positive measurable function U on X such tha t f  Udt J ~ .Í~ d ( t > 0, .f E IA n f i>t}  X In this paper, without assuming the invertibility hypothesis on th e transformations T i , we intend to characterize those finite measures p fa r which (a) holds . 2 . The resul t Theorem . Let (X, .~, p) be afinite measure space and let T i : X—> X (i = 1, 2, . . . , d) be commuting null preserving transformations . lf 1 Ç p < oo, then the following are equivalent . (a) For any f E L p (p,) the sequence {Af} converges to a finite limi t a . e . on X . (b) For any u E L 1 ( l,c ) the average s Añu =(n + 1) -d E T1 ni T~ n2 . . . Tf d u ( oCrz i Cr a converge a .e . on X and also in t he norm topology of L1 (p) ; further to every v E Li (p,) with T i *v = v for all i = 1, 2, . . . , d ther e corresponds a sequence {XN } of invariant sets with XN T X suc h that v E Lq(X N , 11) ; 1/p + 1/q = 1, for all N > 1 . (c) There exists v E L1 (p) with T i *v = v for all i = 1, 2, . . . , d and a sequence {X N } ofinvariant sets with X N T X such that f B v dp, ~ 4 for all nonnull invariant sets B and such that v E L q (X N , ,u) , 1/p + 1/q = 1, for all 1V > 1 . (d) M f < oo for all f E L(i) . > O) PoINTWISE ERGODIC THEOREM  83 (e) There exists a positive measurable function U on X such tha t f  Udµ t–P J ~ f I p dµ (t > 0, f E L p (µ)) . M f>t}  x (f) There exists a positive measurable function U on X, a constan t r > 0, and a subsequence {n(k)} of {n} such tha t / f  \r/ p f{lA n(k) fI>t} Udµ < t –r 1 J ~ f I T ' dµ I  (t > 0, f E L p (µ)) • We begin by proving the following lemma, which deals with the cas e p =oo . Lemma . Let (X, .F, µ) be a finite measure space and let T i : X —> X (i = 1, 2, . . . , d) be commuting vull preserving transformations . Then th e following are equivalent . (a) The sequence {Af} converges a .e . on X for all f E L i (µ) . (b) The sequence {Au} converges in the norm topology of L 1 (p) fo r all u E L 1 (µ) . (c) For any u E Li (µ) with Iluil l > 0 the pointwise límit uo(x) = limAñ (x) exists a .e . on X and satisfies  > O . (d) For any u E Li (µ) with  > 0 we hav e o < II1ininfAuIk C oa . n Proof : (a) (b) follows from a mean ergodic theorem (see e .g . [2, Theorem 2 .1 .51) . (b) ~ (a) and (c) . Let vo = strong-1imAñ 1(E 4 - (II)) . Since T* vo = v o n for alli = 1, 2, . . . , d, we hav e Y C T ? 1 Yfor all i -= 1,2, . . .,d, where Y= {vo~o } . Since the measure v = vo d~c is invariant under the T i ' s, we may regar d the transformations T i as commuting measure preserving transforma - tions on a finite measure space (Y, vo d,u) . Then, by the classical multi - paramete r pointwis e ergodi c theorem , fo r any f E L,,(p, ) th e sequenc e 84 R . SAT O {Ad} converges a . e . on Y . To prove the a . e . convergence of {Af} o n X \ Y, it is sufficient to show tha t (T 1 T 2. . . T d ) rn Y fi X . To do this, let B = lim(T 1 T 2 . . . T d )T n Y . We see easily that T i - 1 B = B n for all i = 1, 2, . . . , d, Le ., B is an invariant set . Henc e p,(X\B)=f A1 d  J  vp dµ J  vp d0 . \B  X\B  { vo=0 } To prove (c), let u E Lt(p,) and I> O . Since IlAñulll = MuIIi ~ 0 and {Au} converges in the norm topology of L 1 (p,), it is sufficien t to prove the a .e . convergence of {Au} . Since the transformations T i preserve the measure v = vo d l .t, the classical pointwise ergodic theore m for multiparameter semigroups of Dunford-Schwartz operators and a n approximation argument imply that {Au} converges a .e . on Y . To prove that limAñ u ( x ) = 0 a .e . on X \Y, we use Brunel ' s Theore m n (see e .g . [2, Theorem 0 .3 .4]} concerning an ergodic inequality far commuting linear contraction operators on L 1 (p,) : there exists a constan t Kd > 0 and a positive linear operator Q on L oo (p) of the for m Qf (x) = E a(n 1 , n2, . . . , nd) f (T l ni T 2 '2 . . . Td d x) , raz ~ o where a(n 1 , n 2 , . . . , n d) > 0 and E a(n 1 , n 2 , . . . , nd) = 1, such that i f n i ~ o Q* denotes the positive linear operator on L 1 Cu} associated with Q, the n n lim supAñu Ç Kd • limsup(n + 1) -1 E Q *i u (u E Lt(p,)} . n  n i = o 0 0 Let C = {x : E Q * i u = oo}\Y . Since 11Q* {1 1 = 1, it follows tha t i = o Q1c > 1c, 1c being the indicator function of C . Thus we have C C T i -1 C for all i = 1, 2, . . . , d, and henc e ,u C  An 1c dp, =  1 d~c --~  vo d ~ =  vo d,u = 0 . x  f  fc  LO O } This proves that limA n *u(x)= 0 a .e . on X \Y . (c) ~ (d) . Obvious . PoINTWISE ERGODIC THEOREM  85 (d) ~ (b) . There exists vo E Li (µ) with T*vo = vo for all i = 1, 2, . . . , d such that if v E Li (µ) satisfies T*v = v for all i = 1, 2, . . . , d then {v > 0} C {vo > 0} . Let Y = {vo > 0} an d B = lim(T 1 T 2 . . .T d ) –n Y . n Since B and X \B are invariant sets, it follows that if u E L1 (X \ B, p, ) and Iuk > o then the functio n izo = lim infAñu satisfies {ü 0 > 0} fl {vo > 0} = n But this is impossible, since T i * ú0 = tú 0 E L1 (p) for all i = 1, 2, . . . , d and (d) implies that {ü 0 > o} > O . We conclude tha t (T 1 T 2 . . . T d ) —n Y T X . Hence by an approximation argument we see that {Au} converges i n the norm topology of L 1 (p) for all u E L 1 (p), completing the proof . ■ Proof of the Theorem : (a) ~ (b) . The first part of (b) follows from th e lemma . To prove the second part, let v E L4 - (II) be such that T : v = = v fa r all i = 1, 2, . . . , d . Putting Y = {v > O}, we see that the transformation s T i can be regarded as commuting null preserving transformations on th e measure space (Y, p) . Since the measure v = v d~u is invariant under th e transformations Ti, it follows that t hese T i are conservative on (Y, ¡ .c ) . By this and the fact that for each f inL p (Y, p,) the sequence {Af } converges to a finite limit a .e . on Y, we can apply Theorem 3 .1 in [4] t o infer that there exists a sequence {Y 1V} of sets in ly, wher e Z y = {BE .F :BCY,B =Y n Tz 1 Bforalli = such that Y N T Y and v E L q ( Y N , p,) for all N ~ 1 . Then, lettin g XN = [him(T l T 2 . . .T d y'Y N ] U [ X\him(T1T2 . . .Td ym Y ] , n we have v E L q (X N , for all N > 1, X N fi X, and X N E 1 wher e 1 ={BE .7~ :B=T i —1 B foralli=1,2, . . .,d } . (b) ~ (c) . It is enough to put v = strong-limAñ 1 . n (c)  (a) . Put Y = {v  It follows (cf . the proof of the lemma ) that ( T 1 T 2 . . . T d )'Y fi X . 86 R . SAT O Hence it is sufficient to prove that for each f in L P (Y, µ) the sequenc e {Af} converges to a finite limit a .e . on Y ; this follows from the equiv - alence of (a) and (f) of Theorem 3 .1 in [4], since the transformations T i may be regarded as commuting conservative vull preserving transforma - tions on the measure space (Y, µ) . (a) ~ (d) . Obvious . (d) ~ (e) . This follows from Nikishin's Theorem (see e .g . [1, p . 536]) . (e) ~ (f) . Obvious . (f) ~ (a) . We may suppose that 0 < U < 1 on X . Using an approximation argument we see (cf . the proof of (d) ~ (a) of Theorem 3 .1 i n [4]) that lím sup  Añ ( k ) 1 d~c = o . 1,tB->O k> 1 B Hence by a mean ergodic theorem we see that the sequence {A1} con - verges in the norm topology of L 1 (II) . Write v = strong-limAñ1 and Y = n {v ~ o} . Since T i *v=v for all i= 1,2, . . .,dand (T 1 T 2 . . .T d )Y T X , it follows from the classical multiparameter pointwise ergodic theore m that for any f E L p + Cu} the limi t f * (x) = lim A n f (x ) n exists a .e . on X (but may be equal to infinity on some subset of X) . To prove that f* < oo a .e . on X, we observe that {f* = oo} C lim inf {An (k) f ~ t} for all t ~ O ; hence by Fatou ' s Lemma and (f ) k r/ p U d p, Ç lim inf  U d p, Ç t —T  f P dp,  . f f*_oo}  k f A n ( k ) f>t}  x Letting t fi oo, we have f {f * =co} U 46 = o and p,{ f* = cc} = O . Th e proof is complete . ■  Since the aboye proofs of the implications (a) ~ (d)  (e) ~ (f ) (a)holdforO<p<oo,wehaveth e Corollary . Let (X, .F, p) be a finite measure space and let T i : X -- > X (i — 1, 2, . . . , d) be cornmuting null preserving transformations . Le t o C p, r < oo . lf there exists a subsequence {n(k)} of {n} such that th e operators An(k) are equicontinuous mappings from L p (p,) to L r (tc) then POINTWISE ERGODIC THEOREM  8 7 for any f E L P (µ) the sequence {Af} converges to a finite límit a . e . o n X . Proof : This follows from the equivalente of (a) and (f) of the theorem . ■ Reference s 1. J . GARCÍACUERVA AND J . L . RUBIO DE FRANCIA, " Weighted Norm lnequalities and Related Topics, " North Holland, Amsterdam , 1985 . 2. U . KRENGEL, " Ergodic Theorems," Walter de Gruyter, Berlin, 1985 . 3. F . J . MARTÍN -REYES, On a pointwise ergodic theorem for multiparameter groups, in "Almost Everywhere Convergence," (G . A . Edgar and L . Sucheston, editors), Academic Press, Boston, 1989 , pp . 267-279 . 4. R . SATO, Multiparameter pointwise ergodic theorems for Marko v operators on  submitted for publication . Department of Mathematic s School of Scienc e Okayama Universit y Okayama 70 0 JAPA N Rebut el 5 de Juliol de 199 3 Added in proof : An obvious argument shows that condition (c) of the theorem ma y be sharpened as follows . (c') There exists v E L E 02 } with Tv v for all 1 Ç i < d such that f $ v d p, > 0 for all nonnull invariant sets B .