Weighted Lp spaces and pointwise ergodic theorems

Publicacions Matem`atiques, Vol 39 (1995), 273–283. WEIGHTED LpSPACES AND POINTWISE ERGODIC THEOREMS Ryotaro Sato Abstract In this paper we give an operator theoretic version of a recent result of F. J. Mart´ın-Reyes and A. de la Torre concerning the problem of finding necessary and sufficient conditions for a nonsingular point transformation to satisfy the Pointwise Ergodic Theorem in Lp. We consider a positive conservative contraction Ton L1of aσ-finite measure space (X, F,µ), a fixed function ein L1with e>0onX, and two positive measurable functions Vand Won X. We then characterize the pairs (V,W) such that for any fin Lp(Vdµ) the averages Rn 0(f,e)= n  k=0 Tkf n  k=0 Tke converge almost everywhere to a function in Lp(Wdµ). The characterizations are given for all p,1≤p<∞. 1. Introduction Let (X,F,µ)beaσ-finite measure space and Ta positive linear contraction of L1(µ). We assume Tto be a conservative operator. (For the usual notation we refer the reader to Krengel’s book [2].) Thus the class (1) I=I(T)={A∈F:T∗1A=1 A} of all invariant sets relative to Tforms a σ-field, where 1Adenotes the indicator function of Aand T∗denotes the adjoint operator of T, acting on L∞(µ). Since Tis positive, we may extend by a canonical manner the domain of Tto the class M+(µ) of all nonnegative extended real valued measurable functions on X. Similarly, this is done for T∗. Now let us fix an e∈L1(µ) with e>0onX. Let 0 <V,W≤∞be two measurable 274 R. Sato functions on X. Previously we observed in [6] that if 1 <p<∞then for any f∈L+ p(Vdµ) the averages (2) Rn 0(f,e)=n  k=0 Tkf n  k=0 Tke converge to a finite limit a.e. on Xif and only if (3) E{e−1V1−p|(X,I,edµ)}<∞a.e. on X, where 1/p +1/p=1. In[6] we also observed implicitely (see especially p. 76–77 in [6]) that for any f∈L+ 1(Vdµ) the averages Rn 0(f,e) converge to a finite limit a.e. on Xif and only if there exists a function U, measurable with respect to I, such that (4) V−1≤U<∞a.e. on X. In this paper we intend to study the problem of charactering the case where the limit function R∞ 0(f,e) belongs to L+ p(Wdµ) for every f∈L+ p(Vdµ). This study was inspired by the work [3] of Mart´ın-Reyes and de la Torre. See also Assani and W´os [1]. As a result, this paper may be considered to be an operator theoretic version of Mart´ın-Reyes and de la Torre’s paper [3]. Using the result obtained we next consider multiparameter pointwise ergodic theorems for commuting positive linear contractions of L1(µ) having a common strictly positive fixed point in L1(µ). 2. The main result Theorem 1. Let Tbe a conservative positive linear contraction of L1(µ).LetV,Wbe two positive real valued measurable functions on X. Fix an e∈L1(µ)with e>0on X.If1<p<∞and 1/p +1/p=1, then the following are equivalent: (a) For any f∈L+ p(Vdµ)the averages Rn 0(f,e)converge a.e. to a function belonging to L+ p(Wdµ). (b) E{e−1W|(X,I,edµ)}1/p ·E{e−1V1−p|(X,I,edµ)}1/p≤Ca.e. on X, where Cis a positive constant. (c) For any f∈L+ p(W1−pdµ)the averages Rn 0(f,e)converge a.e. to a function belonging to L+ p(V1−pdµ). If p=1, then (a) is equivalent to (d) E{e−1W|(X,I,edµ)}≤CV a.e. on X. Pointwise ergodic theorems 275 Proof: Let 1 <p<∞. (a) ⇒(b). By (a) the limit function (5) R∞ 0(f,e) = lim nRn 0(f,e) is finite a.e. on X. Thus by (3) we have E{e−1V1−p|(X,I,edµ)}<∞a.e. on X. Choose Xn∈I,n=1,2,..., so that (6) Xn↑Xand Xn V1−pdµ < ∞. Since V(1−p)p·V=V1−p, it follows that (7) V1−p∈L+ p(Xn,V dµ). On the other hand, since R∞ 0(·,e) is a positive linear operator from Lp(Vdµ)intoLp(Wdµ) by (a), it is bounded, i.e., there exists a constant K>0 such that (8) |R∞ 0(f,e)|pWdµ≤Kp|f|pVdµ (f∈Lp(Vdµ)). Therefore, for any A∈Iwith A⊂Xn, (7) yields (9) A R∞ 0(V1−p,e)pWdµ≤KpA V1−pdµ < ∞. Since R∞ 0(V1−p,e)=E{e−1V1−p|(X,I,edµ)}a.e. on X(cf. p. 73 in [6]), these inequalities imply R∞ 0(V1−p,e)pE{e−1W|(X,I,edµ)}≤KpE{e−1V1−p|(X,I,edµ)} <∞a.e. on X; and (b) follows. (b) ⇒(a). Since e−1f=(e−1/pfV1/p)(e−1/pV−1/p), the H¨older inequality for the conditional expectation operator and (b) imply that if f∈L+ p(Vdµ) then R∞ 0(f,e)=E{e−1f|(X,I,edµ)} ≤E{e−1fpV|(X,I,edµ)}1/p ·E{e−1V1−p|(X,I,edµ)}1/p ≤CE{e−1fpV|(X,I,edµ)}1/p ·E{e−1W|(X, I,edµ)}−1/p 276 R. Sato a.e. on X; and thus R∞ 0(f,e)pWdµ≤CpE{e−1fpV|(X,I,edµ)} E{e−1W|(X,I,edµ)}Wdµ =CpE{e−1fpV|(X,I,edµ)}edµ =CpfpVdµ<∞, which proves (a). (b) ⇔(c). Direct from (a) ⇔(b). Let p=1. (a) ⇔(d). For any f∈L+ 1(Vdµ) we obtain R∞ 0(f,e)Wdµ=E{e−1f|(X,I,edµ)}Wdµ =E{e−1f|(X,I,edµ)}E{e−1W|(X,I,edµ)}edµ =fE{e−1W|(X,I,edµ)}dµ =fV(E{e−1W|(X,I,edµ)}·V−1)dµ. Hence, by (8) with p= 1, (a) is equivalent to (a)fV(E{e−1W|(X,I,edµ)}·V−1)dµ ≤KfV dµ for every f∈L+ 1(Vdµ); and (a) is clearly equivalent to (d). The proof is complete. Corollary 1. In addition to the hypotheses of Theorem 1, if we assume that Tis ergodic, i.e., that Iis trivial, then the following are equivalent, for every 1≤p<∞: (a) For any f∈L+ p(Vdµ)the averages Rn 0(f,e)converge a.e. to a function belonging to L+ p(Wdµ). (b) W∈L1(µ)and V−1∈Lp(Vdµ), where p=∞when p=1. 3. Applications Let d≥1 be an integer and T1,... ,T dbe commuting positive linear contractions of L1(µ). In this section we assume that there exists an e∈L1(µ) with e>0onXsuch that (10) Tie=e(1 ≤i≤d). Pointwise ergodic theorems 277 Thus each Tiis a conservative operator and satisfies the mean ergodic theorem in L1(µ). And by an induction argument we see that for any f∈L1(µ) the averages (11) An(T1,... ,T d)f=An(T1)...A n(Td)f converge in L1-norm, where An(Ti)= 1 n n−1  k=0 Tk i. By Theorem 1 of [5], for any f∈L1(µ) the averages An(T1,... ,T d)fconverge a.e. on X. Let us denote the limit function by A(T1,... ,T d)f;thus (12) A(T1,... ,T d)f= lim nAn(T1,... ,T d)fa.e. on X. If we let (13) T=1 d d  i=1 Ti then Talso satisfies the mean ergodic theorem in L1(µ); and we get the direct decomposition L1(µ)={f∈L1(µ):Tf =f}⊕{g−Tg :g∈L1(µ)}−. Since Tf =fif and only if Tif=ffor each 1 ≤i≤dby the BrunelFalkowitz lemma (cf. p. 82 in [2]) and lim nAn(T1,... ,T d)(g−Tg)1=0 by the equation g−Tg =1 d d  i=1 (g−Tig), it follows that for any f∈L1(µ) the limit function A(T1,... ,T d)fcoincides a.e. with the limit function (14) A(T)f= lim nAn(T)f. Further, since I(T)= d  i=1 I(Ti) (in the sequel Iwill denote this σ-field), it follows that for any f∈L+ 1(µ) A(T1,... ,T d)f= lim nAn(T)f = lim nen  k=0 Tkf n  k=0 Tke (15) =eE{e−1f|(X, I,edµ)}a.e. on X. Hence, by an approximation argument, for any f∈M+(µ) the limit A(T1,... ,T d)f= lim nAn(T1,... ,T d)fexists a.e. on Xand satisfies (15). We are now in position to state the first application of Theorem 1. 278 R. Sato Theorem 2. Let T1,... ,T dbe commuting positive linear contractions of L1(µ)such that Tie=e(1 ≤i≤d)for some e∈L1(µ)with e>0on X.Let0<V,W<∞be two measurable functions on X.If1<p<∞ and 1/p +1/p=1, then the following are equivalent: (a) For any f∈L+ p(Vdµ)the limit function A(T1,... ,T d)fbelongs to L+ p(Wdµ). (b) E{ep−1W|(X,I,edµ)}1/pE{e−1V1−p|(X,I,edµ)}1/p≤Ca.e. on X. (c) For any f∈L+ p(e−pW1−pdµ)the limit function A(T1,... ,T d)f belongs to L+ p(e−pV1−pdµ). If p=1, then (a) is equivalent to (d) E{W|(X,I,edµ)}≤CV a.e. on X. Consequently, in case Iis trivial, (a) is equivalent, for every 1≤p<∞,to (e) epW∈L1(µ)and V−1∈Lp(Vdµ), where p=∞when p=1. Proof: For any f∈L+ p(Vdµ) we have, by (15), A(T1,... ,T d)f= eR∞ 0(f,e). Thus (a) is equivalent to (a)For any f∈L+ p(Vdµ)the limit function R∞ 0(f,e)(relative to T) belongs to L+ p(epWdµ). Therefore, by Theorem 1, we see (a) ⇔(b) when 1 <p<∞, and (a) ⇔(d) when p= 1. When 1 <p<∞, (b) ⇔(c) follows from the equivalence (a) ⇔(b). This completes the proof. Corollary 2. Let T1,... ,T dand ebe the same as in Theorem 2. If 1<p<∞and 1/p +1/p=1, then the following are equivalent: (a) For any f∈L+ p(µ)the limit function A(T1,... ,T d)fbelongs to L+ p(µ). (b) E{ep−1|(X,I,edµ)}1/pE{e−1|(X,I,edµ)}1/p≤Ca.e. on X. Consequently, in case Iis trivial, (a) is equivalent, for every 1<p<∞,to (c) µ(X)<∞and e∈Lp(µ). Remark. We note that (a) of Corollary 2 always holds when p=1. We next consider the adjoint operators T∗ 1,... ,T∗ d. Since (T∗ if)edµ =f(Tie)dµ =fedµ for f∈L+ ∞(µ), T∗ 1,... ,T∗ dcan be regarded as commuting positive linear contractions of L1(edµ). Since (16) T∗ i1=1∈L1(edµ)(1≤i≤d), Pointwise ergodic theorems 279 if we replace the measure µand the function eby edµ and 1, respectively, then the above-given argument shows that for any f∈M+(µ)= M+(edµ) the limit (17) A(T∗ 1,... ,T∗ d)f= lim nAn(T∗ 1,... ,T∗ d)f exists a.e on X; further, since I= d  i=1 I(Ti)= d  i=1 I(T∗ i), it follows that (18) A(T∗ 1,... ,T∗ d)f= lim nAn(T∗)f=E{f|(X,I,edµ)}a.e. on X, where T∗=1 d d  i=1 T∗ i. Theorem 3. Let T1,... ,T dand ebe the same as in Theorem 2. Let 0<V,W<∞be two measurable functions on X.If1<p<∞and 1/p +1/p=1, then the following are equivalent: (a) For any f∈L+ p(Vdµ)the limit function A(T∗ 1,... ,T∗ d)fbelongs to L+ p(Wdµ). (b) E{e−1W|(X,I,edµ)}1/pE{(e−1V)1−p|(X,I,edµ)}1/p≤Ca.e. on X. (c) For any f∈L+ p(epW1−pdµ)the limit function A(T∗ 1,... ,T∗ d)f belongs to L+ p(epV1−pdµ). (d) For any f∈L+ p(W1−pdµ)the limit function A(T1,... ,T d)fbelongs to L+ p(V1−pdµ). (e) For any f∈L+ p(e−pVdµ)the limit function A(T1,... ,T d)fbelongs to L+ p(e−pWdµ). If p=1, then (a) is equivalent to (f) E{e−1W|(X,I,edµ)}≤C(e−1V)a.e. on X. Consequently, in case Iis trivial, (a) is equivalent, for every 1≤p<∞,to (g) W∈L1(µ)and eV −1∈Lp(Vdµ), where p=∞when p=1. Proof: Since L+ p(Vdµ)=L+ p(e−1Vedµ) and L+ p(Wdµ)= L+ p(e−1Wedµ), if we apply Theorem 2 to commuting positive linear contractions T∗ 1,... ,T∗ dof L1(edµ), then (16) yields (a) ⇔(f) when p= 1, and (a) ⇔(b) ⇔(c) when 1 <p<∞. If we write (b) as E{ep−1(e−pW)|(X,I,edµ)}1/pE{e−1(e−pV)1−p|(X,I,edµ)}1/p ≤Ca.e. on X, 280 R. Sato and apply Theorem 2 to commuting positive linear contractions T1,... ,T dof L1(µ), then we obtain (b) ⇔(e) ⇔(d) when 1 <p<∞. The proof is complete. Corollary 3 (cf. [3] and [4]). Let T1,... ,T dand ebe the same as in Theorem 2. If 1<p<∞and 1/p +1/p=1, then the following are equivalent: (a) For any f∈L+ p(µ)the limit function A(T∗ 1,... ,T∗ d)fbelongs to L+ p(µ). (b) E{e−1|(X,I,edµ)}1/pE{ep−1|(X,I,edµ)}1/p≤Ca.e. on X. (c) For any f∈L+ p(µ)the limit function A(T1,... ,T d)fbelongs to L+ p(µ). If p=1, then (a) is equivalent to (d) E{e−1|(X,I,edµ)}≤Ce−1a.e. on X. Consequently, in case Iis trivial, (a) is equivalent, for every 1≤p<∞,to (e) µ(X)<∞and e∈Lp(µ), where p=∞when p=1. Corollary 4. Suppose (X,F,µ)is a finite measure space. Let T1,... ,T dbe commuting positive linear contractions of L1(µ), and assume that µis invariant under T1,... ,T d, i.e., that Ti1=1∈L1(µ) (1 ≤i≤d).Let0<V,W<∞be two measurable functions on X.If 1<p<∞and 1/p +1/p=1, then the following are equivalent: (a) For any f∈L+ p(Vdµ)the limit function A(T∗ 1,... ,T∗ d)fbelongs to L+ p(Wdµ). (b) E{W|(X,I,µ)}1/pE{V1−p|(X,I,µ)}1/p≤Ca.e. on X. (c) For any f∈L+ p(W1−pdµ)the limit function A(T∗ 1,... ,T∗ d)f belongs to L+ p(V1−pdµ). If p=1, then (a) is equivalent to (d) E{W|(X,I,µ)}≤CV a.e. on X. Consequently, in case Iis trivial, (a) is equivalent, for every 1≤p<∞,to (e) W∈L1(µ)and V−1∈Lp(Vdµ), where p=∞when p=1. Remark. Under the hypotheses of Corollary 4, it follows (see (15) and (18)) that for any f∈M+(µ) A(T1,... ,T d)f=A(T∗ 1,... ,T∗ d)f=E{f|(X,I,µ)}a.e. on X, so that the function A(T∗ 1,... ,T∗ d)fcan be replaced by the function A(T1,... ,T d)fin Corollary 4, without any influence. Pointwise ergodic theorems 281 4. Concluding remarks Throughout this section, (X, F,µ)isaσ-finite measure space, and T1,... ,T dare commuting positive linear contractions of L1(µ) such that Tie=e(1 ≤i≤d) for some e∈L1(µ) with e>0onX. Here we briefly discuss the problem of characterizing a positive measurable function V on Xsuch that if f∈L+ p(Vdµ) then the limit function A(T1,... ,T d)f (or A(T∗ 1,... ,T∗ d)f) is finite a.e. on X. As in the preceding section, we will denote I= d  i=1 I(Ti). The results may be stated as follows. (For a related result we refer the reader to [7].) Theorem 4. Let 0<V ≤∞be a measurable function on X.If 1<p<∞and 1/p +1/p=1, then the following are equivalent: (a) For any f∈L+ p(Vdµ)the limit function A(T1,... ,T d)fis finite a.e. on X. (b) E{e−1V1−p|(X,I,edµ)}<∞a.e. on X. If p=1, then (a) is equivalent to (c) V−1≤U<∞a.e. on Xfor some U, measurable with respect to I. Proof: By virtue of (15) and the result mentioned in Introduction (see especially (3) and (4)), Theorem 4 follows immediately. Corollary 5. If 1<p<∞, then the following are equivalent: (a) For any f∈L+ p(µ)the limit function A(T1,... ,T d)fis fintie a.e. on X. (b) There exist Xn∈I,n=1,2,..., such that Xn↑Xand µ(Xn)< ∞. (c) For any f∈ 1≤r≤∞ L+ r(µ)the limit function A(T1,... ,T d)fis finite a.e. on X. Proof: Since the implications (b) ⇒(c) ⇒(a) are obvious, we only prove (a) ⇔(b). To do this we apply Theorem 4 with V=1onXand see that (a) is equivalent to E{e−1|(X,I,edµ)}<∞a.e. on X, which is clearly equivalent to (b). The proof is complete.