Pointwise multipliers on martingale Campanato spaces

Transcription

1 arxiv: v2 [math.pr] Oct 203 Pointwise multipliers on martingale Campanato spaces Eiichi Nakai and Gaku Sadasue Abstract Weintroducegeneralized CampanatospacesL onaprobability space (Ω,F,P), where p [, ) and φ : (0,] (0, ). If p = and φ, then L = BMO. We give a characterization of the set of all pointwise multipliers on L. Introduction We consider a probability space (Ω,F,P) such that F = σ( n F n), where {F n } n 0 is a nondecreasing sequence of sub-σ-algebras of F. For the sake of simplicity, let F = F 0. We suppose that every σ-algebra F n is generated by countable atoms, where B F n is called an atom (more precisely a (F n,p)-atom), if any A B with A F n satisfies P(A) = P(B) or P(A) = 0. Denote by A(F n ) the set of all atoms in F n. The expectation operator and the conditional expectation operators relative to F n are denoted by E and E n, respectively. Let X be a normed space of F-measurable functions. We say that an F-measurable function g is a pointwise multiplier on X, if the pointwise multiplication fg is in X for any f X. We denote by PWM(X) the set of all pointwise multipliers on X. If X is a Banach space and has the following property, then every g PWM(X) is a bounded operator on X. (.) f n f in X (n ) = {n(j)} s.t. f n(j) f a.s. (j ). 200 Mathematics Subject Classification. Primary 60G46; Secondary 46E30, 42B35. Key words and phrases. martingale, pointwise multiplier, Campanato space, bounded mean oscillation.

2 Actually, from (.) we see that g is a closed operator. Therefore, g is a bounded operator by the closed graph theorem. It is known that PWM(L p ) = L for p (0, ]. More generally, if X is a (quasi) Banach function space, then PWM(X) = L (see [4, 7]). For Banach function spaces, see Kikuchi [2]. In this paper we consider the pointwise multipliers on generalized Campanato spaces which are not Banach function spaces in general. We always assume that F 0 = {,Ω}, that is, the operator E 0 coincides with E. Then we introduce generalized Campanato spaces L and L as the following: Definition.. Let p [, ) and φ be a function from (0,] to (0, ). For f L, let (.2) f L = sup n 0 and sup B A(F n) (.3) f L = f L + Ef. Define ( /p f E n f dp) p, φ(p(b)) P(B) B L = {f L : f L < } and L = {f L : f L < }. If φ(r) = r λ, λ (, ), we simply denote L and L by L p,λ and L p,λ, respectively, which were introduced by [9]. Note that L and L coincide as sets of measurable functions. We regardl = (L, L )isaseminormedspaceandl = (L, L) is a normed space. Then L is a Banach space, but it is not a Banach function space in general. It is easy to see that L has the property (.), since f L E[ f Ef ]+ Ef max(,φ()) f L. For g PWM(L ), let g Op = sup f 0 fg L f L We also define BMO and Lip α as the following: 2.

3 Definition.2. For φ, denote L,φ and L,φ by BMO and BMO, respectively. For φ(r) = r α, α > 0, denote L,φ and L,φ by Lip α and Lip α, respectively. Let L,0 = {f L : Ef = 0}. Then BMO L,0 = BMO L,0 and Lip α L,0 = Lip α L,0. These spaces coincide with BMO and Lip α defined by Weisz [2, 3], respectively, under the assumption that every σ-algebra F n is generated by countable atoms, see [9] for details. We say {F n } n 0 is regular if there exists R 2 such that (.4) f n Rf n for all non-negative martingales f = (f n ) n 0. A function θ : (0,] (0, ) is said to satisfy the doubling condition if there exists a constant C > 0 such that C θ(r) θ(s) C for r,s (0,], 2 r s 2. A function θ : (0,] (0, ) is said to be almost increasing (almost decreasing) if there exists a constant C > 0 such that θ(r) Cθ(s) (θ(r) Cθ(s)) for 0 < r s. Our main result is the following: Theorem.. Let {F n } n 0 be regular, F 0 = {,Ω}, p [, ) and φ : (0,] (0, ). Assume that φ satisfies the doubling condition and that (.5) Let r 0 (.6) φ (r) = + Then φ(t) p dt Crφ(r) p for all r (0,]. r φ(t) t dt. PWM(L ) = L /φ L. Moreover, for g PWM(L ), g Op is equivalent to g L/φ + g L. 3

4 See [, 6, 0,, 4] for pointwise multipliers on BMO and Campanato spaces defined on the Euclidean space. Our basic idea comes from [, 0]. Remark.. (i) Ifφsatisfiesthedoublingconditionand(.5),thenrφ(r) p is almost increasing. (ii) If φ is almost increasing, then φ/φ is also. (iii) Let (.7) f L,F = sup n 0 sup A F n φ(p(a)) ( /p f E n f dp) p. P(A) A Then f L f L,F by the definition. If φ is almost increasing, then f L and f L,F are equivalent. Actually, for any A F n, there exists a sequence of atoms B l A(F n ), l =,2,, such that A = l B l and P(A) = l P(B l). Then f E n f p dp = f E n f p dp A l B l l φ(p(b l )) p P(B l ) f p L C p φ(p(a)) p P(A) f p L. This shows f L,F C f L. If φ is not almost increasing, then f L is not equivalent to f L,F in general, see [9]. The norm (.7) was introduced by [5] for general {F n } n 0. By Theorem. we have the next two corollaries immediately: Corollary.2. Let {F n } n 0 be regular and F 0 = {,Ω}. Then PWM(BMO ) = L,ψ L, where ψ(r) = /log(e/r). Moreover, for g PWM(BMO ), g Op is equivalent to g L,ψ + g L. Corollary.3. Let {F n } n 0 be regular, F 0 = {,Ω} and α > 0. Then PWM(Lip α) = Lip α L. Moreover, for g PWM(Lip α ), g Op is equivalent to g Lipα + g L. 4

5 Example.. Let{F n } n 0, pandφsatisfytheassumptionintheorem.. For a sequence let (.8) g = sinh, where h = B 0 B B n, B n A(F n ), n= ( ) φ(p(b n )) P(Bn ) φ (P(B n )) P(B n ) χ B n χ Bn. ThenhisinL /φ, seelemma2.4andremark2.. Henceg L /φ L, sincesinθ islipschitzcontinuous, seeremark2.3. Thatis, g PWM(L ). If φ, then φ(r)/φ (r) = /log(e/r) and g PWM(BMO ). Example.2. The following function satisfies the doubling condition and the property (.5): φ(r) = r α (log(e/r)) β (α ( /p, ), β (, )). If α ( /p,0) and β (, ), then φ φ, that is, there exists a positive constant C such that C φ(r) φ (r) Cφ(r) for all r (0,]. In general, under the assumption of Theorem., if φ φ, then L,φ/φ = BMO and then PWM(L ) = BMO L = L. If α [0, ) and β (, ), then φ φ and φ(r)/φ (r) 0 as r 0. In this case L,φ/φ L L in general (see also Remark 2.2). In particular, if α = 0 and β (, ), or if α (0, ) and β (, ), then φ. In general, under the assumption of Theorem., if φ, then L,φ/φ = L,φ L by Lemma 2.2 below, and then PWM(L ) = L,φ L = L,φ. Moreover, if φ is almost increasing, then we can use the John-Nirenberg type inequality in [5, Theorem 2.9], that is, PWM(L ) = L. We can also take the function φ(r) = r α (log(e/r)) β (loglog(e/r)) γ (α ( /p, ), β,γ (, )), and so on. 5

6 Next, for a martingale (f n ) n 0 relative to {F n } n 0, it is said to be L p,λ - bounded if f n L p,λ (n 0) and sup n 0 f n Lp,λ <. Similarly, the martingale (f n ) n 0 is said to be L p,λ -bounded if f n L p,λ (n 0) and sup n 0 f n L <. p,λ Let and L (F n ) = {f L : f is F n -measurable and f L < } L (F n) = {f L : f is F n -measurable and f L < }. Then we have the following: Theorem.4. Let {F n } n 0 be regular, F 0 = {,Ω}, p [, ) and φ : (0,] (0, ). Assume that φ satisfies the doubling condition and (.5). Let g L and (g n ) n 0 be its corresponding martingale with g n = E n g (n 0). If g PWM(L ), then g n PWM(L (F n)). Conversely, if g n PWM(L (F n)) and sup n 0 g n Op <, then g PWM(L ). We show several lemmas in Section 2 to prove Theorem. in Section 3. We prove Theorem.4 in Section 4. At the end of this section, we make some conventions. Throughout this paper, we always use C to denote a positive constant that is independent of the main parameters involved but whose value may differ from line to line. Constants with subscripts, such as C p, is dependent on the subscripts. If f Cg, we then write f g or g f; and if f g f, we then write f g. 2 Lemmas To prove Theorem. we show several lemmas in this section. The first lemma was proved in [9]. Lemma 2. ([9, Lemma3.3]). Let {F n } n 0 be regular. Then every sequence B 0 B B n, B n A(F n ) 6

7 has the following property; for each n, B n = B n or ( + ) P(B n ) P(B n ) RP(B n ), R where R is the constant in (.4). For a function f L and an atom B A(F n ), let f B = f dp. P(B) B For a function φ : (0,] (0, ), let φ be defined by (.6). If φ satisfies the doubling condition, then φ(r) Cφ (r) for all r (0,]. Lemma 2.2. Let {F n } n 0 be regular, F 0 = {,Ω}, p [, ) and φ : (0,] (0, ). Assume that φ satisfies the doubling condition. For f L and B n 0A(F n ), (2.) f B Cφ (P(B)) f L. Proof. By Lemma 2., we can choose B kj A(F kj ), 0 = k 0 < k < < k m n, such that B k0 B k B k2 B km = B and that ( + /R)P(B kj ) P(B kj ) RP(B kj ). Then, we have f Bkj f Bkj = f(ω)dp f(ω)dp P(B kj ) B kj P(B kj ) B kj = [f E kj f](ω)dp P(B kj ) B kj ( ) /p f E kj f p dp P(B kj ) B kj ( ) /p f E kj f p dp P(B kj ) B kj φ(p(b kj )) f L. 7

8 Since φ satisfies the doubling condition, f B f B0 = m f Bkj f Bkj j= m φ(p(b kj )) f L j= P(Bkj ) m j= P(B) P(B kj ) φ(t) t φ(t) t dt f L dt f L = {φ (P(B)) } f L. On the other hand, Therefore, we have (2.). f B0 = Ef f L. Lemma 2.3. Let F 0 = {,Ω}, p [, ) and φ : (0,] (0, ). Assume that rφ(r) p is almost increasing. For any atom B n 0 A(F n ), the characteristic function χ B is in L and there exists a positive constant C, independent of B, such that (2.2) χ B L C φ(p(b)). Proof. Let B A(F n ) and B A(F k ). Let B j A(F j ), 0 j n, such that B 0 B B n = B. If k n, then χ B E k χ B = 0 and B χ B E k χ B p dp = 0. If k < n and B B k, then B B k = and Hence, we have B χ B E k χ B p dp = 0. ( ) /p χ B L = sup χ B E k χ B p dp. k<n φ(p(b k )) P(B k ) B k 8

9 For k < n, since rφ(r) p is almost increasing, χ φ(p(b k )) p B E k χ B p dp P(B k ) B k = φ(p(b k )) p P(B k ) { ( P(B n ) P(B ) p ( ) p } n) P(Bn ) +(P(B k ) P(B n )) P(B k ) P(B k ) φ(p(b n )) p P(B n ) { ( P(B n ) P(B ) p ( ) p } n) P(Bn ) +(P(B k ) P(B n )) P(B k ) P(B k ) { ( = P(B ) p n) +( P(B )( ) } p n) P(Bn ) φ(p(b n )) p P(B k ) P(B k ) P(B k ) φ(p(b n )) = p φ(p(b)) p. Therefore, we have (2.3) χ B L φ(p(b)). On the other hand, since rφ(r) p is almost increasing, (2.4) Eχ B = P(B) P(B) /p Combining (2.3) and (2.4), we have (2.2). φ(p(b)). Lemma 2.4. Let {F n } n 0 be regular, F 0 = {,Ω}, p [, ) and φ : (0,] (0, ). Assume that φ satisfies the doubling condition and (.5). For a sequence let and let B 0 B B n, B n A(F n ), ( ) P(Bk ) f 0 = χ B0, u k = φ(p(b k )) P(B k ) χ B k χ Bk, (2.5) f n = f n u k. k=

10 Then (f n ) n 0 is a martingale and L -bounded. The sum f f 0+ k= u k converges a.s. and in L p, and E n f = f n for n 0. Moreover, there exist positive constants C and C 2, independent of the sequence of atoms, such that (2.6) f L C and f Bn C 2 φ (P(B n )), n 0. Proof. Since E n [u k ] = 0 for k > n, (f n ) n 0 is a martingale. We show that the sum f 0 + k= u k converges in L p. If lim k P(B k ) > 0 then the convergence is clear because there exists m such that B m = B n for all n m. We assume that lim k P(B k ) = 0. By Lemma 2., we can take a sequence of integers 0 = k 0 < k < < k j < that satisfies (2.7) (+/R)P(B kj ) P(B kj ) RP(B kj ), and B kj = B k if k j k < k j. In this case we can write f n = χ B0 + ( ) P(Bkj ) φ(p(b kj )) P(B kj ) χ B kj χ Bkj. k j n Note that, by Remark. and [8, Lemma 7.], the doubling condition and (.5) implies (2.8) r 0 φ(t)t /p dt C p φ(r)r /p for all r (0,]. Using the doubling condition and (2.8), we have φ(p(b kj )) P(B kj ) P(B kj ) χ (2.9) B kj χ Bkj k j >n L p k j >nφ(p(b kj ))(R χ Bkj Lp + χ Bkj Lp ) 2R k j >n C k j >n C P(Bn) 0 φ(p(b kj ))P(B kj ) /p P(Bkj ) P(B kj ) φ(t)t /p dt φ(t)t /p dt CC p φ(p(b n ))P(B n ) /p. 0

11 We can deduce from (2.9) that f f 0 + k= u k converges in L p. By the martingale convergence theorem, f 0 + k= u k also converges almost surely. Moreover, we have E n f = f n and ( ) /p (2.0) f E n f p dp CC p φ(p(b n )). P(B n ) B n For B A(F n ), we have f E n f (B = B n ) (2.) (f E n f)χ B = 0 (B B n ). Combining (2.0) and (2.), we have f L C where C is a positive constant independent of the sequence of atoms. Moreover, since B 0 = Ω, Ef = f 0 =. Therefore, f L C where C is a positive constant independent of the sequence of atoms. We now show f Bn C 2 φ (P(B n )). On the atom B n, we have f n = + ( ) P(Bkj ) φ(p(b kj )) P(B kj ) + φ(p(b kj )). R k j n Therefore, we have f Bn = f n dp P(B n ) B n + φ(p(b kj )) R + = + k j n P(Bkj ) k j n P(B n) P(B kj ) φ(t) t φ(t) t dt dt = φ (P(B n )) k j n That is, f Bn C 2 φ (P(B n )) where C 2 is a positive constant independent of the sequence of atoms. Remark 2.. From the proof of Lemma 2.4 we see that, for n (2.2) h = u k, h 0 = 0, h n = u k (n ), k= k=

12 h is in L and (h n ) n 0 is its corresponding martingale with h n = E n h (n 0). Remark 2.2. Let (Ω,F,P) be as follows: Ω = [0,), A(F n ) = {I n,j = [j2 n,(j +)2 n ) : j = 0,,,2 n } F n = σ(a(f n )), F = σ( n F n ), P = the Lebesgue measure. If φ(r) = /log(e/r), then h in (2.2) is unbounded. Actually, and h = n k= u k = +klog2 (2χ B k χ Bk ), +klog2 +(n+)log2 on B n \B n+. Remark 2.3. If F : C C is Lipschitz continuous, that is, F(z ) F(z 2 ) C z z 2, z,z 2 C, then, for B F n, F(f) E n [F(f)] dp 2C B Actually, F(f) E n [F(f)] dp B F(f) F(E n f) dp + B = F(f) F(E n f) dp + B 2 F(f) F(E n f) dp B 2C f E n f dp. B B B B f E n f dp. F(E n f) E n [F(f)] dp E n [F(E n f) F(f)] dp Lemma 2.5. Let p [, ) and φ : (0,] (0, ). Suppose that f L and g L. Then fg L if and only if (2.3) F(f,g) := sup n 0 In this case, (2.4) sup B A(F n) f B φ(p(b)) ( P(B) F(f,g) fg L 2 f L g L. B g E n g p dp) /p <. 2

13 Proof. Let f L and g L. Let B A(F n ). Since E n f = f B on B, we can use the same method as in [6, Lemma 3.5] and we have (2.5) ( /p ( /p fg E n [fg] dp) p f B g E n g dp) p P(B) B P(B) B ( /p 2 (f E n f)g dp) p 2φ(P(B)) f L P(B) g L. B Therefore, fg L if and only if F(f,g) <. In this case, we can deduce (2.4) from (2.5). Lemma 2.6. Let {F n } n 0 be regular, F 0 = {,Ω}, p [, ) and φ : (0,] (0, ). Assume that rφ(r) p is almost increasing and that φ satisfies the doubling condition. If g PWM(L ), then g L and g L C g Op for some positive constant C independent of g. Proof. Let g PWM(L ). Since the constant function is in L, the pointwise multiplication g = g is in L, which implies g L. Then E[ g ] E[ g Eg ]+ Eg max(,φ()) g L g Op L = g Op. Since {F n } n 0 is regular, we also have E n g L as follows: E n [ g ] RE n [ g ] R n E 0 [ g ] = R n E[ g ]. Next we shall show that there exists a positive constant C such that g L C g Op. Then we have the conclusion. Let B A(F n ) such that g B E n g L /2. By Lemma 2. there exists B A(F n ) with B B such that (+/R)P(B) P(B ) RP(B). Then, we have gχ B L = ( φ(p(b )) ( φ(p(b )) φ(p(b )) ) /p gχ P(B B E n [gχ B ] p dp ) B gχ P(B B E n [gχ B ] p dp ) B \B ( /p E P(B n [[E n g]χ B ] dp) p. ) 3 B \B ) /p

14 Since [E n g]χ B = g B χ B E n g L χ B /2, we have ( ) p ( ) p E n [[E n g]χ B ] p En g L P(B) dp P(B \B). 2 P(B ) B \B Hence, we have (2.6) gχ B L E n g L 2R(R+) /p φ(p(b )). Using (2.6), Lemma 2.3 and the doubling condition on φ, we have Therefore, This shows the conclusion. E n g L 2R(R+) /p φ(p(b )) gχ B L g Op φ(p(b )) φ(p(b)) g Op. g L = sup E n g L C g Op. n 0 3 Proof of Theorem. We first show that (3.) L /φ L PWM(L ) and g Op C( g L/φ + g L ). Let g L /φ L and f L. Let F(f,g) be as in Lemma 2.5. Then, by the definition of F(f,g) and Lemma 2.2 we have F(f,g) C f L g L/φ <. Therefore, by Lemma 2.5, we have fg L and (3.2) fg L C f L g L/φ +2 f L g L. On the other hand, we have (3.3) E[fg] g L E[ f ] g L max(,φ()) f L. 4

15 Combining (3.2) and (3.3), we obtain (3.). We now show the converse, that is, (3.4) PWM(L ) L /φ L and g L/φ + g L C g Op. Let g PWM(L ). ByLemma 2.6, wehave g L and g L C g Op. Let B A(F n ). We take B j A(F j ) with B n = B such that B 0 B B n. Let f be the function described in Lemma 2.4. Then, combining Lemma 2.4 and Lemma 2.5, we have C 2 φ (P(B)) φ(p(b)) f B φ(p(b)) F(f,g) Therefore, we have (3.4). ( P(B) ( P(B) fg L +2 g L f L B B ) /p g E n g p dp ) /p g E n g p dp g Op f L +2C g Op f L g Op f L C g Op. 4 Proof of Theorem.4 To prove Theorem.4 we use the following proposition. It can be shown by the same way as [9, Proposition 2.2] which deals with the case φ(r) = r λ, λ (, ). Proposition 4.. Let p < and φ : (0,] (0, ). Let f L and (f n ) n 0 be its corresponding martingale with f n = E n f (n 0).. If f L, then (f n ) n 0 is L -bounded and f L sup f n L. n 0 Conversely, if (f n ) n 0 is L -bounded, then f L and f L sup f n L. n 0 5

16 2. If f L, then (f n) n 0 is L -bounded and f L sup f n L. n 0 Conversely, if (f n ) n 0 is L -bounded, then f L and f L sup f n L. n 0 Remark 4.. In general, for f L L,0 (res. f L ), its corresponding martingale (f n ) n 0 with f n = E n f does not always converge to f in L (res. L ). See Remark 3.7 in [9] for the case φ(r) = rλ. Proof of Theorem.4. Let g PWM(L ) and f L (F n). Then, using Proposition 4., we have E n [g]f L = E n [gf] L gf L g Op f L. Therefore, we have E n g PWM(L (F n)). Conversely, assume that E n g PWM(L (F n)) and sup n 0 E n g Op <. Then, using Proposition 4. and Theorem., we have g L/φ + g L sup n 0 E n g L/φ +sup n 0 Using Theorem. again, we have g PWM(L ). E n g L sup E n g Op <. n 0 Acknowledgement The authors would like to thank the referees for their careful reading and useful comments. The first author was supported by Grant-in-Aid for Scientific Research (C), No , Japan Society for the Promotion of Science. The second author was supported by Grant-in-Aid for Scientific Research (C), No , Japan Society for the Promotion of Science. References [] S. Janson, On functions with conditions on the mean oscillation, Ark. Math. 4 (976),

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18 [4] K. Yabuta, Pointwise multipliers of weighted BMO spaces, Proc. Amer. Math. Soc. 7 (993), Eiichi Nakai Department of Mathematics Ibaraki University Mito, Ibaraki , Japan enakai@mx.ibaraki.ac.jp Gaku Sadasue Department of Mathematics Osaka Kyoiku University Kashiwara, Osaka , Japan sadasue@cc.osaka-kyoiku.ac.jp 8