arXiv:1703.00851v5 [math.CA] 15 Nov 2017 1.2. c domains. uct h olrSdsysaeadtesto utpir rmtelittle the fin algebr from To multiplier little multipliers same. the of mean the set the characterize are the we and spaces space method; both mea Cotlar-Sadosky indirect if the the an ask into use to continuously we natural embeds was out, BMO it little and misleading. the somehow BMO that is little this clear and pretty ([2]) the is Sadosky BMO to and It spirit Cotlar product in of close the BMO very little of is the definition algebra Its multiplier 6]). th 5, in the ([1, author Chang-Fefferman of the and characterization Pott the S. by to literature the in recently introduced pc oayo h w aibe suiomyi BMO( in uniformly is interva variables two any over the mean of any their to that such spect functions those of consists 1.1. and When ok n eoe bmo( denoted and dosky e od n phrases. and words Key 2010 h ocle ml M pc on space BMO small so-called The NEBDIGRLTO O ONE MEAN BOUNDED FOR RELATION EMBEDDING AN Y entosadresults. and Definitions Introduction. h pc fpitiemlilesfrom multipliers pointwise of space the , X ahmtc ujc Classification. Subject epoeta h olrSdsysaeo ucin fbounded of Chang-Fefferman. functions of of BMO space bmo( product Cotlar-Sadosky oscillation re the the in that of author prove the algebra This We and to multiplier oscillation. Pott respect S. mean the by with bounded with introduced and uniformly recently been has any has variables over space two mean the its of if any BMO, little mean the Abstract. = Y esml write simply we , BMO SILTO NRECTANGLES. ON OSCILLATION M ntetoprmtrstig esyafnto eog to belongs a say we setting, two-parameter the In ( 1. T ,Y X, ntetoprmtrcs,tema iteBOspace BMO little mean the case, two-parameter the In . N one enoclain oaihi enoclain prod- oscillation, mean logarithmic oscillation, mean Bounded sasrc usaeo h enltl BMO. little mean the of subspace strict a is ) nrdcinadresults and Introduction T = ) N ossso functions of consists ) BENO { φ M : ie w aahsae ffunctions of spaces Banach two Given φf ˆ TF SEHBA F. IT ( ,X X, 1 ∈ rmr 21,3A7 eodr 42B35. Secondary 32A37; 42B15, Primary Y T = ) N o all for nrdcdb olradSa- and Cotlar by introduced , M X ( X f b to ). ∈ ∈ Y L X sdfie sfollows as defined is 2 T } ( T . .Ti pc was space This ). N uhta the that such ) lation mean ihre- with l BMO i way eir n of one of a X to of d n 2 BENOˆIT F. SEHBA quantity 1 (1.1) ||b||∗,N := sup |b(t1, ··· , tN ) − mRb|dt1 ··· dtN TN |R| R⊂ , rectangle ZR 1 is finite, mRb = |R| R b(t1, ··· , tN )dt1 ··· dtN . Seen as a quotient space with N T b TN the set of constants,R bmo( ) is a Banach space with || ||bmo( ) := ||b||∗,N . N Note that in the above definition, since R is a rectangle in T , mRf is a constant. We will sometimes consider the case where R is a rectangle in M T with M an integer, 0

2 N N Definition 1.1. A function b ∈ L (T ) belongs to bmom(T ) if there is a constant C > 0 such that for any integers 0 < N1, N2 < N, N1 + N2 = N and any rectangle R ⊂ TN1 ,

||mRb||∗,N2 ≤ C.

N The space bmom(T ) seen as a quotient space by the set of constants is a Banach space under the norm ∗ N ||b||bmom(T ) := C where C∗ stands for the smallest constant in the above definition. It is clear from the definitions above that bmo(TN ) embeds continuously N into bmom(T ). We will be calling bmom the mean little BMO. Our main result is the following.

N N Theorem 1.2. bmo(T ) is strictly continuously embedded into bmom(T ). To prove the above theorem, we first prove the following. Theorem 1.3. The only pointwise multipliers of bmo(TN ) are the con- stants. We say a function b ∈ L2(TN ) has bounded logarithmic mean oscillation on rectangles, i.e b ∈ lmo(TN ) if N log 4 j=1 |Ij| ||b||∗,log,N := sup |b(t) − mRb|dt TN |R| R=I1×···×IN ⊂ P ZR < ∞. Let us introduce also the mean little LMO space in product domains. BOUNDED MEAN OSCILLATION ON RECTANGLES. 3

2 N N Definition 1.4. A function b ∈ L (T ) belongs to lmom(T ) if there is a constant C > 0 such that for any decomposition 0 < N1, N2 < N, N1 +N2 = N, and any rectangle R ⊂ TN1 ,

||mRb||∗,log,N2 ≤ C.

If C∗ stands for the smallest constant in the Definition 1.4, then seen as N a quotient space by the set of constants, lmom(T ) is a Banach space with the following norm ∗ N ||b||lmom(T ) := C . In terms of multipliers, to get close to the one parameter situation, we N N need to start from bmo(T ) and take bmom(T ) as the target space.

Theorem 1.5. Let φ ∈ L2(TN ). Then the following assertions are equiva- lent. N N (i) φ is a multiplier from bmo(T ) to bmom(T ). N ∞ N (ii) φ ∈ lmom(T ) ∩ L (T ). Moreover,

N N ∞ N N kMφkbmo(T )→bmom(T ) ≃kφkL (T ) + kφklmom(T )

N N where kMφkbmo(T )→bmom(T ) is the norm of the multiplication operator from N N bmo(T ) to bmom(T ).

Theorem 1.3 and Theorem 1.5 clearly establish Theorem 1.2 since N ∞ N lmom(T ) ∩ L (T ) contains more than constants. The proofs are given in the next section. The last section of this note also states that the only multiplier from a Banach space of functions (strictly) containing bmo(TN ) to bmo(TN ) is the constant zero. As we are dealing only with little spaces of functions of bounded mean oscillation, we essentially make use of the one parameter techniques. This is not longer possible when considering the multipliers of the product BMO of Chang-Fefferman for which one needs more demanding techniques ([4, 5, 6]).

2. Comparison via multiplier algebras 2.1. Proof of Theorem 1.3. The space bmo(TN ) has the following equiv- alent definitions ([2, 3]) that we need here.

Proposition 2.1. The following assertions are equivalent. (1) b ∈ bmo(TN ). (2) b ∈ L2(TN ) and there exists a constant C > 0 such that for any

decomposition N1 + N2 = N, 0 < N1, N2 < N, 4 BENOˆIT F. SEHBA

TN2 (i) kb(·, t)k∗,N1 ≤ C, for all t ∈ . TN1 (ii) kb(s, ·)k∗,N2 ≤ C for all s ∈ .

Proof. The proof was given in the two-parameter case in [2]. It is essentially the same proof in the multi-parameter setting. We follow the simplified two-parameter proof from [8]. We first suppose that b ∈ bmo(TN ) that is we have that for any S ⊂ TN1 , N2 K ⊂ T , N1 + N2 = N, 1 1 |b(s, t) − m b|dsdt ≤kbk . |S| |K| S×K ∗,N ZS ZK

If S = S1 ×···× SN1 , then letting |S1|→ 0 we get that 1 |b(s, t) − mK×S′ b|dsdt ≤kbk∗,N , |S′||K| ′ ZK ZS ′ TN1−1 for any S = S2 ×···× SN1 ⊂ .

Repeating this process for S2, ··· ,SN1 , we obtain that

1 |b(s, t) − m b|dt ≤kbk |K| K ∗,N ZK and consequently that

sup kb(s, ·)k∗,N2 ≤kbk∗,N . s∈TN1 The same reasoning leads to

sup kb(·, t)k∗,N1 ≤kbk∗,N . t∈TN2

For the converse, we write b(s, t) − mS×Kb as follows

b(s, t) − mS×K b =(b(s, t) − mK b(s))+(mK b(s) − mS×Kb) . Hence

(2.1) |b(s, t) − mS×Kb|≤|b(s, t) − mKb(s)| + |mK b(s) − mS×Kb|. Integrating both sides of (2.1) over S × K and with respect to the measure dsdt |S||K|, we obtain 1 L := |b(s, t) − m b|dsdt |S||K| S×K ZS ZK 1 ≤ |b(s, t) − m b(s)|dsdt |S||K| K ZS ZK 1 + |m b(s) − m b|dsdt |S||K| K S×K ZS ZK = L1 + L2. BOUNDED MEAN OSCILLATION ON RECTANGLES. 5

Clearly, 1 L := |b(s, t) − m b(s)|dsdt 1 |S||K| K ZS ZK 1 1 ≤ |b(s, t) − m b(s)|dt ds |S| |K| K ZS  ZK  1 ≤ kb(s, ·)k ds ≤ C. |S| ∗,N2 ZS On the other hand, 1 L := |m b(s) − m b|dsdt 2 |S||K| K S×K ZS ZK 1 = |m b(s) − m b|ds |S| K S×K ZS 1 1 = (b(s, t) − m b(t)) dt ds |S| |K| S ZS ZK 1 ≤ |b(s, t) − m b(t)|dsdt |S||K| S ZS ZK 1 1 = |b(s, t) − m b(t)|ds dt |K| |S| S ZK  ZS  1 ≤ kb(·, t)k dt ≤ C. |K| ∗,N1 ZK Thus for any S ∈ TN1 and K ∈ TN2 , 1 |b(s, t) − m b|dsdt ≤ 2C. |S||K| S×K ZS ZK Hence kbk∗,N < ∞. The proof is complete. 

Note that if C∗ is the smallest constant in the equivalent definition above, ∗ then C is comparable to kkbmo(TN ). We make the following observation that can be proved exactly as in the one parameter case.

Lemma 2.2. Let b ∈ L2(TN ). Then

∗ 1 kbkbmo(TN ) ≃kbkN := sup inf |b(t) − λ|dt. TN λ∈C |R| R⊂ ZR Let us also observe the following.

Lemma 2.3. The following assertions hold. (i) Given an interval I in T, there is a function in BMO(T), denoted

logI such that 4 – the restriction of logI to I is log |I| . 6 BENOˆIT F. SEHBA

– k logI kBMO(T) ≤ C where C is a constant that does not depend on I.

(ii) For any f1, ··· , fN ∈ BMO(T), the function N TN b(t1, ··· , tN )= j=1 fj(tj) belongs to bmo( ). Moreover, P N kbkbmo(TN ) ≤ kfjkBMO(T). j=1 X N (iii) There is a constant C > 0 such that for any b ∈ bmom(T ) and any N rectangle R = I1 ×···× IN ⊂ T , 4 4 (2.2) |m b| ≤ C log + ··· + log kbk TN , R |I | |I | bmom( )  1 N  and this is sharp.

Proof. Assertion (ii) follows directly from the definition of bmo(TN ). (i) is surely well known, we give a proof here for completeness: let J be a fixed interval in T. Let J0 = J and Jk be the intervals in T with the same k center as J and such that |Jk| = 2 |J|, here k = 1, 2, ··· , N − 1 and N is N the smallest integer such that 2 |J| ≥ 1. We define JN = T. Thus, 4 N ≤ log ≤ N +2. 2 |J|

Next, we define U0 = J0 = J, Uk = Jk \Jk−1, for k =1, ··· , N. Now consider T the function logJ defined on by N T (2.3) logJ (t)= (N +2 − k)χUk (t), t ∈ . k X=0 Clearly, 4 log (t)= N +2 ≃ log for all t ∈ J. J 2 |J| T Lemma 2.4. For each interval J ⊂ , the function logJ defined by (2.3) belongs to BMO(T).

2 Proof. We start by estimating the L -norm of logJ . We have N N+2 2 2 2 || logJ ||2 = (N +2 − k) |Uk| = k |JN+2−k| k k X=0 X=2 N+2 N+2 ≤ k22N+2−k|J| ≤ k22N+2−k21−N k k X=1 X=1 N+2 = 8 k22−k. k X=1 BOUNDED MEAN OSCILLATION ON RECTANGLES. 7

It is clear that the last sum in the above equalities is finite and so logJ ∈ L2(T). For any dyadic interval I ⊂ T, let m ∈{0, ··· , N +1} be minimal such that I ∩Um =6 ∅, and l ∈{0, ··· , N +1} be maximal such that I ∩Um+l =6 ∅.

Let us estimate the length of I ∩ Uj for any m ≤ j ≤ m + l.

If l = 1 then I ∩ Um = I and there is nothing to say. If l = 2 then

|I ∩ Um|≤|I| and |I ∩ Um+1|≤|I|. Next, we consider the case l ≥ 2. We remark that in this case, half of

Um+l−1 is contained in I. Consequently, for any m ≤ j < m + l, we have 1 |I ∩ Uj| ≤ 2 2m+l−j−1 |I|. Finally, we have

|I ∩ Um+l| ≤ 2|I ∩ Um+l−1| ≤ 2|I|.

Hence, 1 L := | log −(N +2 − m − l)|dt |I| J ZI m l 1 + = | (m + l − k)χ |dt |I| Uk I k m Z X= m l 1 + ≤ (m + l − k)|I ∩ U | |I| k k m X= m l 1 + ≤ 4 (m + l − k)2−m−l+k|I| |I| k m X= k=l k = 4 ≤ 6. 2k k X=0 T Thus, for each interval J ⊂ , the function logJ given by (2.3) belongs to BMO(T) and there exists a positive constant C independent of J such that

|| logJ ||BMO(T) ≤ C. N To prove (iii), we observe that by definition, given b ∈ bmom(T ), for K any rectangle S ⊂ T , 0

4 |m (m b)| . log ||m b|| I Q |I| Q ∗,1   4 . log ||b|| TN . |I| bmom( )   8 BENOˆIT F. SEHBA

N In particular, for any rectangle R = I1 ×···× IN ⊂ T , we have N 4 N |mRb| ≤ C log kbkbmom(T ). |Ij| j=1 ! X The sharpness follows by applying the last inequality to the function N logR(t1, ··· , tN )= j=1 logIj (tj), R = I1 ×···× IN , and using (ii). The proof is complete.  P Lemma 2.4 and its proof complete the proof of Lemma 2.3. 

We now reformulate and prove Theorem 1.3.

Theorem 2.5. Let φ ∈ L2(TN ). Then the following assertions are equiva- lent. (a) φ is multiplier of bmo(TN ). (b) φ is a constant.

Proof. Clearly, (b) ⇒ (a). We prove that (a) ⇒ (b). Assume that φ ∈ L2(TN ) is a multiplier of bmo(TN ). Then for any f ∈ TN bmo( ), and any integer 0 < N1 < N, N2 = N −N1, ||(φf)(., t)||∗,N1 is uni- TN2 formly bounded for all t ∈ fixed and k(φf)(., t)k∗,N1 ≤kφfkbmo(TN ). Let f f s, t s, t N1 s N2 t us take as the function ( ) = logR( )= k=1 logSk ( )+ j=1 logQj ( ), R = S × Q ⊂ TN1 × TN2 , S = S ×···× S , Q = Q ×···× Q , 1 P N1 1 P N2 Sk ⊂ T, Qj ⊂ T. Then it follows that

1 N1 |φ(s, t)f(s, t) − m (φf) |ds ≤kφfk TN , for all S ⊂ T . |S| S bmo( ) ZS But from assertion (i) of Lemma 2.3 we have that for any t ∈ Q ⊂ TN2 fixed,

N2 4 N1 4 j=1 log |Q | + k=1 log |S | L := j k |φ(s, t) − m φ|ds |S| S P P ZS 1 . |φ(s, t) log (s, t) − m (φ log )|ds |S| R S R ZS ≤ ||φ logR ||bmo(TN )

. kMφk, where kMφk is the norm of the multiplication by φ, Mφ(f)= φf. Hence for any S ⊂ TN1 , Q ⊂ TN2 and t ∈ Q, N N 2 4 1 4 1 (2.4) log + log |φ(s, t) − mSφ|ds < ∞. |Qj| |Sk| |S| j=1 k ! S X X=1  Z  BOUNDED MEAN OSCILLATION ON RECTANGLES. 9

Letting for example |Q1|→ 0 in (2.4), we see that necessarily, N1 φ(s, t)= φ(t) for any s ∈ S ⊂ T . As N1 runs through (0, N), we obtain N that for any (t1, ··· , tN ) ∈ T ,

φ(t1, ··· , tN )= φ(tj)= φ(tj1 , ··· , tjk ), j, jl ∈{1, ··· , N}, 0 < k < N. The latter gives that φ is a constant. 

We have the following consequence which says that the only bounded functions in lmo(TN ) are the constants. This is pretty different from the one parameter case ([7]).

Corollary 2.6. Assume that φ ∈ L∞(TN ) and N log 4 j=1 |Ij | (2.5) ||φ||∗,log,N := sup |φ(t) − mRφ|dt < ∞. TN |R| R=I1×···×IN ⊂ P ZR Then φ is a constant.

Proof. Following Theorem 1.3 we only need to prove that any bounded function φ which satisfies (2.5) is a multiplier of bmo(TN ). For this we first N N recall that if f ∈ bmo(T ), then for any rectangle R = I1 ×···× IN ⊂ T , we have the estimate

4 4 |m f| . log + ··· + log ||f|| TN . R |I | |I | bmo( )  1 N  Now assume that φ ∈ L∞(TN ) and satisfies (2.5), and let f ∈ bmo(TN ). N Then using the above estimate, we obtain for any R = I1 ×···× IN ⊂ T , 1 |(fφ)(t) − m φm f|dt |R| R R ZR 1 ≤ |φ(t)||f(t) − m f|dt + |R| R ZR 1 |m f||φ(t) − m φ|dt |R| R R ZR ||φ||L∞ TN ≤ ( ) |f(t) − m f|dt + |R| R ZR 4 4 ||f|| TN log + ··· + log bmo( ) |I1| |I | N |φ(t) − m φ|dt  |R|  R ZR ≤ ||φ||L∞(TN ) + ||φ||∗,log,N ||f||bmo(TN ). It follows from the latter and Lemma 2.2 that if φ is bounded and satisfies (2.5), then for any f ∈ bmo(TN ), φf belongs to bmo(TN ). That is φ is a multiplier of bmo(TN ). The proof is complete.  10 BENOˆIT F. SEHBA

Remark 2.7. Let us first recall that in the one parameter case, it is a result of D. Stegenga [7] that L∞(T) ∩ LMO(T) is the exact range of pointwise multipliers of BMO(T). Let us define another little LMO space in the two- 2 parameter case lmoinv(T ) as follows.

2 2 2 Definition 2.8. A function b ∈ L (T ) is in lmoinv(T ) if there is a constant

C > 0 such that kb(·, t)k∗,log,1 ≤ C for all t ∈ T and kb(s, ·)k∗,log,1 ≤ C for all s ∈ T.

2 2 Clearly, lmoinv(T ) is a subspace of lmom(T ). The one parameter in- tuition and the equivalent definition of bmo(T2) in Proposition 2.1 may ∞ 2 2 lead one to claim that any function φ ∈ L (T ) ∩ lmoinv(T ) is a multi- plier of bmo(T2). This is not the case as the above results show and since ∞ 2 2 L (T ) ∩ lmoinv(T ) contains more than constants. For example, for any ∞ φ1,φ2 ∈ L (T) ∩ LMO(T), the function φ : (s, t) 7→ φ1(s)φ2(t) belongs to ∞ 2 2 L (T ) ∩ lmoinv(T ).

2.2. Proof of Theorem 1.5. We prove Theorem 1.5 in this section.

Proof of Theorem 1.5. (i) ⇒ (ii): we start by proving that any multiplier N N from bmo(T ) to bmom(T ) is a bounded function. We recall the following N estimate of the mean over a rectangle of functions in bmom(T ):

4 4 N |m b| . log + ··· + log ||b|| TN , R = I ×···× I ⊂ T . R |I | |I | bmom( ) 1 N  1 N  N N It follows that if φ is multiplier from bmo(T ) to bmom(T ), then for any N N b ∈ bmo(T ) and for any rectangle R = I1 ×···× IN ⊂ T ,

N 4 N (2.6) |mR (bφ) | . log ||bφ||bmom(T ) |Ij| j=1 ! X N 4 N N N ≤ C log kMφkbmo(T )→bmom(T )kbkbmo(T ). |Ij| j=1 ! X

Applying (2.6) to b = logI1 + ··· + logIN and using assertions (i) and (ii) of Lemma 2.3, we see that there is a constant C > 0 such that

N N N T |mRφ| ≤ CkMφkbmo(T )→bmom(T ), for any R = I1 ×···× IN ⊂ . We conclude that φ ∈ L∞(TN ).

N N To prove that φ ∈ lmom(T ), we only need by the definition of lmom(T ) to check that for any integer 0

TK T in and logS(t1, ··· , tK) = logS1 (t1)+ ··· + logSK (tK), Sj ⊂ be again the associated sum of functions which are uniformly in BMO(T). We have

K 4 j=1 log |S | L := j |m φ(t) − m φ|dt |S | R S×R P j ZS 1 = |m (φ log )(t) − m (φ log )|dt |S| R S S×R S ZS K ≤ ||mR(φ logS)||bmom(T )

N ≤ ||φ logS ||bmom(T )

N N N . ||Mφ||bmo(T )→bmom(T )|| logS ||bmo(T )

N N K = ||Mφ||bmo(T )→bmom(T )|| logS ||bmo(T )

N N . ||Mφ||bmo(T )→bmom(T ).

M Hence for any integer 0

∞ N N (ii) ⇒ (i): Let φ ∈ L (T ) ∩ lmom(T ). To prove that N N φ ∈ M(bmo(T ), bmom(T )), we only need to check that for any inte- ger 0

To estimate the first term, we only use that φ ∈ L∞(TN ) to obtain

1 L := |m [(φ − m φ)(f − m f)](t)|dt 1 |S| R S×R S×R ZS 1 ≤ | [(φ − m φ)(f − m f)](t , ··· , t )|dt ··· dt |S||R| S×R S×R 1 N 1 N ZS×R ||φ||L∞ TN ≤ ( ) |f(t , ··· , t ) − m f|dt ··· dt |S||R| 1 N S×R 1 N ZS×R ≤ ||φ||L∞(TN )||f||bmo(TN ). 12 BENOˆIT F. SEHBA

For the second term, we use the fact that as kmRfk∗,K is uniformly bounded, K 4 |mS×Rf| = |mS(mRf)| . log ||mRf||∗,K |Sj| j=1 ! X K 4 N ≤ log ||f||bmom(T ), |Sj| j=1 ! X K S = S1 ×···× SK ⊂ T ,K = N − M. Consequently, 1 L := | (m f)(m φ)(t) − m φm f|dt 2 |S| S×R R S×R S×R ZS K 4 j=1 log |S | ||f||bmo(TN ) . j |(m φ)(t) − m φ|dt  |S| R S×R P ZS N ≤ ||f||bmom(T )||mRφ||∗,log,K

N N ≤ ||f||bmom(T )||φ||lmom(T ). The last term only uses the fact that φ ∈ L∞(TN ). 1 L := | (m φ)(m f)(t) − (m φ)(m f) |dt 3 |S| S×R R S×R S×R ZS 1 ≤ kφk ∞ TN |(m f)(t) − m f|dt L ( ) |S| R S×R ZS ≤ kφkL∞(TN )kmRfk∗,K

∞ N N ≤ ||φ||L (T )||f||bmom(T ).

The estimates of L1, L2 and L3, and Lemma 2.2 allow to conclude that

N ∞ N N N ||φf||bmom(T ) . ||φ||L (T ) + ||φ||lmom(T ) ||f||bmo(T ). This complete the proof of the theorem.  

3. Multipliers to bmo(TN )

We would like to deduce some consequences of the above approach. We consider multipliers from any Banach space of functions on TN (strictly) containing bmo(TN ) to bmo(TN ). We have the following general result.

Theorem 3.1. Let X be any Banach space of functions on TN that strictly contains bmo(TN ). Then M(X, bmo(TN )) = {0}.

Proof. Clearly, 0 sends any function of X to bmo(TN ) by multiplication. Now let φ be any multiplier from X to bmo(TN ), then φ is also a multiplier from bmo(TN ) to itself. It follows from Theorem 1.3 that φ is a constant C. Suppose that C =6 0 and recall that bmo(TN ) is a proper subspace of 1 1 TN X. Then for any f ∈ X, we have that f = C( C f) = φ( C f) ∈ bmo( ). BOUNDED MEAN OSCILLATION ON RECTANGLES. 13

This contradicts the fact that bmo(TN ) is a strict subspace of X. Hence C is necessarily 0. The proof is complete. 

N Taking as X, the Chang-Fefferman BMO space or bmom(T ) we have as corollary the following.

Corollary 3.2. We have N N N N M(BMO(T ), bmo(T )) = M(bmom(T ), bmo(T )) = {0}.

The author would like to thank the referee for comments and observa- tions that improved the presentation of this note.

References

[1] S-Y. A. Chang and R. Fefferman, A continuous version of H1 with BMO on the bidisc, Ann. of Math. (2) 112 (1980), no. 1, 179-201. [2] M. Cotlar, C. Sadosky, Two distinguished subspaces of product BMO and Nehari-AAK theory for Hankel operators on the torus, Equations Operator Theory 26 (1996), no. 3, 273304. [3] S. Ferguson, C. Sadosky, Characterizations of bounded mean oscillation on the polydisk in terms of Hankel operators and Carleson measures, J. Anal. Math. 81 (2000), 239-267. [4] S. Pott, B. Sehba, Logarithmic mean oscillation on the polydisc, end- point results for multi-parameter paraproducts, and commutators on BMO, J. Anal. Math. 117 (2012), No. 1, 1-27. [5] S. Pott, B. Sehba, The multiplier algebra of product BMO, preprint. [6] B. F. Sehba, Operators on some analytic function spaces and their dyadic counterparts, thesis, University of Glasgow 2009. [7] D. A. Stegenga, Bounded Toeplitz operators on H1 and applications of the duality between H1 and the functions of bounded mean oscillation, Amer. J. Math. 98 (1976), No. 3, 573–589 [8] B. Wick, Little BMO and Muckenhoupt weights in product domains, unpublished notes. E-mail address: [email protected]