Q SPACES OF SEVERAL REAL VARIABLES

Q SPACES OF SEVERAL REAL VARIABLES MATTS ESSE N, SVANTE JANSON, LIZHONG PENG, AND JIE XIAO Abstract. For  2 (?1; 1), let Q (Rn ) be the space of all measurable

functions with

sup[`(I )]2?n

Z Z I

jf (x) ? f (y)j2 dx dy < 1; n+2 I jx ? y j

where the supremum is taken over all cubes I with the edge length `(I ) and the edges parellel to the coordinate axes in Rn . If  2 (?1; 0), then Q (Rn ) = BMO(Rn ), and if  2 [1; 1), then Q (Rn ) = fconstantsg. In the present paper, we discuss the case  2 [0; 1). These spaces are new subspaces of BMO(Rn ) containing some special Besov spaces. We characterize functions in Q (Rn ) by means of the Poisson extension, pCarleson measures, mean oscillation and wavelet coecients, and give a dyadic counterpart. Finally, we pose some open problems.

Contents

1. Introduction 2. Basic properties 3. Poisson extension 4. Generalized Carleson measures 5. Mean oscillation 6. Wavelets 7. The dyadic counterpart 8. Some problems References

1 3 10 16 19 25 30 34 35

1. Introduction For each p 2 (0; 1), the space Qp was introduced in [3] as the Banach space of all analytic functions f in the unit disk 4 satisfying sup

w24

ZZ

4

jf 0(z)j2 [g(z; w)]pdm(z) < 1;

(1.1)

Date : December 1, 1998. 1991 Mathematics Subject Classi cation. 46E15. Key words and phrases. Q (Rn ), Poisson extension, Carleson measure, Local analysis, Wavelets, Dyadic counterpart. This research was carried out when J. Xiao visited the Department of Mathematics, Uppsala University during 1998. He was supported by a grant from the department and from the Swedish Institute, as well as from the NNSF and SEC of China. 1

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MATTS ESSE N, SVANTE JANSON, LIZHONG PENG, AND JIE XIAO

where g(z; w) = log j(1 ? wz)=(w ? z)j is the Green function of 4 and m is the Lebesgue measure. As in [3], Qp is a proper subspace of BMOA (obtained by taking p = 1 in (1.1)) and Qp ( Qp if 0 < p1 < p2 < 1. Essen and Xiao [5] showed that an analytic function f in the Hardy space H 2 on the unit disk belongs to Qp if and only if its boundary values on the unit circle @ 4 satisfy Z Z jf (ei ) ? f (ei')j2 d d' < 1; ? p sup jI j (1.2) I I I jei ? ei' j2?p where the supremum is taken over all subarcs I  @ 4. Janson [9] used (1.2) to de ne a dyadic analogue Qdp of Qp and to prove that Qp is the intersection of Qdp and BMOA. Observe that (1.2) makes sense even if f is not analytic and it becomes possible to consider an extension to Harmonic Analysis over Euclidean spaces. This is our starting point. Throughout this paper, we always let R n be n-dimensional Euclidean space, and let R n++1 be the upper half space based on R n . A cube means always a cube in R n with edges parallel to the coordinate axes. We denote the edge length of a cube I by `(I ), and the Lebesgue measure of I by jI j; thus `(I ) = jI j1=n. Also, for t > 0, tI means the cube which has the same center as I and the edge length t`(I ). We let jxj denote the usual Euclidean norm for x 2 R n . For  2 (?1; 1), in analogy with (1.2), we de ne Q(R n ) to be the space of all measurable functions on Rn that satisfy Z Z jf (x) ? f (y)j2 dx dy < 1; 2 2  ? n (1.3) kf kQ(Rn ) = sup[`(I )] I I I jx ? y jn+2 where I ranges over all cubes in R n . Note that we have changed the parameter from p in (1.2) to  in (1.3); the relation between them is p = 1 ? 2 (for n = 1 as in (1.2)). We will henceforth use Q (R n ) as de ned by (1.3) exclusively, hopefully avoiding any possible confusion. Note that kf kQ(Rn ) = 0 if and only if f is constant a.e.; we thus regard Q (R n ) as a Banach space of functions modulo constants. (It is immediate that kf kQ(Rn ) is a norm; completeness also is easily veri ed, see Section 2.) Remark 1.1. Since every cube I is contained in a cube J with dyadic edge length (i.e. `(J ) 2 f2k : k 2 Zg) such that `(J ) < 2`(I ), it is obvious that we obtain an equivalent de nition, with an equivalent norm, if we consider only cubes of dyadic edge lengths in (1.3). Similarly, one can consider balls instead of cubes (with `(I ) replaced by the radius). We rst observe that if  = ? n2 , then Q (R n ) = BMO(R n ), which can be de ned for example as the space all functions in L1loc (R n ) satisfying 1

kf k2

2

?1 BMO(Rn ) = sup jI j I

Z

I

jf (x) ? f (I )j2 dx < 1;

(1.4)

Q SPACES OF SEVERAL REAL VARIABLES

3

where the supremum is taken over all cubes I in R n and Z f (I ) = jI j?1 f (x) dx I

stands for the mean value of f over the cube I , cf. [10]. In fact, we will prove below that Q(R n ) = BMO(Rn ) for all  2 (?1; 0). It is well known that the important space BMO(R n ) can be described by Poisson integrals, Carleson measures, wavelet coecients and dyadic cubes. The purpose of this paper is to give analogues for Q (R n ),  2 (0; 1), which are given in Sections 2{4 and 6{7. In Section 7 we also consider a dyadic version of Q (R n ). In Section 5, we will provide a local analysis of Q (R n ) which sheds further light on the relation between Q (R n ) and BMO(R n ). Finally, in Section 8 we will pose some open problems. Remark 1.2. One can similarly de ne Q spaces of functions de ned on a cube in R n or a torus Tn . Many of the results below extend to these situations, but we leave the details to the reader. Remark 1.3. It is also possible to study Qp spaces in several complex variables, see [2] and [15]. Some notations. Throughout the paper,  is a xed number in (?1; 1); usually we assume  2 (0; 1). C and c will denote unspeci ed positive constants, possibly dierent at each occurence; the constants may depend on  and the dimension n, but not on the functions or cubes involved. (They may sometimes depend on other xed parameters, for example, in Section 6, on the choice of wavelets.) We write X  Y , meaning cX  Y  CX . We sometimes consider dyadic cubes: Let D0 = D0(R n ) be the set of unit cubes whose vertices have integer coordinates, and let, for S any integer k 2 Z, 1 n ? k Dk = Dk (R ) = f2 I : I 2 D0 g; then the cubes in D = ?1 Dk are called dyadic. Furthermore, if I is any cube, we let Dk (I ), k  0, denote the set of the 2kn subcubes of edge length 2?k `(I ) obtained by k successive bipartitions S D ( I of each edge of I . Moreover, put D(I ) = 1 0 k ). E is the characteristic function of the set E . We let, for x 2 R n , jxj1 be the l1-norm on R n : j(x1; : : : ; xn)j1 = maxk jxk j. 2. Basic properties This section is devoted to some simple properties of Q (R n ) and to relations between Q(R n ) and the Besov spaces. We rst observe that, by simple changes of variables in (1.3), kf kQ(Rn ) is not aected by translations or dilations of R n , i.e. by replacing f (x) by f (x ? x0 ), x0 2 R n or f (tx), t > 0; if we use the norm de ned using balls, cf. Remark 1.1, the same holds for rotations. Thus, Theorem 2.1. Q (R n ) is invariant under translations, rotations and dilations, and thus under all similarities of R n ; moreover, there exists an equivalent norm on the space such that all similarities preserve the norm. We note the following alternative characterization of Q(R n ).

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MATTS ESSE N, SVANTE JANSON, LIZHONG PENG, AND JIE XIAO

Lemma 2.2. Let ?1 Z<  < 1Z . Then f 2 Q (R n ) if and only if jf (x + y) ? f (x)j2 dx jyjdy sup[`(I )]2?n n+2 < 1: I

jyj 1, and to the case  2 [0; 1=2] for n = 1.

Theorem 2.3. (i) Q (R n ) is decreasing in , i.e. Q (Rn )  Q (Rn ) if   . (ii) If n  2 and   1, or if n = 1 and  > 1=2, then Q (R n ) contains only functions that are a.e. constant, and thus Q (R n ) = f0g (as a Banach space ). (iii) If  < 0, then Q (R n ) = BMO(R n ). Remark 2.4. The inclusion in (i) is strict if  < except in the cases 1   < (n  2), 1=2 <  < (n = 1), and  < < 0, where equality holds by (ii) or (iii); see Example 2.10 and Remarks 2.8 and 2.11 below. In particular, if n  2 and 0   < 1 or n = 1 and 0    1=2, then f0g ( Q(R n ) ( BMO(R n ). Proof. (i). Suppose  < . If f 2 Q (R n ), then for any cube I in R n , we have Z Z jf (x) ? f (y)j2 dx dy I ZI Z jx ? y jn+2 jf (x) ? f (y)j2 jx ? yj2( ?) dx dy = I I jx ? y jZn+2Z jf (x) ? f (y)j2 dx dy  C [`(I )]2( ?) I I jx ? y jn+2  C [`(I )]n?2kf k2Q (Rn ) ; that is to say, f 2 Q(R n ). So, Q (Rn )  Q (R n ). (ii). First assume  > n=2. If f 2 Q (R n ), then by (1.3), for any cube I in Rn , Z Z jf (x) ? f (y)j2 dx dy  [`(I )]n?2kf k2 n : Q(R ) I I jx ? y jn+2 Since n ? 2 < 0, letting I grow to R n in the last inequality produces Z Z jf (x) ? f (y)j2 dx dy = 0: n+2 Rn Rn jx ? y j So, f is constant a.e. on R n . Secondly, assume   1 and assume f 2 Q (R n ). Assume rst that f 2 1 C (R n ), and that f is non-constant. By considering either the real or imaginary part, we may further assume that f is real. Then there exists a point x0 with rf (x0) =6 0, and by the rotation invariance (Theorem 2.1) we may assume that rf (x0 ) is directed along the positive x1-axis. Then there exist  > 0 and

Q SPACES OF SEVERAL REAL VARIABLES

5

a small cube I about x0 on which @f=@x1 > 2 and j@f=@xk j < , k  2. Let D be the cone fx : jx2 j +


+ jxnj < x1 < `(I )=2g; then if x; y 2 I and x ? y 2 D, by the mean value theorem, f (x) ? f (y) > (x1 ? y1 ). Consequently, Z Z jf (x) ? f (y)j2 dx dy I ZI jZx ? y jn+2 2z2  1 dz dx  n +2 I=2 z2D jz j  Z `(I )=2 z12 dz = 1; 2 = c jI j z11+2 1 0 a contradiction. Thus, if f 2 Q(R n ) \ C 1(R n ), then f is constant. Now, Q (R n ) is translation invariant by Theorem 2.1, and it follows by Minkowski's inequality that if f 2 Q (R n ) and g 2 L1 (R n ), then f  g 2 Q (R n ). In particular, if g 2 C01(R n ), then f  g 2 Q(R n ) \ C 1(R n ), and thus (assuming   1) f  g is constant. Finally, choosing a sequence gn  0 R with gn = 1 and supp gn shrinking to 0, f  gn ! f a.e., and it follows that f is a.e. constant, which completes the proof of (ii). (Note that the rst case uses large cubes and the second case small cubes to show that f has to be constant.) (iii). Case 1: ?n=2   < 0. On the one hand, by (i), Q (R n )  Q?n=2 (Rn ) = BMO(R n ). On the other hand, if f 2 BMO(R n ) and I is a cube, then for every y 2 R n with jyj < `(I ) Z

jf (x + y) ? f (x)j2 dx

IZ

?




 2 jf (x + y) ? f (2I )j2 + jf (x) ? f (2I )j2 dx 4

IZ

2I

and thus, since  < 0, Z

jf (x) ? f (2I )j2 dx  C jI jkf k2BMO(Rn ) Z

jyj c`(I )?2?njI j = c`(I )?2:

MATTS ESSE N, SVANTE JANSON, LIZHONG PENG, AND JIE XIAO

6

Consequently,

jI j?2

Z Z

jf (x) ? f (y)j2 dx dy

I I Z Z Z ?
2  ? 2 n min jx ? zj?2?n ; jy ? zj?2?n jf (x) ? f (y)j2 dx dy dz  C`(I ) ZI ZI ZI ?
min jx ? zj?2?n ; jy ? zj?2?n  C`(I )2?2n I I I ?


 jf (x) ? f (z)j2 + jf (y) ? f (z)j2 dx dy dz  Z Z 2 2 j f ( y ) ? f ( z ) j j f ( x ) ? f ( z ) j 2  ? n dx dz + dy dz  C`(I ) I I jy ? z j2+n I I jx ? z j2+n Z Z jf (x) ? f (y)j2 dx dy 2  ? n = C`(I ) I I jx ? y j2+n Z Z

(2.2)

Hence the left hand side of (2.2) is bounded as I ranges over all cubes, which means that f 2 BMO(R n ), cf. (5.2). Thus Q (R n )  BMO(R n ), and we have shown Q (Rn ) = BMO(R n ). We may now easily verify that Q (R n ) is a Banach space.

Theorem 2.5. Q (R n ) is complete, and thus a Banach space. Proof. Let ffmg be a Cauchy sequence in Q (R n ). By Theorem 2.3 and its proof, Q (R n )  BMO(R n ) with the inclusion map bounded. Hence, ffm g is a Cauchy sequence in BMO(R n ) too, and fm ! f in BMO(R n ) for some f . It follows easily, using Fatou's lemma, that for every k  1, kf ? fk kQ(Rn )  lim supm!1 kfm ? fk kQ(Rn ), which implies that fk ! f in Q (R n ) too. The following result relates Q spaces de ned in dierent dimensions.

Theorem 2.6. Let ?1 <  < 1. Let f be a function on Rn , n  1, and de ne F on R n+1 by F (x; t) = f (x), x 2 R n , t 2 R . Then F 2 Q (R n+1 ) () f 2 Q (R n ). Proof. A cube in R n+1 can be written I  [a; a + `(I )] for a cube I in R n and a real number a. Thus,

kF k2Q(Rn

+1

)

= sup[`(I )]2?(n+1) I;a

Z Z Z `(I )+a Z `(I )+a

I I a

a

jf (x) ? f (y)j2 dt du dx dy: j(x; t) ? (y; u)jn+1+2

Q SPACES OF SEVERAL REAL VARIABLES

7

The multiple integral is independent of a, so we may take a = 0. Moreover, letting s = t ? u, and assuming  > ?n=2, as we may by Theorem 2.3(iii), Z `(I ) Z `(I ) dt du j(x; t) ? (y; u)jn+1+2 0 0 Z `(I ) Z `(I ) dt du  n +1+2 jx ? yj  + jt ? ujn+1+2 0 Z 0 1  `(I ) jx ? yjn+1+2ds + jsjn+1+2 ?1 = C`(I )jx ? yj?n?2; while a similar opposite inequality follows by considering only t and u with jt ? uj < jx ? yj. Consequently, Z Z jf (x) ? f (y)j2 dx dy = kf k2 n : 2 2  ? n kF kQ(Rn )  sup[`(I )] Q(R ) I I I jx ? y jn+2 +1

Connection with Besov spaces. Denote the homogeneous Besov spaces on n R n by p;q  (R ). We refer to e.g. [16] for a general de nition; if 0 <  < 1 and n 1  p; q < 1, then p;q  (R ) consists of all measurable functions f such that Z

Z

Rn

q=p

Rn

jf (x + y) ? f (x)jp dx

dy < 1; jyjn+q

(2.3)

n and if 0 <  < 1 and 1  p < q = 1, then p;q  (R ) consists of the functions f such that

kf kp; 1(Rn ) = supn jyj? y2R

Z

1=p

Rn

jf (x + y) ? f (x)jp dx

< 1:

(2.4)

Theorem 2.7. Let n  2 and 0 <  < 1, or n = 1 and 0 <  < 1=2. (i) If q  2, then n=;q  (R n )  Q (R n ). n n (ii) If >  and q  1, then n= ;q (R )  Q (R ). Remark 2.8. In the case n = 1 and  = 1=2, it is seen by (1.3) and (2.3) that



 2 j f ( x ) ? f ( y ) j Q1=2 (R ) = f : dx dy < 1 = 21;=22 (R ); 2 j x ? y j R R which coincides with the Sobolev space L21=2 (R ). Thus (i) holds in this case too, while (ii) holds only for q  2. In particular, Q1=2 (R ) contains non-constant functions, and it is thus clear that the inclusion Q1=2 (R )  Q (R), > 1=2, is strict (cf. Theorem 2.3 and

Remark 2.4).

Z Z

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MATTS ESSE N, SVANTE JANSON, LIZHONG PENG, AND JIE XIAO

n=;2 Proof. (i). Since n=;q  (R n )   (R n ) [16, Chapter 3, Theorem 4], we may 2 assume that q = 2. Thus, assume that f 2 n=;  (R n ). By Holder's inequality with exponents n=2 and n=(n ? 2) we get, for any cube I in R n , Z

Z

jyj [16, Chapter 3, Theorem 5], we may assume that  < < 1, and in the case n = 1 further  1=2. If n= ;1(R n ), then for any cube I in R n we apply Holder's n f 2 n= ;q (R )   inequality with exponents n=2 and n=(n ? 2 ) together with (2.4) to get Z

Z

jyj k, there are at most C 2(j?k)n cubes J 2 Dj (I ) with  2 2(J ), each contributing 2?j(n?2+2?")?k("?2?n)jI j?1ja()j2 to the second double sum. Together these yield at most k 1  X X ? 1 2 2 j 2?(2?2?")(j?k)+2k C jI j ja()j 2 + j =0

 C 22k jI j?1ja()j2:

j =k+1

As a consequence, (6.7) yields f; (I )

C =C

1 X X

X

I 0 2E (I ) k=0 I ()2Dk (I 0 ) X

I 0 2E (I )

22k jI j?1ja()j2 + C

Ta;(I 0) + C  C:

We have proved that f; (I )  C for every cube I of dyadic edge length. Since the same estimate applies to every translate I + t, Lemma 5.4 shows that for every cube I of dyadic edge length, Z Z jf (x) ? f (y)j2 dx dy  C; 2  ? n [`(I )] I I jx ? y jn+2 nally, Remark 1.1 yields f 2 Q (R n ) and kf kQ(Rn )  C . Uniqueness of f follows from the uniqueness in BMO(R n ); if f; g 2 Q(R n )  BMO(R n ) have the same wavelet coecients, then they de ne the same linear functional on H 1(R n ) and thus f = g as elements of BMO(R n ) (i.e. modulo constants), see again [13, Section 5.6]. 7. The dyadic counterpart We de ne a dyadic analogue of Q (R n ) and give some results relating the two spaces. Once again, recall that D is the set of all dyadic cubes in R n , and de ne the dyadic distance (x; y) between two points in R n by (x; y) = inf f`(I ) : x; y 2 I 2 Dg: (The dyadic distance is in nite between points in dierent octants.) Note that

p jx ? yj  d (x; y):

(7.1)

Q SPACES OF SEVERAL REAL VARIABLES

31

The space Qd (R n ) is de ned as the space of all (measurable) functions f on such that Z Z jf (x) ? f (y)j2 dx dy < 1: (7.2) 2 2  ? n kf kQd(Rn ) = sup[`(I )] I 2D I I  (x; y )n+2 Note that kf kQd(Rn ) = 0 if and only if f is a.e. constant in each octant. Qd (R n ) is a Banach space if we regard it as a space of functions modulo such functions. We have analogues of Lemma 5.8 and Theorem 5.5. Lemma 7.1. Let ?1 <  < 1. Then, for any cube I 2 D and f 2 L2 (I ), Z Z jf (x) ? f (y)j2 dx dy: 2  ? n f;(I )  [`(I )] I I  (x; y )n+2 Proof. Suppose rst  > ?n=2. If I is a dyadic cube and x; y 2 I , then x; y 2 J for some J 2 Dk (I ) if and only if (x; y)  2?k `(I ), and thus, by (5.9), I (x; y) X = 12 2(2+n)k jI j?2

Rn



2k `(I )=(x;y) `(I ) 2+n

 (x; y)

jI j?2

= `(I )2?n(x; y)?2?n: The result follows by (5.8). If   ?n=2, this argument yields the inequality Z Z jf (x) ? f (y)j2 dx dy: 2  ? n f; (I )  C [`(I )] I I  (x; y )n+2 The opposite inequality follows by Lemma 5.3 and (7.1). Theorem 7.2. Let ?1 <  < 1. Then f 2 Qd (R n ) if and only if sup f;(I ) < 1:

(7.3)

I 2D

Moreover, supI 2D [ f; (I )]1=2 is a norm on Qd (R n ), equivalent to kf kQd(Rn ) as de ned above. Proof. Immediate by Lemma 7.1. We have also an analogue of Theorem 2.3. Let BMOd (R n ) be the dyadic BMO space de ned by ff 2 L2loc (R n ) : supI 2D f (I ) < 1g.

Theorem 7.3. (i) Qd (R n ) is decreasing in , i.e. Qd (Rn )  Qd (Rn ) if  < . (ii) If  > n=2, then Qd (Rn ) contains only functions that are a.e. constant in each octant, i.e. Qd (Rn ) = f0g (as a Banach space ). (iii) If  < 0, then Qd (R n ) = BMOd (R n ).

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MATTS ESSE N, SVANTE JANSON, LIZHONG PENG, AND JIE XIAO

Proof. (i) follows directly by (x; y)  `(I ) for x; y 2 I 2 D. For (ii), suppose that f 2 Qd (R n ), with  > n=2. If J is a dyadic cube, and Jk the dyadic cube containing J with `(Jk ) = 2k `(J ), then f;(Jk )  2(2?n)k f (J ), and by letting k ! 1 we obtain f (J ) = 0, J 2 D. Finally, (iii) follows by Lemmas 7.1 and 5.2. Remark 7.4. The inclusion in (i) is strict unless n=2 <  < or  < < 0, as can be seen by the following example. Example 7.5. For a cube I , let I? and I+ denote its left and right halves, separated by aPhyperplane x of I . Let f a 1 = a through thePcenter k g1 1 be a P 1 1 2 sequence with 1 jak j < 1 and de ne f = k=1 J 2Dk(I ) ak (J ? J? ), where I0 is any xed cube. It is then easily seen (cf. Theorem 7.11 below) that 8P1 2  < 0; > 1 jak j < 1; > > P < 1 2  = 0; f 2 Qd (R n ) () >P11 k2jakk j 1 2 jak j < 1; 0 <   n=2; > : every ak = 0; n=2 < : In particular, Qdn=2 (R n ) 6= f0g, so there is no cut-o at  = 1 as for the non-dyadic spaces (Theorem 2.3). Remark 7.6. If one instead considers Qd de ned on a cube, cf. Remark 1.2, then the cut-o at  = n=2 in Theorem 7.3(ii) disappears, and the space is non-trivial for arbitrarily large . Relations between Q (Rn ) and Qd (R n ). We rst observe that if we consider all cubes I with dyadic edge lengths in Theorem 7.2, instead of just dyadic cubes, we obtain instead Q (R n ). Lemma 7.7. Let ?1 <  < 1. Then Q (R n ) equals the space of all functions f on R n such that supI f; (I ) < 1, where I ranges over the set of all cubes in R n with dyadic edge lengths. Moreover, kf kQ(Rn )  supI f; (I )1=2 . Proof. Immediate by Remark 1.1 and Lemmas 5.3 and 5.4. Let t denote the translation operator t f (x) = f (x ? t). Theorem 7.8. Let ?1 <  < 1. Then f 2 Q (R n ) if and only if t f 2 Qd (R n ) for all t 2 R n and supt ktf kQd(Rn) < 1. Moreover, kf kQ(Rn )  supt kt f kQd(Rn ) . Proof. Since every cube of dyadic edge length is the translate of a dyadic cube, Lemma 7.7 and Theorem 7.2 show that 0

kf k2Q(Rn )  supn sup f; (I ? t) t2R I 2D

= supn sup tf;(I )

and the result follows.

t2R I 2D  supn kt f k2Qd(Rn ) ; t2R

+

Q SPACES OF SEVERAL REAL VARIABLES

33

Theorem 7.9. Let ?1 <  < 1=2. Then Q(R n ) = Qd (R n ) \ BMO(R n ). Proof. The inclusion Q(R n )  Qd (R n ) \ BMO(R n ) follows directly from the de nitions, (7.1) and Theorem 2.3(iii). Conversely, suppose that f 2 Qd (R n ) \ BMO(R n ). Let I be a cube of dyadic n edge length. Then there exist S a family of 2 dyadic cubes Ii of the same size as I , such that the union i Ii is a cube J with `(J ) = 2`(I ) and I  J . (J is not necessarily a dyadic cube.) By Lemma 5.7 and (5.5), f; (I )  C f;(J ) =C

2n X

i=1

f;(Ii) + C f (J )

 C kf k2Qd(Rn ) + C kf k2BMO(Rn ):

Hence f;(I ) is bounded uniformly for all cubes I of dyadic edge length, and the result follows by Lemma 7.7. Remark 7.10. We do not know whether the condition  < 1=2 in Theorem 7.9 is necessary. The theorem fails at least for 1    n=2, since then, cf. Example 7.5, I ? I? 2 Qd (R n ) \ BMO(R n ) for any cube I , while Q (R n ) = f0g by Theorem 2.3. The Haar system. Let h ,  2 , be the n-dimensional Haar system. Functions in BMOd(R n ) can be described by the Haar system, and the characterization coincides with one given above of BMO(R n ) using a wavelet basis [13, p. 157]. Similarly, for Qd (R n ) we have Theorem 7.11. Let  > 0. If f 2 Qd (R n ), then the sequence of its Haar coecients a() = (f; h),  2 , satis es sup Ta;(I ) < 1: (7.4) +

I 2D

Conversely, every sequence a() satisfying (7.4) is the sequence of Haar coef cients of a unique f 2 Qd (R n ). Proof. If a() = (f; h ) and I is a dyadic cube, then X (f ? f (I ))I = a()h

and thus

I ()I

f (I ) = jI j?1

X

I ()I

ja()j2 = Ta;0(I ):

It follows by the de nition (5.5) and Lemma 6.1 that f;(I )  Ta;(I ); and the result follows by Theorem 7.2. Let U be the isometry of L2 (R n ) given by U (  ) = h . Then Theorems 6.2 and 7.11 show that the operator U can be extended to an isomorphism between Q(R n ) and Qd (R n ).

34

MATTS ESSE N, SVANTE JANSON, LIZHONG PENG, AND JIE XIAO

n d n Corollary 7.12. Let 0 <  < 1. Then the ?Banach spaces
Q P P (R ) and Q (R )

are isomorphic; more precisely, the map U a()  = a()h is an isomorphism (with the sums interpreted formally or as converging in suitable weak topologies ).

8. Some problems In this section, we would like to mention some open problems.

John{Nirenberg type inequality. The John{Nirenberg distribution inequality [10] says that if f 2 BMO(R n ) then for any cube I and any t > 0, (8.1) mI (t) = jfx 2 I : jf (x) ? f (I )j > tgj  C jI j exp( kf k ?ct n ): BMO(R )

We hope to explore the following Problem 8.1. Let  2 (0; 1). Give a John{Nirenberg type inequality for Q (R n ). With (8.1) and the local behaviour of Q (R n ), we may conjecture that if f 2 Q (R n ) then 1 X mJ (t) X ?1 exp( ?ct ): (8.2)  Ct 2(2?n)k n) j J j k f k Q ( R  k=0 J 2Dk (I )

Duality. Corresponding to the important result BMO(R n )=[H 1(R n )], a problem of Q (R n )-type is Problem 8.2. Let  2 (0; 1). Find a predual of Q (R n ). Feerman{Stein type decomposition. Feerman and Stein [6] proved that f 2 BMO(Rn ) if and only if there are 'j 2 L1(R n ) such that f = '0 +

n X j =1

Rj ('j );

where Rj ' = '  (xj =jxjn+1) are the Riesz transforms. So, we naturally pose Problem 8.3. Let  2 (0; 1). Give a Feerman{Stein type decomposition of Q (R n ). Note that the space Qp(@ 4) has a decomposition of Feerman{Stein type [14, Theorem 1.2].

Quasiconformal homeomorphism. Reimann [17] proved that if ' is an ACL-homeomorphism from R n , n  2 to itself, then composition operator C', de ned by C'f = f  ', is bounded on BMO(R n ). With Theorem 2.1, we therefore have the following Problem 8.4. Let  2 (0; 1). Prove or disprove C' is bounded on Q (R n ).

Q SPACES OF SEVERAL REAL VARIABLES

35

Quasiconformal extension. We can introduce the concept of Q( )|a Q space on a general domain  R n , by considering only cubes I 

in (1.3). Jones's theorem [11] tells us that for a Jordan domain  R2 , BMO( ) = BMO(R2)j (the restriction of BMO(Rn ) to ) if and only if @

(the boundary of ) is a quasi-circle. So, our question is

Problem 8.5. Let  2 (0; 1). Find a geometric property of @ such that Q ( ) = Q (R2)j . Further problems can be posed via [4].

References [1] H. Aikawa and M. Essen, Potential Theory - Selected Topics. Lecture Notes Math. 1633, Springer-Verlag 1996. [2] M. Andersson and H. Carlsson, Spectral properties of multipliers on Qp spaces in strictly pseudoconvex domains (in preparation). [3] R. Aulaskari, J. Xiao and R.H. Zhao, On subspaces and subsets of BMOA and UBC. Analysis 15 (1995), 101{121. [4] L. Carleson, BMO{10 years' development, 18th Scandinavian Congress of Mathematicians, Birkhauser 15 1980, 3{21. [5] M. Essen and J. Xiao, Some results on Qp spaces, 0 < p < 1. J. reine angew. Math. 485 (1997), 173{195. [6] C. Feerman and E.M. Stein, H p spaces of several variables. Acta Math. 129 (1972), 137{193. [7] J. Garnett, Bounded analytic functions. Academic Press, New York 1981. [8] J.B. Garnett and P.W. Jones, BMO from dyadic BMO. Paci c J. Math. 99 (1982), 351{371. [9] S. Janson On the space Qp and its dyadic counterpart. Proc. Symposium \Complex Analysis and Dierential Equations", June 15{18, 1997, Uppsala, Sweden, to appear. [10] F. John and L. Nirenberg, On functions of bounded mean oscillation. Comm. Pure Appl. Math., 18 (1965), 415{426. [11] P. Jones, Extension theorems for BMO. Indiana Math. J., 29 (1980), 511{530. [12] H. Leutwiler, On BMO and the torsion function. Complex Analysis, Ed. J. Hersch and A. Huber, Birkhauser Verlag, Basel 1988, 157{179. [13] Y. Meyer, Wavelets and operators. Cambridge Univ. Press, 1992. [14] A. Nicolau and J. Xiao, Bounded functions in Mobius invariant Dirichlet spaces. J. Funct. Anal. 150 (1997), 383{425. [15] C. Ouyang, W. Yang and R. Zhao, Mobius invariant Qp spaces associated with the Green's function on the unitball of C n . Paci c J. Math. 182 (1998), 68{100. [16] J. Peetre, New thoughts on Besov spaces. Duke Univ. Math. Series, 1976. [17] H.M. Reimann, Functions of bounded mean oscillation and quasiconformal mappings. Comm. Math. Helv. 49 (1974), 260{276. [18] D. Stegenga, Multipliers of the Dirichlet space. Ill. J. Math. 24 (1980), 113{139. [19] E.M. Stein, Singular integrals and dierentiability properties of functions, Princeton Univ. Press, New Jersey, 1970.

36

MATTS ESSE N, SVANTE JANSON, LIZHONG PENG, AND JIE XIAO

Department of Mathematics, Uppsala University, PO Box 480, S-751 06 Uppsala, Sweden

E-mail address :

[email protected]

Department of Mathematics, Uppsala University, PO Box 480, S-751 06 Uppsala, Sweden

E-mail address :

[email protected]

Department of Mathematics, Peking University, Beijing 100871, P. R. China

E-mail address :

[email protected]

Department of Mathematics, Peking University, Beijing 100871, P. R. China

E-mail address :

[email protected]