45 K. M. Dyaknv Garsia nrm kfk G = sup zd jfj d z ;jf(z)j = dened riginally fr f L (T m), is in fact an equivalent nrm n BMO. We shall als be cncerned

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1 Revista Matematica Iberamericana Vl. 5, N. 3, 999 Abslute values f BMOA functins Knstantin M. Dyaknv Abstract. The paper cntains a cmplete characterizatin f the mduli f BMOA functins. These are described explicitly by a certain Muckenhupt-type cnditin invlving Pissn integrals. As a cnsequence, it is shwn that an uter functin with BMO mdulus need nt belng t BMOA. Sme related results are btained fr the Blch space.. Intrductin. Let D dente the disk fz C : jzj < g, T its bundary, m the nrmalized arclength measure n T. Further, let z be the harmnic measure assciated with a pint z D, s that d z () = ;jzj j ; zj dm() T : The space BMO cnsists, by denitin, f all functins f L (T m) satisfying kfk = sup zd jf() ; f(z)j d z () < where f(z) sts fr R fd z. Alternative characterizatins f BMO, as well as a systematic treatment f the subject, can be fund in [G, Chapter VI] r [K, Chapter X]. Meanwhile, let us nly recall that the 45

2 45 K. M. Dyaknv Garsia nrm kfk G = sup zd jfj d z ;jf(z)j = dened riginally fr f L (T m), is in fact an equivalent nrm n BMO. We shall als be cncerned with the analytic subspace BMOA = BMO \ H (as usual, we dente by H p, 0 < p, the classical Hardy spaces f the disk). It is well knwn that \ H BMOA H p : 0<p< Nw ne f the basic facts abut H p spaces (see e.g. [G, Chapter II]) is this: In rder that a functin ' 0, living almst everywhere n T, cincide with the mdulus f sme nnzer H p functin, it is necessary sucient that ' L p (T m) (.) 'dm>; : On the ther h, the very natural ( perhaps n less imprtant) prblem f characterizing the mduli f functins in BMOA seems t have been unslved (r unpsed?) until nw, the present paper is intended t ll that gap. Thus, we lk at a measurable functin ' 0nT ask whether (.) ' = jfj fr sme f BMOA f 6 0 : The tw immediate necessary cnditins are (.) (.3) ' BMO : (T see that (.) implies (.3), use the fllwing simple fact: If fr any z D there is a number c(z) such that (.4) sup zd j'() ; c(z)j d z () <

3 Abslute values f BMOA functins 453 then ' BMO. Nw, given that (.) hlds, (.4) is bviusly fullled with c(z) = jf(z)j.) Hwever, we shall see that (.) (.3) tgether are nt yet sucient fr (.) t hld. Assuming that (.) hlds true, we cnsider the uter functin O ' given by O ' (z) + z = exp ; z '() dm() nte that (.) is equivalent t saying that (.5) O ' BMOA : z D Indeed, since jo ' j = ' almst everywhere n T, the implicatin (.5) implies (.) is bvius. The cnverse is als true, because the uter factr f a BMOA functin must itself belng t BMOA (in fact, if f = FI with F H I an inner functin, then it is easy t see that kfk G kf k G ). The prblem has thus been reduced t ascertaining when (.5) hlds. In this paper we pint ut a new crucial cnditin (reminiscent, t sme extent, f the Muckenhupt (A p ) cnditin, cf. [G, Chapter VI]) which characterizes, tgether with (.) (.3), the nnnegative functins ' with O ' BMOA this is cntained in Sectin belw. Further, in Sectin 3, we exhibit an example f a BMO functin ' 0 with ' L (T m)frwhich ur Muckenhupt-type cnditin fails. In ther wrds, we shw that the bvius necessary cnditins (.) (.3) alne d nt ensure the inclusin O ' BMOA. Finally, in Sectin 4 we nd ut when an uter functin with BMO mdulus lies in the Blch space B.. Outer functins in BMOA. Given a functin ' L (T m), ' 0, we recall the ntatin = 'd z intrduce, fr a xed M > 0, the level set z D (' M) = fz D : Mg :

4 454 K. M. Dyaknv In rder t avid cnfusin, let us pint ut the ntatinal distinctin between p p = () p = 'd z ' p (z) = (' p )(z) = ' p d z (here p>0 z D ). Finally, we need the functin Our main result is ; t = ( t 0 <t< 0 t : Therem. Suppse that ' BMO, ' 0, 'dm>; : The fllwing are equivalent. i) O ' BMOA. ii) Fr sme M > 0, ne has sup n ; 'd z : z (' M) < : Remark. The latter is vaguely reminiscent f the well-knwn Muckenhupt (A p ) cnditin [G, Chapter VI] which can be written in the frm sup n ' ; d z : z D < where ==(p ; ) <p<. The prf f Therem makes use f the fllwing elementary fact. Lemma. The functin R(u) = u + u ; u>0

5 Abslute values f BMOA functins 455 is nnnegative satises R(u) (u ; ) fr u : Indeed, since R(u) is the remainder term in the rst rder Taylr frmula fr =u, when exped abut the pint u =,ne has R(u) = (u ; ) where = (u) isa suitable pint between u. We als cite, as Lemma, the \harmnic measure versin" f the classical Jhn-Nirenberg therem (see Sectin Exercise 8 in [G, Chapter VI]). Lemma. There are abslute cnstants C > 0 c>0 such that z f T : jf() ; f(z)j >gc exp ; c kfk whenever z D, f BMO >0 (here again f(z) = R fd z ). Prf f Therem. Since ' BMO, we knw that (.) k'k G = sup zd (' (z) ; ) < : Similarly, cnditin i) f Therem is equivalent t hence, in view f (.), t ko ' k G = sup zd (' (z) ;jo ' (z)j ) < (.) sup zd ( ;jo ' (z)j ) < : In rder t ascertain when (.) hlds, we nte that jo ' (z)j = exp 'd z = e ;J(z)

6 456 K. M. Dyaknv where J(z) = ; 'd z rewrite (.) in the frm (.3) sup zd ( ; e ;J(z) ) < : We remark that J(z) 0 by Jensen's inequality. Further, we claim that (.3) is equivalent t the fllwing cnditin (.4) sup f J(z) : z (' M)g < fr sme M > 0 : Indeed, t deduce (.3) frm (.4), ne uses the inequality ; e ;x x the bvius fact that sup f ( ; e ;J(z) ): z D n (' M)g M : Cnversely, t shw that (.3) implies (.4), let K be the value f the supremum in (.3) put M = p K. It then fllws frm (.3) that sup fj(z) : z (' M)g < s ; e ;J(z) is cmparable t J(z) as lng as z (' M). We have thus reduced cnditin i) t (.4), we nw prceed by lking at (.4) mre clsely. T this end, we x a pint z (' ) intrduce the sets E = E (z) = n T : '() E = E (z) = T n E : Using the functin R(u) frm Lemma, we write (.5) J(z) = = = '() d z() '() '() R = I (z)+i (z) + '() ; d z () d z ()

7 where I j (z) = E j Abslute values f BMOA functins 457 '() R d z () j = : Nw if E then '()= =, Lemma tells us that '() '() ; R : Integrating, we get I (z) ('() ; ) d z () k'k G s that (.6) I (z) =O : In rder t estimate I (z), we bserve that (.7) z (E )= z n : '() < = z n : ; '() > n z : j ; '()j > C exp ; c k'k as fllws frm Lemma. Besides, fr E ne bviusly has (.8) '() ; =; '() (.9) '() > 0 : Further, we set E + E ; = f E : '() g = f E : '() < g

8 458 K. M. Dyaknv S(z) = E '() ; d z ()+ E + '() d z() + z (E ; ) : We have then (.0) I (z) =S(z)+ E ; '() d z() : Using (.8) (.9), we see that Cnsequently, 0 E '() ; E + d z () z (E ) '() d z() z (E + ) : (.) js(z)j z (E )+( z (E + )+ z(e ; )) = z (E )(+) C exp ; c ( + ) k'k where the last inequality relies n (.7). The functin t 7;! t exp (;at) ( + t) t being bunded fr any xed a>0, we cnclude frm (.) that S(z) =O : Tgether with (.0), this means that (.) I (z) =O + E ; '() d z() :

9 Abslute values f BMOA functins 459 A juxtapsitin f (.5), (.6) (.) nw yields (.3) J(z) =O + '() d z() : Finally, recalling the assumptin z (' ), we nte that E ; E ; = f T : '() < g (indeed, if T '() <, then '() < =, s that E ). Thus, (.3) can be rewritten as J(z) =O + ; 'd z this relatin has been actually veried fr z (' ). It nw fllws that cnditin (.4) (in which ne can safely replace the wrds \fr sme M > 0" by \fr sme M > ") hlds if nly if sup n ; 'd z : z (' M) < fr sme M > 0 we have thus arrived at ii). On the ther h, we have seen that (.4) is a restatement f i). The desired equivalence relatin is therefre established. We prceed by pinting ut a few crllaries f Therem. Crllary. Let ' BMO, ' 0, R 'dm > ;. If O ' BMOA 0 <p<, then O ' p(= O p ') BMOA. Prf. Since ' BMO, we have als ' p BMO (this is easily deduced frm the inequality ja p ; b p jja ; bj p, valid fr a, b 0 0 <p<). By Therem, the inclusin O ' BMOA yields (.4) sup n ; 'd z : z (' M) < fr sme M > 0, hence als fr sme M. Hlder's inequality gives ' p (z) p z D

10 460 K. M. Dyaknv whence ' p (z) z (' ) (' p M p ) (' M) : Therefre, (.4) with M implies the cnditin sup n(' p (z)) ; ' p d z : z (' p M p ) < which in turn means, by Therem, that O ' p BMOA. R Crllary. Let ' BMO, ' 0, 'dm > ;. Assume, in additin, that ' pssesses (after a pssible crrectin n a set f zer measure) the fllwing prperty: Fr sme " > 0, the set f T : '() "g is clsed cnsists f cntinuity pints fr '. Then O ' BMOA. Prf. We may put " = (therwise, cnsider the functin ' = '="). Thus, we are assuming that the set K = f T : '() g is clsed, while ' is cntinuus at every pint fk. We nw claim that (.5) K \ cls (' ) =? : Indeed, if 0 K \ cls (' ), then ne culd nd a sequence fz n g D such that '(z n ) z n ;! 0. On the ther h, since ' is cntinuus at 0, we wuld have lim n! '(z n ) = '( 0 ), a cntradictin. Frm (.5) it fllws that Hence, fr z (' ), ne has (.6) = dist (K (' )) > 0 : ; 'd z = K ;jzj j ; zj '() dm() ;jzj k 'k L (T m) :

11 Abslute values f BMOA functins 46 An easy estimate fr the Pissn integral f a BMO functin gives (.7) =O z D : ;jzj Cmbining (.6) (.7) yields ; 'd z (.8) ;jzj cnst k 'k ;jzj L (T m) fr all z (' ). Since ( ;jzj ) = O() ;jzj z D the right-h side f (.8) is bunded by a cnstant independent f z. Thus, sup n ; 'd z : z (' ) < the desired cnclusin fllws by Therem. Crllary 3. If ' BMO ess inf T '() > 0, then O ' BMOA. Prf. Fr a suitable " > 0 ne has f T : '() "g =?, s it nly remains t apply Crllary. 3. An uter functin with BMO mdulus that des nt belng t BMOA. Althugh Therem prvides a cmplete characterizatin f the mduli f BMOA functins, ne may still ask whether the bvius necessary cnditins (.) (.3) are als sucient fr O ' t be in BMOA (equivalently, whether cnditin ii) f Therem fllws autmatically frm (.) (.3)). An armative answer might parhaps seem plausible in light f crllaries 3 abve. Hwever, we are nw ging t cnstruct an example that settles the questin in the negative. In ther wrds, we prve

12 46 K. M. Dyaknv Therem. There is a nnnegative functin ' BMO with 'dm>; such that O ' = BMOA. Actually, we nd it mre cnvenient t deal with the space BMO(R) f the real line, dened as the set f functins f L (R dt=(+ t )) with kfk = sup jf(t) ; f(z)j d z(t) < : zc + R Here C + dentes the upper half-plane fim z > 0g, the harmnic measure z is nw given by d z (t) = Im z jt ; zj dt z C + t R f(z) sts fr R R fd z. The subspace BMOA(C + ) cnsists, by denitin, f thse f BMO(R) fr which f(z) ishlmrphic n C +. Using the cnfrmal invariance f BMO (see [G, Chapter VI]), ne can restate Therem as fllws. Therem 0. There is a nnnegative functin ' BMO(R) with R '(t) dt > ; +t such that the uter functin O ' (z) = exp i R z ; t + t '(t) dt t + z C + fails t belng t BMOA(C + ). The prf will rely n the fllwing auxiliary result. Lemma 3. Let E I be tw (nite nndegenerate) subintervals f R having the same center satisfying jej jij = <

13 Abslute values f BMOA functins 463 (here j j dentes length). Then there exists a functin BMO(R) such that (3.) (3.) 0 almst everywhere n R j E = j RnI =0 (3.3) k k C ; where C > 0 is sme abslute cnstant. Prf f Lemma 3. By means f a linear mapping, the general case is reduced t the special ne where E = [; ] I =[; ]. This dne, we dene the functin by (3.) by (t) = jtj <jtj : Nw (3.) is bvius, while (3.3) fllws frm the well-knwn facts that jtj BMO(R) that BMO(R) is preserved by truncatins (see Sectin Exercise in [G, Chapter VI]). Remark. A mre general ( much mre dicult) versin f Lemma 3, where E is an arbitrary measurable set cntained in the middle third f I, is due t Garnett Jnes [GJ] see als Exercise 9 in [G, Chapter VI]. We have, nnetheless, fund it wrthwhile t include a shrt prf f the versin required. Prf f Therem 0. Fr k = :::, set k = exp (;k ) let the numbers 0=a <b <a <b < be such that b k ; a k = k a k+ ; b k = k ;5=4 k : Cnsider the intervals I k = [a k b k ] J k = [b k a k+ ]. Further, let (3.4) x k = a k + b k y k = k

14 464 K. M. Dyaknv E k = h x k ; y k x k + i y k : Since je k j = k = k ji k j, Lemma 3 prvides, fr every k N, a functin k BMO(R) such that 0 k n R kj Ek = k j RnIk =0 Finally, we set k k k C ; : k k = k 3=4 k = exp ; k dene the sught-after functin ' by X ' = Rn[k + J k k ( k k + k Jk ) (here, as usual, A sts fr the characteristic functin f the set A). In rder t shw that ' enjys the required prperties, we have tverify several claims. Claim. ' BMO(R). This fllws at nce frm the inclusins ' ; X k k k L (R) X where the latter hlds true because X X k k k k C k k k k k k BMO(R) k ; = C X k k ;5=4 < :

15 Claim. ' L (R dt=( + t )). Abslute values f BMOA functins 465 Indeed, since '(t) < if nly if t S k J k, we have ; 'dt= X k J k ' dt = X k jj k j k = X k k ;5=4 < : Thus ; ' L (R dt). Observing, in additin, that ' = 0 utside the nite interval [ [ S = I k [ J k k k nting that Claim implies ' L (S dt), whence als + ' (= j 'j; ; ') L (S dt) we eventually cnclude that ' L (R dt): A strnger versin f Claim is thus established. Claim 3. Fr every M > 0, ne has (3.5) sup n ; 'd z : z C + M = : T verify (3.5), we set z k = x k + iy k (here x k y k are dened by (3.4)) shw that bth (3.6) lim k! '(z k)= (3.7) lim '(z k) k! ; 'd zk = : T this end, we rst nte that zk (E k ) = cnst, s (3.8) '(z k )= 'd zk 'd zk =( k +) zk (E k ) cnst k E k

16 466 K. M. Dyaknv which prves (3.6). Further, we write (3.9) ; 'd zk Tgether with the simple fact that the inequality (3.9) gives (3.0) J k ; 'd zk = zk (J k ) k : zk (J k ) cnst jj k j ; 'd zk cnst jj k j k : Finally, cmbining (3.8) (3.0), we btain '(z k ) ; 'd zk cnst k jj k j k = cnst k =4 : This prves (3.7), hence als Claim 3. In view f Therem (which admits an bvius restatement fr BMO(R)), Claim 3 is equivalent t saying that O ' = BMOA(C + ) s the prf is cmplete. 4. Outer functins with BMO mduli lying in the Blch space. Recall that the Blch space B is dened t be the set f analytic functins f n D with kfk B = sup ( ;jzj) jf 0 (z)j < zd (see [ACP] fr a detailed discussin f this class). We nw supplement Therem frm Sectin with the fllwing result. Therem 3. Let (4.) ' BMO ' 0 'dm>; :

17 Abslute values f BMOA functins 467 Suppse that, fr sme M > 0, n (4.) sup ; 'd z : z (' M) < : Then O ' B. The prf hinges n Lemma 4. If ' satises (4.), then (4.3) ( ;jzj) jo 0 j cnst + ; 'd z whenever z (' ) the cnstant n the right depends nly n '. Prf f Lemma 4. Dierentiating the equality O ' (z) = exp + z ; z '() dm() z D gives (4.4) O 0 =O ' (z) = O ' (z) '() dm() ( ; z) '() ( ; z) dm() where we have als used the fact that dm() =0: ( ; z) Frm (4.4) ne gets (4.5) ( ;jzj) jo 0 j jo ' (z)j '() d z () we prceed by lking at the integral n the right. Fllwing the strategy emplyed in the prf f Therem, we set E = E (z) = n T : '()

18 468 K. M. Dyaknv Using the elementary inequality E = E (z) = T n E : j uj ju ; j u we btain (4.6) '() E d z () '() E ; dz () k'k : j'() ; j d z () Repeating again sme steps frm the prf f Therem, we intrduce the sets E + E ; = f E : '() g = f E : '() < g nte that, since z (' ) (which is assumed frm nw n), we actually have (4.7) E ; = f T : '() < g : This dne, we write (4.8) '() E d z () = E '() d z() = z (E )+ + E ; E + '() d z() : The estimate (.7) frm Sectin says (4.9) z (E ) C exp ; c k'k '() d z()

19 Abslute values f BMOA functins 469 where C > 0 c > 0 are certain abslute cnstants. Besides, we bviusly have (4.0) E + '() d z() 0 (4.) E ; '() d z() = ; '() d z () (the latter relies n (4.7)). Using (4.9), (4.0) (4.) t estimate the right-h side f (4.8), we get (4.) '() E d z () C exp ; c + k'k ; 'd z : Since fr any a>0, (4.) implies (4.3) '() E sup te ;at t< t d z () cnst + Cmbining (4.6) (4.3) yields (4.4) '() d z () cnst T + ; 'd z : ; 'd z : Finally, substituting (4.4) int the right-h side f (4.5) nting that jo ' (z)j (say, by Jensen's inequality), ne eventually arrives at (4.3). Prf f Therem 3. Let M beanumber fr which (4.) hlds. Further, set = p '. Then BMO, L (T m), (z) z D :

20 470 K. M. Dyaknv In particular, ( p M) (' M) (similar bservatins were made in the prf f Crllary in Sectin ). Cnditin (4.) therefre yields n sup (z) ; d z : z ( p M) < : By Therem, it fllws that O BMOA. Since BMOA B, we als knw that O B. In rder t derive the required estimate (4.5) jo 0 j cnst ( ;jzj) ; we distinguish tw cases. Case. z D n (' M). s We have then jo (z)j (z) = p M jo 0 j = j(o ) 0 (z)j =jo (z)jjo 0 (z)j p M ko k B ( ;jzj) ; : Case. z (' M). Since (' M) (' ), a juxtapsitin f (4.3) (4.) immediately yields ( ;jzj) jo 0 j cnst < : Thus, (4.5) is established fr all z D, the prf is cmplete. Befre prceeding with ur nal result, we pint ut tw elementary facts. Lemma 5. Let ' satisfy (4.). equivalent. (a) (b) sup n n sup ' (z) Fr any M > 0, the fllwing are ; 'd z : z (' M) < : ; 'd z : z (' M) < :

21 Abslute values f BMOA functins 47 Prf. Since ' (z), the implicatin (b) implies (a) is bvius. Cnversely,letC be the value f the supremum in (a). Fr z (' M), cnditin (a) implies ; 'd z C M hence (' (z) ; ) ; 'd z CM ; k'k G which leads t (b). Lemma 6. Let BMO, 0. Suppse the numbers M > 0 M > 0 are related by (4.6) M = M + k k G : Then ( M ) ( M). Prf. If (z) M, then s that (z) M. (z) = (z) ; ( (z) ; (z) ) M ;k k G = M Nw we are in a psitin t prve Therem 4. If f BMOA is an uter functin with jfj then f B. BMO, Prf. Set = jfj ' =,sthatf = O f = O '. Since O BMOA, Therem yields n (4.7) sup (z) ; d z : z ( M) < with sme M > 0. By Lemma 5, we can replace (z) by (z) (= ) by Lemma 6, the arising cnditin will remain valid if we replace ( M) by the smaller set (' M ), where M is dened by (4.6). Cnsequently, (4.7) implies n sup ; 'd z : z (' M ) < :

22 47 K. M. Dyaknv Since ' BMO, the desired cnclusin that O ' B nw fllws by Therem 3. Remarks. ) Of curse, there are uter functins f BMOA with f =B. Fr example, this happens fr f(z) =(; z), where is the branch determined by = i. ) Let ' 0 n T. Recalling Muckenhupt's (A p ) cnditin (see Sectin abve), we have the implicatins ' BMO \ (A 3= ) implies O ' BMOA ' BMO \ (A ) implies O ' B: T seewhy, use Therems 3 tgether with the inequality ; ' ' ; ( > 0). It wuld be interesting t determine the full range f p's fr which ' BMO \ (A p ) implies O ' BMOA r O ' B. 3) There used t be a questin whether there existed a functin lying in all H p classes with 0 < p < in B, but nt in BMOA. Varius cnstructins (based n dierent ideas) f such functins were given in [CCS], [HT] [D]. Our current results shw hw t cnstruct an uter functin with these prperties. Namely, it suces t nd a functin ' satisfying (4.) (4.), with sme M > 0, but such that sup n ; 'd z : z (' M) = fr all M > 0. (An explicit example can be furnished in the spirit f Sectin 3 abve.) This dne, ne has O ' T 0<p< Hp (because ' T 0<p< Lp ) O ' BnBMOA, as readily seen frm Therems 3. 4) While this paper deals with uter functins nly, in [D] [D] wehave studied the interactin between the uter inner factrs f BMOA functins. Besides, we have characterized in [D3], [D4], [D5] the mduli f analytic functins in sme ther ppular classes, such as Lipschitz Besv spaces. In this cnnectin, see als [Sh, Chapter II]. Finally, wementin the recent paper [D6], which is clse in spirit t the current ne.

23 Abslute values f BMOA functins 473 References. [ACP] Andersn, J. M., Clunie, J., Pmmerenke, Ch., On Blch functins nrmal functins. J. Reine Angew. Math. 70 (974), -37. [CCS] Campbell, D., Cima, J., Stephensn, K., A Blch functin in all H p classes, but nt in BMOA. Prc. Amer. Math. Sc. 78 (980), [D] Dyaknv, K. M., Divisin multiplicatin by inner functins embedding therems fr star-invariant subspaces. Amer. J. Math. 5 (993), [D] Dyaknv, K. M., Smth functins cinvariant subspaces f the shift peratr. St. Petersburg Math. J. 4 (993), [D3] Dyaknv, K. M., The mduli f hlmrphic functins in Lipschitz spaces. Michigan Math. J. 44 (997), [D4] Dyaknv, K. M., Equivalent nrms n Lipschitz-type spaces f hlmrphic functins. Acta Math. 78 (997), [D5] Dyaknv, K. M., Besv spaces uter functins. Michigan Math. J. 45 (998), [D6] Dyaknv, K. M., Multiplicative structure in weighted BMOA spaces. J. Analyse Math. 75 (998), [G] Garnett, J. B., Bunded Analytic Functins. Academic Press, 98. [GJ] Garnett, J. B., Jnes, P. W., The distance in BMO t L. Ann. f Math. 08 (978), [HT] Hll, F., Twmey, J. B., Explicit examples f Blch functins in every H p space, but nt in BMOA. Prc. Amer. Math. Sc. 95 (985), 7-9. [K] Ksis, P., Intrductin t H p Spaces. Cambridge Univ. Press, 980. [Sh] Shirkv, N. A., Analytic Functins Smth up t the Bundary. Lecture Ntes in Math. 3 Springer-Verlag, 988. Recibid: de agst de.997 Knstantin M. Dyaknv Steklv Institute f Mathematics St. Petersburg Branch (POMI) Departament de Matematica Fntanka 7 Aplicada i Analisi St. Petersburg, 90, RUSSIA Universitat de Barcelna dyaknv@pdmi.ras.ru Gran Via, 585 E-0807 Barcelna, SPAIN