The Bers Spaces and Sequence Spaces

Transcription

1 Internatonal Journal of Contemorary athematcal Scences Vol. 12, 2017, no. 6, HIARI Ltd, htts://do.org/ /jcms The Bers Saces and Sequence Saces Akhro Hoshda Suzuya, Chuo-ku, Satama-sh, Satama, , Jaan Coyrght c 2017 Akhro Hoshda. Ths artcle s dstrbuted under the Creatve Commons Attrbuton Lcense, whch ermts unrestrcted use, dstrbuton, and reroducton n any medum, rovded the orgnal work s roerly cted. Abstract In ths aer, we show that the Bers sace A q() on a hyerbolc Remann surface s somorhc to l as normed saces when t has nfnte dmenson. As a corollary, we gve an alternatve roof of a dualty between A q() and A q (), and we rove that A q() s a subsace of the bounded Bers sace A q (), on such a Remann surface. athematcs Subject Classfcaton: 30F30, 30F35, 30H05, 46E15 eywords: Bers sace, Bergman rojecton, comlemented subsace 1 Introducton In ths aer, let q 2 be an nteger, 1 < and 1 < be wth = 1. Let D be the unt dsc n the comlex lane and be a Remann surface whose unversal cover s D. Let π : D be an unversal holomorhc coverng ma and ρ be the hyerbolc densty for the surface defned by ρ (π(z)) π (z) = 1. The L sace L 1 z q() and the Bers sace 2 A q() are defned by L q() := { ϕ : measurable q form on ϕ L q () < } where ϕ L q () := ( ρ2 q (z) ϕ(z) dxdy) 1, A q() := { ϕ L q() ϕ : holomorhc }.

2 266 Akhro Hoshda The L sace L q () and the bounded Bers sace A q () are smlarly defned by { } L q () := ϕ : measurable q form on ϕ L q () < where ϕ L q () := su z (ρ q (z) ϕ(z) ), A q () := { ϕ L q () ϕ : holomorhc }. Let l and l be the Banach saces of sequences of comlex numbers defned by l := {ϕ = {ϕ n } n N ϕ l := ( } n N ϕ n ) 1 <, l := { ϕ = {ϕ n } n N ϕ l := su n N ϕ n < }. Let l be the normed sace defned by l := {ϕ l ϕ n = 0 for n > } wth the norm l for any N, and e j be the sequence defned by e j := {δ j,n } n N (δ : ronecker s delta) for j N. The Bers sace A q() s well nvestgated (cf. [10], [2], [6], [7], [8], [13], etc.). On the other hand, Lndenstrauss and Pelczynsk [11] showed that A 2(D) s somorhc to l as normed saces. Fletcher [4] showed that A 1 2() s somorhc to l 1 as normed saces when t has nfnte dmenson. Therefore t s a natural queston whether A q() s somorhc to l as normed saces. In ths aer, we show that A q() s somorhc to l as normed saces when t has nfnte dmenson (Theorem 3.2.2). As a corollary, we gve an alternatve roof of a dualty between A q() and A q () on such a Remann surface (Corollary 3.3.2), whch was showen by Bers [1] and ra [9], or Earle [3]. Also, as a corollary, A q() s a subsace of A q () on such a Remann surface (Corollary 3.3.3). It may be of nterest because Pommerenke [15] showed that there s a Remann surface 0 such that A 2( 0 ) A 2 ( 0 ) for any 1 <. 2 Prelmnary Notes In ths secton, we shall gve some basc facts. For 2 N ℵ 0 and Banach saces X 1, X 2,, X N, t s easy to check that X1 X N defned by (x 1,, x N ) X1 X N := ( x 1 X x N X N ) 1 s a norm on X1 X N. And f there s a Banach sace X havng subbanach saces X 1,, X N, then t s also easy to see that X1 S S X N defned by (x 1,, x N ) X1 S S X N := x x N X s a semnorm on X 1 X N. For two normed saces X and Y, we say a ma f from X to Y a normed embeddng f f s a somorhsm as normed saces between X and f(x). Prooston 2.1. Let X be a Banach sace, and Y be a normed sace. The followng condtons are equvalent.

3 The Bers saces and sequence saces There exst an oerator P from X to Y and a normed embeddng ι from Y to X satsfyng that P ι = d Y. 2. There exst an oerator P from X onto Y and a normed embeddng ι from Y to X satsfyng that P ι P = P. 3. There exst a vector sace Z, a norm Y Z on Y Z, an somorhsm µ between X and Y Z as normed saces, a ostve constant Y satsfyng that (y, 0) Y Z Y y Y for any y Y, and a ostve constant satsfyng that Y comonent of µ(x) Y x X for any x X. Proof. It s obvous that 1 2 holds. 2 3: We set Z := {x X P (x) = 0} whch s a subvector sace of X, and (y, z) Y Z := (ι(y), z) Y S Z for any (y, z) Y Z. The semnorm Y Z become a norm on Y Z. Indeed, let (y, z) Y Z = 0. By the assumton 2, there s an x X wth y = P (x). So, y = P (x) = P ι P (x) = P ι(y) = P ( z) = 0 and z = ι(0) = 0. The ma µ : X Y Z : x (P (x), x ι P (x)) s a lnear bjectve ma and µ 1 : Y Z X : (y, z) ι(y) + z. oreover, the ma µ s a sometrcally somorhsm because µ(x) Y Z = (P (x), x ι P (x)) Y S Z = ι P (x)+(x ι P (x)) X = x X. Now we can check that (y, 0) Y Z = ι(y) X ι y Y for y Y, and Y comonent of µ(x) Y = P (x) Y P x X for x X. 3 1: We can see that the ma ι : Y X : y µ 1 (y, 0) s a lnear njectve ma and ι(y) X µ 1 (y, 0) Y Z µ 1 Y y Y. So, the ma ι s a normed embeddng by the oen mang theorem. We can also check that the ma P defned by P : X Y : x (Y comonent of µ(x)) s a oerator wth P (x) Y = Y comonent of µ(x) Y x X. That the equaton P ι = d Y hold s obvous. A normed sace Y satsfyng the equvalent condtons stated n Prooston 2.1 for a Banach sace X shall be called a comlemented subsace of X (wth the rojecton P and the embeddng ι). The followng roertes are basc to our argument. Prooston Let X be a Banach sace, and Y be a comlemented subsace of X wth the rojecton P and the embeddng ι. If there exsts an oerator T from Y to X satsfyng that T ι < 1, P then T s a normed embeddng.

4 268 Akhro Hoshda 2. Let be a Remann surface whose unversal cover s D. Then there exsts an oerator θ (whch s called the Bergman rojecton) from L q() to A q() such that θ A q () = d A q (). Proof. 1. We set the oerator S := d X (ι T ) P from X to tself. We can see that S has an nverse oerator S 1 from X to tself such that S <, because d 1 d X S 1 P T ι X S = (ι T ) P P ι T < 1 by the assumton. So, the ma S s an somorhsm as normed saces from X to tself, and thus, the ma S ι s a normed embeddng from Y to X. Now, the equalty S ι = d X ι (ι T ) P ι = ι (ι T ) = T mly the concluson. 2. See [10] Theorem and Theorem See also [5] Lemma and Theorem , or [4] Lemma 2.1 and Theorem an Result Frst, we state a roerty on a bounded doman n the unt dsc. Prooston 3.1. Let Ω be an oen subset n C satsfyng Ω D and vol(ω) > 0, where vol(e) := E dxdy for a measurable subset E of R2. We denote by Θ q(d) := { f A q(d) f L q (D) 1 }. For any ɛ > 0, there exst an N N and subsets Ω 1,, Ω N Ω satsfyng followng all condtons: 1. Ω and Ω j are dsjont for any j. 2. vol(ω \ 1 k N Ω k ) = vol(ω k ) > 0 for any 1 k N. 4. For any f Θ q(d), any 1 k N, any z, z 0 Ω k, f(z) f(z 0 ) < ɛ. Proof. Let U(z 0, t) be the oen ball n the comlex lane centered at z 0 and radus t. We denote by d(z 0, B) := nf z B z 0 z (z 0 C, B C), d(a, B) := nf z0 A d(z 0, B) (A, B C), and by Θ q(ω) := { f Ω f Θ q(d) }. Let be a smly connected oen subset n C satsfyng Ω and D. For f Θ q(d) and z 0 Ω and 0 < t < d(z 0, ), the Cauchy ntegral fomula gves πt 2 f(z 0 ) = U(z 0 f(z)dxdy. Therefore, by the Hölder,t) nequalty, ( = ( πt 2 f(z 0 ) = f L q (D)( f(z)dxdy 2 q ρ f dxdy) 1 ( f dxdy) 1 ( ρ 2 q q 2 ρ dxdy) 1 ( q 2 ρ dxdy) 1 q 2 ρ dxdy) 1 ρ q 2 dxdy) 1.

5 The Bers saces and sequence saces 269 Thus, q 2 su f(z 0 ) ( ρ dxdy) 1 <, z 0 Ω π d(ω, ) 2 and so, Θ q(ω) s unformly bounded. Hence Θ q(ω) s equcontnuous, whch means that for ɛ > 0, there s a δ > 0 satsfyng the condton: f(z) f(z 0 ) < ɛ for any f Θ q(d) and z, z 0 Ω wth z z 0 < δ. Snce Ω s a relatvely comact subset of D, there s an N N and z 1,, z N Ω such that Ω 1 j N U(z j, δ ). So, { } Ω 3 j defned { 1 j N by Ω U(z 1, δ 3 j := ) Ω, j = 1 (U(z j, δ ) Ω) \ satsfes that Ω = 3 (Ω 1 Ω j 1), 2 j N 1 j N Ω j and z z 0 < δ for 1 k N and z, z 0 Ω k. Thus, we can fnd a subclass {Ω k } 1 k N of { } Ω j we needed. 1 j N Now, we show the man result. Theorem A q() s a comlemented subsace of l. 2. If A q() has nfnte dmenson, then A q() and l are somorhc as normed saces. Proof. We show only the frst statement because the second one s mmedate from the frst one and the Pelczynsk theorem that any nfnte dmensonal comlemanted subsace of l s somorhc to l as normed saces (cf. [14] Theorem 1 or [12] Theorem 2.a.3). For every, there exsts an oen subset U contanng, and a chart ψ such that ψ (U ) = D and ψ () = 0. Let V be an oen smly connected set n whose closure s contaned n U and V. Now, {V } forms an oen cover of, and by usng a countable bass of an oen Remann surface, t s ossble to fnd a countable { subclass {V } N such that = N V. So, { k } k N defned by k := V k, k = 1 V k \ 1 k 1, k 2 satsfes that k U k for k N and vol( \ k N k ) = 0, where vol(e) := dxdy for a measurable subset E of. Thus, we fnd a counterble or fnte E subclass { } N of { k } k N satsfyng that vol( ) > 0 and U for N and vol( \ N ) = 0. Let Ω := ψ ( ). By Prooston 2.2.2, there exsts an oerator θ : L q() A q() satsfyng θ A q () = d A q (). Let 0 < ɛ < 1 and ɛ ɛ θ n := ((n+1) 2 (su µ n ρ 2 q (µ)) vol(n)) 1 (n N). We denote by Θ q := { f A q() f L q () 1 }. For any N, by Prooston 3.1, there exst an N N and subsets Ω,1,, Ω,N Ω satsfyng that vol(ω \ 1 k N Ω,k ) = 0 and vol(ω,k ) > 0 for 1 k N and

6 270 Akhro Hoshda f ψ 1 (z) f ψ 1 (z 0 ) < ɛ for f Θ q and 1 k N and z, z 0 Ω k. Let z,k Ω,k and w,k := ψ 1 (z,k ) and,k := ψ 1 (Ω,k ). Let I E be the ndcator for a subset E of and,k A q() be the subnormed sace of L q() defned by,k A q() := { N 1 k N f,k I,k f,k A q() for N and 1 k N It s obvous that A q() s an sometrcally subnormed sace of,k A q() by the normed embeddng ι : A q(),k A q() : f f I,k. N 1 k N }. So, the oerator θ defned by θ :,k A q() A q() : f,k I,k θ( f,k I,k ) N 1 k N N 1 k N wth θ θ < tells us that A q() s a comlemented subsace of,k A q() wth the rojecton θ and the embeddng ι. Let T be the lnear ma defned by T :,k A q(),k A q() : f,k I,k f,k (w,k ) I,k N 1 k N N 1 k N and S := d,k A q() T. We can see that two normed saces (l N 1 l N ) and T (,k A q()) are sometrcally somorhc. Indeed, the ma A defned by A : T (,k A q()) (l N 1 l N ) f,k (w,k ) I,k (, f,k (w,k ) ( ρ 2 q,k ) 1 ek, ) N 1 k N 1 k N,k and the ma B defned by B : (l N 1 l N ) T (,k A q()) (, a,k e k, ) a,k ( 1 k N N 1 k N ρ 2 q,k,k ) 1 I,k are sometrcally somorhsms and A B = d and B A = d. T (,k A q()) s a comlemented subsace of l. Thus,

7 The Bers saces and sequence saces 271 The estmate S ι(f),k A q() = (f f(w,k )) I,k L q () N 1 k N = ( f f(w,k ) ρ 2 q I,k ) 1 N 1 k N,k ( ρ N(su 2 q ) f ψ 1 (z) f ψ 1 (z,k ) ) 1 z Ω,k 1 k N < ( ρ N(su 2 q ) ɛ vol(,k )) 1 1 k N = ( ρ N(su 2 q ) ɛ vol( )) 1 < ɛ for f Θ q leads that S ι s an oerator wth S ι < ɛ. So, T ι = ι S ι s also an oerator and T ι ι = S ι < ɛ < 1 1. Thus, by θ θ Prooston 2.2.1, T ι s a normed embeddng. Smlarly, d A q () θ T ι < θ ɛ < 1 and that θ T ι s an somorhsm as normed saces from A q() to tself. Therefore, A q() s a comlemented subsace of T (,k A q()) wth the rojecton ( θ T ι) 1 ( θ T (,k A q())) and the embeddng T ι. Hence, A q() s a comlemented subsace of l. Corollary 3.3. Let q 2 be an nteger, and be an oen Remann surface whose unversal cover s D. 1. If A q() and A q ( ) have nfnte dmenson, then A q() and A q ( ) are somorhc as normed saces. 2. Let 1 < <. If A q() and A q ( ) have nfnte dmenson, then (A q()) and A q ( ) are somorhc as normed saces. 3. Let < r. If A q() and A r q ( ) have nfnte dmenson, then there s a normed embeddng F from A q() to A r q ( ). Proof. 1. Theorem says that A q() = l = A q ( ). 2. Theorem also says that (A q()) = (l ) = l = A q ( ). 3. It s well known that the ncluson ma from l to l r s a normed embeddng. On the other hand, by Theorem 3.2.2, there are somorhsms G 1 : A q() l and G 2 : A r q ( ) l r for 1 r <. For r =, by the Bers theorem ([1]) that (A 1 q ( )) = A q ( ), we ut G 2 : A q ( ) = (A 1 q ( )) = (l 1 ) = l. Thus, for < r, F := G 1 2 G 1 s a normed embeddng from A q() to A r q ( ).

8 272 Akhro Hoshda References [1] L. Bers, Automorhc forms and Poncare seres for nfntely generated Fuchsan grous, Amer. J. ath., 87 (1965), htts://do.org/ / [2] P. Duren and A. Schuster, Bergman Saces, Amer. ath. Soc., htts://do.org/ /surv/100 [3] C. J. Earle, Some remarks on Poncare seres, Comos. ath., 21 (1969), [4] A. Fletcher, Local rgdty of nfnte dmensonal Techmüller saces, J. London ath. Soc., 74 (2006), no. 2, htts://do.org/ /s [5] A. Fletcher and V. arkovc, Quasconformal as and Techmüller Theory, Oxford Unv. Press, [6] F. P. Gardner, Techmüller Theory and Quadratc Dfferentals, John Wley and Sons, New-York, [7] F. P. Gardner and N. Lakc, Quasconformal Techmüller Theory, Amer. ath. Soc., htts://do.org/ /surv/076 [8] H. Hedenmalm and B. orenblum and. Zhu, Theorey of Bergman Saces, Srnger, htts://do.org/ / [9] I. ra, Echler cohomology and the structure of fntely generated lenan grous, Chater n Advances n the Theory of Remann Surfaces, Prnceton Unv. Press and Unv. Tokyo Press, Prnceton, New Jersey, 1971, htts://do.org/ / [10] I. ra, Automorhc Forms and lenan Grous, W. A. Benjamn [11] J. Lndenstrauss and A. Pelczynsk, Contrbutons to the theory of the classcal Banach saces, J. Func. Anal., 8 (1971), htts://do.org/ / (71) [12] J. Lndenstrauss and L. Tzafrr, Classcal Banach Saces I, Srnger- Verlag Berln Hedelberg, New-York, [13] S. Nag, The Comlex Anlytc Theory of Techmüller Saces, John Wley and Sons, New-York, [14] A. Pelczynsk, Projectons n certan Banach saces, Studa. ath., 19 (1960), htts://do.org/ /sm

9 The Bers saces and sequence saces 273 [15] Ch. Pommerenke, On ncluson relatons for saces of automorhc forms, Chater n Lecture Notes n ath., Srnger, Vol. 505, (1976), htts://do.org/ /bfb Receved: Setember 27, 2017; Publshed: October 17, 2017