On new estimates for distances in analytic function spaces in the unit disk, the polydisk and the unit ball.

Transcription

1 Boletín de la Asociación Matemática Venezolana, Vol. XVII, No. 2 (21) 89 On new estimates for distances in analytic function saces in the unit disk, the olydisk and the unit ball. Romi Shamoyan, Olivera Mihic. Abstract. We rovide various new shar estimates for distances of fixed analytic functions of a certain classical analytic class (analytic Besov sace, Bloch tye sace) to its subsaces in the unit disk, the unit olydisk and the unit ball. We substantially enlarge the list of reviously known assertions of this tye. Resumen. Ofrecemos varias nuevas estimaciones fuerte ara las distancias de funciones analiticas fijas de una cierta clase de funciones analíticas clasicas (esacios analiticos de Besov, esacios de tio Bloch) a sus subesacios en el disco unidad, el olydisco unidad y la bola unidad. Amliamos sustancialmente la lista de afirmaciones reviamente conocidas de este tio. 1 Introduction and main notations Let be, as usual, the unit disk on the comlex lane, da(z) be the normalized Lebesgue measure on so that A() = 1 and dξ be the Lebesgue measure on the circle = {ξ : ξ = 1}. Let further H() be the sace of all analytic functions on the unit disk. For f H() and f(z) = k a kz k, define the fractional derivative of the function f as usual in the following manner α f(z) = (k + 1) α a k z k, α R. k= We will write f(z) if α = 1. Obviously, for all α R, α f H() if f H(). Suorted by MNZŽS Serbia, Project 1441

2 9 R. Shamoyan, O. Mihic 1 For a, let g(z, a) = log( ϕ a(z) ) be the Green s function for with ole at a, where ϕ a (z) = a z 1 az. For < <, 2 < q <, < s <, 1 < q + s <, we say that f F (, q, s), if f H() and f F (,q,s) = su f(z) (1 z 2 ) q g(z, a) s da(z) <. a As we know [15], if < <, 2 < q <, < s <, 1 < q + s <, f F (, q, s) if and only if f(z) (1 z 2 ) q (1 ϕ a (z) ) s da(z) <. su a It is known (see [15]) that F (2,, 1) = BMOA. We recall that the weighted Bloch class B α (), α >, is the collection of the analytic functions on the unit disk satisfying f B α = su f(z) (1 z 2 ) α <. z Sace B α () is a Banach sace with the norm f B α.note B 1 () = B() is a classical Bloch class (see [2], [8] and the references there). For k > s, <, q, the weighted analytic Besov sace Bs q, () is the class of analytic functions satisfying (see [8]) f q B q, s = 1 ( k f(rξ) dξ ) q (1 r) (k s)q 1 dr <. Quasinorm f B q, does not deend on k. If min(, q) 1, the class B q, s s () is a Banach sace under the norm f B q,. If min(, q) < 1, then we have a s quasinormed class. he well-known so called duality aroach to extremal roblems in theory of analytic functions leads to the following general formula dist Y (g, X) = su l(g) = inf g ϕ Y, l X, l 1 ϕ X where g Y, X is subsace of a normed sace Y, Y H() and X is the ortogonal comlement of X in Y, the dual sace of Y and l is a linear functional on Y (see [7]). Various extremal roblems in H Hardy classes in based on duality aroach we mentioned were discussed in [3, Chater 8]. In articular for a function K L q 1 () the following equality holds (see [3]), 1 <, + 1 q = 1, dist L q(k, H q ) = inf K g g H q,k L q H q = su 1 f H, f H 1 2π ξ =1 f(ξ)k(ξ)dξ.

3 On new estimates for distances in analytic function saces. 91 It is well known that if > 1 then the inf-dual extremal roblem in the analytic H Hardy classes has a solution, it is unique if an extremal function exists (see [3]). Note also that extremal roblems for H saces in multily connected domains were studied before in [1], [9]. Various new results on extremal roblems in A Bergman class and in its subsaces were obtained recently by many authors (see [6] and the references there). In this aer we will rovide direct roofs for estimation of dist Y (f, X) = inf g X f g Y, X Y, X, Y H(), f Y, not only in unit disk, but also in higher dimension. Let further Ω k α,ε = {z : k f(z) (1 z 2 ) α ε}, α, ε >, Ω α,ε = Ω α,ε. Alying famous Fefferman duality theorem, P. Jones roved the following heorem A. ([4], [15]) Let f B. hen the following are equivalent: (a) d 1 = dist B (f, BMOA); (b) d 2 = inf{ε > : χ Ω 1 1,ε (f)(z) da(z) 1 z is a Carleson measure}, 2 where χ denotes the characteristic function of the mentioned set. Recently, R. Zhao (see [15]) and W. Xu (see [14]), reeating arguments of R. Zhao in the unit ball, obtained results on distances from Bloch functions to some Möbius invariant function saces in one and higher dimensions in a relatively direct way. he goal of this aer is to develo further their ideas and resent new shar theorems in the unit disk and higher dimension. In next sections various shar assertions for distance function will be given. We will indicate roofs of some assertions in details, short sketches of roofs in some cases will be also rovided. hroughout the aer, we write C (sometimes with indexes) to denote a ositive constant which might be different at each occurrence (even in a chain of inequalities) but is indeendent of the functions or variables being discussed. Given two non negative real numbers A, B we will write A B if there is a ositive constant C such that A < CB. 2 New shar assertions on dist X (f, Y ) function in the unit disk For the roof of one of the main results of this aer we will need the following estimate which can be found in [8]. Lemma 1. (see [8]) Let s > 1, r >, t > and r +t s > 2. If t < s+2 < r (1 z then we have 2 ) s da(z) C 1 wz r 1 az t (1 w 2 ) r s 2 1 aw, a, w. t

4 92 R. Shamoyan, O. Mihic Note that F (, q, s) B q+2, s (, 1], (see [15]). Hence for α q+2, the roblem of finding dist B α(f, F (, q, s)) aears naturally. In the following theorem we show that in Zhao s theorems (see[15]) Möebius invariant Bloch classes can be relaced by Bloch classes with general weights. heorem 1. Let 1 <, α >, < s 1, α q+2, q > α( 1) s 1, q > s 2 + α( 1) and f B α. hen the following are equivalent: (a) d 1 = dist B α(f, F (, q, s)); da(z) (b) d 2 = inf{ε > : χ Ω 1 α,ε (z) is an s Carleson measure}. (1 z 2 ) α q s Proof. First we show d 1 Cd 2. According to the Bergman reresentation formula (see [2]), we have f(z) = C(α) f(w)(1 w 2 ) α 1 1 = C(α) f(w)(1 w 2 ) α 1 1 da(w)+ Ω (1 wz) α+2 1 α,ε (1 wz) α+2 da(w) +C(α) f(w)(1 w 2 ) α 1 1 \Ω 1 α,ε (1 wz) da(w)) = f α+2 1 (z)+f 2 (z), where C(α) is the constant of the Bergman reresentation formula (see [2]). By f 1 (z) = C(α) Ω 1 α,ε f(w)(1 w 2 ) α (1 wz) 2+α f(w) (1 w 2 ) α 1 wz 2+α da(w), 1 (1 w ). α f 1 (z) C da(w) C f Ω 1 B α α,ε hen f 1 B α. By Lemma 1, f 1 (z) (1 z 2 ) q (1 ϕ a (z) 2 ) s da(z) C f 1 1 B α f 1 (z) (1 z 2 ) q ( 1)α (1 ϕ a (z) 2 ) s da(z) C f 1 1 f(w) (1 w 2 ) α B α Ω 1 wz 2+α da(w)(1 z 2 ) q ( 1)α (1 ϕ a (z) 2 ) s da(z) 1 α,ε By χ Ω 1 α,ε C f 1 1 B f α B α (1 a 2 ) s (1 z 2 ) q ( 1)α+s da(z)da(w) Ω 1 α,ε da(z) (1 z 2 ) α q s C Ω 1 α,ε (1 a 2 ) s 1 w 2 α q s da(w). 1 aw 2s is an s-carleson measure, f 1 F (, q, s). Also we have f(w) (1 w 2 ) α da(w) f 2 (z) C \Ω 1 wz 2+α da(w) Cε 1 α,ε 1 wz 2+α Cε (1 z ) α. So, dist B α(f, F (, q, s)) f f 1 B α = f 2 B α < ε.

5 On new estimates for distances in analytic function saces. 93 It remains to show that d 1 d 2. If d 1 < d 2 then we can find two numbers ε, ε 1 such that ε > ε 1 >, and a function f ε1 F (, q, s), f f ε1 B α ε 1, and χ Ω 1 α,ε (z) (1 z 2 ) α q s is not a s-carleson measure. Since ( f(z) f ε1 (z) )(1 z 2 ) α ε 1, we can easily obtain (ε ε 1 )χ Ω 1 α,ε (z)da(z) C f ε1 (z) (1 z 2 ) α, (1) where χ Ω 1 α,ε is defined above. Hence from (1) and the fact that f ε1 F (, q, s) we arrive at a contradiction. he theorem is roved. Remark 1. heorem 1 can be exanded similarly to more general analytic classes with quasinorms su z <1 γ f(z) (1 z ) α, α >, with some restrictions on α, γ. Let B t = 1 B t = { f H() : 1 f B t}, t <. It is well-known that Bs q,q () B t (), t = s 1 q, t <, s < (see [8]). In the following theorem we calculate distances from a weighted Bloch class to Bergman saces for q 1. heorem 2. Let < q 1, s <, t s 1 q, β > 1 sq q 2 and β > 1 t. Let f B t. hen the following are equivalent: (a) l 1 = dist B t(f, Bs q,q ); (b) l 2 = inf{ε > : ( (1 w ) β+t q Ω ε, t(f) da(w)) 1 zw (1 z ) sq 1 da(z) < 2+β }. Proof. First ( we show that l 1 Cl 2. For β > 1 t, we have f(w)(1 w ) f(z) = C(β) β da(w) + ) f(w)(1 w ) β \Ωε, t (1 wz) β+2 Ω ε, t da(w) = f (1 wz) β+2 1 (z)+ f 2 (z), where C(β) is a well-known Bergman reresentation constant (see [2], [8]). For t <, f 1 (z) C f(w) (1 w ) \Ωε, t β (1 w ) β+t 1 wz β+2 da(w) Cε 1 wz β+2 da(w) Cε 1 (1 z ) t. So su z f 1 (z) (1 z ) t < Cε. For s <, t <, we have ( f 2 (z) q (1 z ) sq 1 da(z) C Ω ε, t So we finally have dist B t(f, B q,q s ) C f f 2 B t = C f 1 B t Cε. ) q (1 w ) β+t da(w) (1 z ) sq 1 da(z) C. 1 wz β+2

6 94 R. Shamoyan, O. Mihic It remains to rove that l 2 l 1. Let us assume that l 1 < l 2. hen we can find two numbers ε, ε 1 such that ε > ε 1 >, and a function f ε1 Bs q,q, f f ε1 ε B t 1, and ( (1 w ) β+t q da(w)) Ωε, t 1 zw (1 z ) sq 1 da(z) =. β+2 Hence as above we easily get from f f ε1 ε B t 1 that (ε ε 1 )χ Ωε, t(f)(z)(1 z ) t C f ε1 (z), and hence ( ) q χ Ωε, t(f)(z)(1 w ) β+t M = 1 wz β+2 da(w) (1 z ) sq 1 da(z) ( f ε1 (w) (1 w ) β ) q C 1 wz β+2 da(w) (1 z ) sq 1 da(z). Since for q 1, (see [2], [8]) ( f ε1 (z) (1 z ) α da(z) 1 wz t ) q C f ε1 (z) q (1 z ) αq+q 2 da(z) 1 wz tq, (2) where α > 1 q q, t >, f ε 1 H(), w, and (1 z ) sq 1 1 wz da(z) C, (3) q(β+2) (1 w ) q(β+2)+sq 1 where s <, β > 1 sq q 2, w. We get M C f ε1 (z) q (1 z ) sq 1 da(z). So as in the roof of the revious theorem we arrive at a contradiction. he following theorem is a version of heorem 2 for the case q > 1. heorem 3. Let q > 1, s <, t s 1 q, β > 1 sq q f B t. hen the following are equivalent: (a) l 1 = dist B t(f, Bs q,q ); }. (b) l 2 = inf{ε > : ( Ω ε, t(f) and β > 1 t. Let (1 w ) β+t 1 zw 2+β da(w)) q (1 z ) sq 1 da(z) < he roof of this theorem is similar to the roof of heorem 2 but here we will use (4) (see below) instead of (2). For ε >, q > 1, β >, α > 1 q, (see [8]) ( f(z) (1 z ) α ) q f(z) q (1 z ) αq 1 wz β+2 da(z) C 1 wz βq εq+2 da(z)(1 w ) εq, w, (4) which follows immediately from Hölder s inequality and (3) (see [8]).

7 On new estimates for distances in analytic function saces. 95 Remark 2. In heorems 2 and 3 we considered only the linear case ( = q). Estimates for distances of more general mixed norm Bs,q classes, s <, can be obtained similarly. We give an examle in this direction. It is known that B,q s () B (s 1 q ) (), s < (see [8]). heorem 4. Let o < q 1, q 1, s <, t s 1 q, β > sq + 1 q 2. Let f B t. hen the following are equivalent: (a) l 1 = dist B t(f, Bs,q ); (b) l 2 = inf{ε > : ( 1 ( Ωε, t(f) ) (1 w ) β+t q ) da(w) q 1 zw dξ (1 z ) s 1 d z < }. 2+β Let X be a quasinormed class in the unit disk, X H(). Let also su f(z) (1 z ) α+τ C α,τ f X, (5) z where C α,τ (α >, τ > ) is an absolute constant. It is well known that for many classes the above estimate (5) holds (see [2], [8] and the references there). he following result follows directly from arguments we rovided above during the roof of revious theorem. heorem 5. Let X, Y H(), X Y, α >, τ >. Let f Y () and su z f(z) (1 z ) τ C τ f Y, su z f(z) (1 z ) α+τ Ĉα,τ f X. hen B(f, X) Cdist Y (f, X), C = C(α, τ), where { } B(f, X) = inf ε > : su χ Ωτ,ε (z)(1 z ) α < z <1 Remark 3. heorem 5 rovides various new results for dist- function for different analytic classes. he general theorem we have resented is true even if the unit disk is relaced by the olydisk or the unit ball, since uniform estimates like (5), which is the base of roof, are well known in unit disk, unit ball and olydisk for various concrete classes of analytic functions (Bergman, Hardy, Bloch, BMOA, Q, etc.), see [2]. For < < and α >, let as above { ( B () = f H() : su r<1 ( { B(),1 = f H() : 1 f(rξ) dξ ) } (1 r) α <, f(rξ) dξ ) (1 r) α 1 dr < It is easy to see that B(),1 B (), < <, α >. We now define a new subset of the unit interval and then using its characteristic function we will give a new shar assertion concerning distance function. For ε >, f H(), let L ε,α (f) = {r (, 1) : (1 r) α f(rξ) dξ ε}.. }.

8 96 R. Shamoyan, O. Mihic heorem 6. Let f B, α >, 1 <. hen the following are equivalent: (a) s 1 = dist B (f, B,1 ); (b) s 2 = inf{ε > : 1 (1 r) 1 χ Lε,α(f)(r)dr < }. Proof. First we rove s 1 s 2. Let as assume that s 1 < s 2. hen we can find two numbers ε, ε 1 such that ε > ε 1 >, and a function f ε1 B,,1 f f ε1 B ε 1, and 1 (1 r) 1 χ Lε,α(f)(r) =. Hence we have (1 r) α (1 r) α f ε1 (rξ) dξ (1 r) α f(rξ) dξ ε 1. Hence for any s [ 1, ), 1 (ε ε 1 ) (1 r) s χ Lε,α(f)(r)dr C f(rξ) dξ su(1 r) α f(rξ) f ε1 (rξ) dξ r<1 1 ( f ε1 (rξ) dξ ) (1 r) α+s dr. hus we have a contradiction. It remains to show s 1 Cs 2. Let I = [, 1). We argue as above and obtain from the classical Bergman reresentation formula (see [16]). f(rξ)(1 r) t f(rξ)(1 r) t f(ρζ) = f(z) = C(t) dξdr+c(t) dξdr (1 rξρζ) t+2 (1 rξρζ) t+2 L ε,α(f) I\L ε,α(f) = f 1 (z) + f 2 (z), where t is large enough. hen we have (1 ρ) α f 2 (ρζ) dζ C(1 ρ) α f(rξ) (1 r) t dξ dr dζ I\L ε,α(f) 1 rξρζ t+2 ( ) C(1 ρ) α f(rξ) (1 r) t 1 dζ dξ dr I\L ε,α(f) 1 rξρζ t+2 C(1 ρ) α (1 r) t 1 (1 r) t f(rξ) dξ dr Cε(1 ρ)α dr Cε. I\L ε,α(f) (1 rρ) t+1 (1 rρ) t+1 For α >, (1 ρ)α 1 f 1 (ρζ) da(ρζ) C (1 ρ) α 1 f(rξ) (1 r) t dξ drda(ρζ) L ε,α(f) 1 rξρζ t+2 ) C su ((1 r) α (1 r) t f(rξ) dξ dr r<1 L ε,α(f) (1 r) t+1

9 On new estimates for distances in analytic function saces. 97 ) = C su ((1 r) α 1 f(rξ) dξ r<1 L ε,α(f) (1 r) dr. Note that the imlication f 1 B,1 known estimate (see [8]) ( 1 ( (1 ρ) α 1 α >, 1, f 1 H(). Hence inf g B,1 f g B he theorem is roved. < for 1 follows directly from the ) f 1 (ρξ) dξ) 1 dρ C (1 ρ) α 1 f 1 (ρξ) da(ρξ), C f f 1 B = f 2 B Cε. For < < and α >, let as above { 1 } B, () = f H() : (M (f, r)) (1 r) α 1 dr <, where M (f, r) = max ξ f(rξ), r (, 1), f H(). It is easy to see that B, () B α (), < <, α >. For ε >, f H(), let L ε,α (f) = {r (, 1) : (1 r) α M (f, r) ε}. heorem 7. Let f B α, α >, 1 <. hen the following are equivalent: (a) s 1 = dist Bα(f, B, ); (b) s 2 = inf{ε > : 1 (1 r) 1 χ Lε,α(f) (r)dr < }. he roof of heorem 7 is reetition of arguments rovided in heorem 6. We now rovide a shar version of heorem 6 for 1 case. heorem 8. Let f B, 1, t > α 1, α >. hen the following are equivalent: (a) s 1 = dist B (f, B);,1 (b) ŝ 2 = inf{ε > : 1 ( 1 χ L ε,α(f)(r) (1 r)t (1 rρ) t+1 dr) (1 ρ) α 1 dρ < }. Proof. he roof of heorem 8 is similar to the one rovided in heorem 6. One art of the theorem follows directly from the estimate ( q χ Lε,α(f)(r) C(ε, ε 1 ) f ε1 (rξ) dξ ) (1 r) αq, r (, 1), < q <, α >, (6) which were given in the roof of the revious theorem. Indeed, from (6) for q = 1, we get 1 ( 1 (1 r) t ) χ Lε,α(f)(r)dr (1 rρ) t+1 (1 ρ) α 1 dρ

10 98 R. Shamoyan, O. Mihic C 1 1 ( f ε1 (rξ) dξ ) (1 r) α+ 1+(t) (1 ρ) α 1 drdρ C f (1 rρ) (t+1) ε1 B,1. he rest is clear. It remains to argue as in the revious theorem to arrive to a contradiction. o rove the second art we note that as in heorem 6 we get f(z) = f 1 (z) + f 2 (z) and f 2 (z) C(1 r) α f 2 (rξ) dξ Cε. And moreover, arguing similarly as for 1 in heorem 6 we will have 1 ( f 1 (rξ) dξ ) (1 r) α 1 dr ) 1 ( 1 ) C su ((1 r) α (1 r)t f(rξ) dξ χ Lε,α(f)(r) dr (1 ρ) α 1 dρ. r<1 (1 rρ) t+1 Hence inf g B,1 f g B C f f 1 B = f 2 B Cε. he theorem is roved. Remark 4. Proofs of heorem 6, 7 and 8 can be easily extended to B q, saces with more general w(1 r) weights under some natural restrictions on the function w(r). For α > 1, β > and < <, let Mβ α () = {f H() : su(1 r) β f(w) (1 w ) α da(w) < } r<1 w r and M α,β() = {f H() : 1 ( (1 r) β 1 f(w) (1 w ) da(w)) α dr < }. w r Mβ α() and M,β α () for 1 are Banach saces and they were studied by various authors (see for examle [5]). It is easy to show that M,β α () M β α(), where (, ), β >, α > 1. In the following result we rovide another shar result on the dist function using the characteristic function of a new set. For f H() and ε >, let Gε,β α (f) = {r (, 1) : (1 r)β w r f(w) (1 w )α da(w) ε}, β >, α > 1. heorem 9. Let 1, α > 1, β >, f Mβ α. hen the following are equivalent: (a) t 1 = dist M α β (f, M,β α ); (b) t 2 = inf{ε > : 1 (1 r) 1 χ G α ε,β (f)(r)dr < }. he roof of heorem 9 will be omitted. It can be obtained by a small modification of the roof of the revious theorem.

11 On new estimates for distances in analytic function saces Shar assertions for the ist function in the unit ball and in the olydisk he goal of this section is to rovide straightforward generalizations of some of the results of the revious section to the case of the unit ball and the olydisk in C n : Practically all results of the revious section can be generalized to the case of the olydisk and to the unit ball. he roofs of these assertions are mostly based on the same ideas as in case of one variable, though some technical difficulties arise on that way. For the roofs of the theorems we will formulate below in higher dimensions we simly relace the well- known Bergman integral reresentation in the unit disk that were used in the revious section by the corresonding known integral reresentation version in the unit ball or in the olydisk (see [2], [16]) and then we define aroriate sets which will allow us to estimate such integral reresentation. o formulate our results we will need some standard definitions (see [2], [16]). We denote the oen unit ball in C n by B = {z C n : z < 1}. he boundary of B will be denoted by S, S = {z C n : z = 1}. By dv we denote the volume measure on B, normalized so that v(b) = 1, and by dσ we denote the surface measure on S normalized so that σ(s) = 1. As usual, we denote by H(B) the class of all holomorhic functions on B. We denote the unit olydisk by n = {z C n : z k < 1, 1 k n} and the distinguished boundary of n by n = {z C n : z k = 1, 1 k n}. By da 2n we denote the volume measure on n and by dm n we denote the normalized Lebesgue measure on n. Let H( n ) be the sace of all holomorhic functions on n. We refer to [2] and [1] for further details. For every function f H( n ) and f(z 1,..., z n ) = k 1,...k n a k1,...,k n z k1 1 zkn n, we define the oerator of fractional differentiation by n α f(z 1,..., z n ) = (k j + 1) α a k1,...,k n z k1 1 zkn n, α R. k 1= k 1= j=1 We will write f(z) if α = 1. For any α, α is an oerator acting from H( n ) to H( n ) (see [2]). We formulate now direct generalization of heorem 6 in the unit ball, its roof is a simle reetition of arguments we rovide above for the unit disk and will be omitted. For < < and α >, let B (B) = { f H(B) : su r<1 ( { B(B),1 = f H(B) : 1 ( f(rξ) dσ(ξ) S ) } (1 r) α <, f(rξ) dσ(ξ) ) (1 r) α 1 dr < S }.

12 1 R. Shamoyan, O. Mihic It is easy to see that B(B),1 B (B), < <, α >. We now define a new set on the unit interval and then using its characteristic function we will give a new shar assertion concerning the distance function. For ε >, f H(B), let L ε,α (f) = {r (, 1) : (1 r) α S f(rξ) dσ(ξ) ε}. heorem 1. Let f B (B), α >, 1 <. hen the following are equivalent: (a) ŝ 1 = dist B (B)(f, B,1 (B)); (b) ŝ 2 = inf{ε > : 1 (1 r) 1 χ Lε,α(f)(r)dr < }. Let I = (, 1). We denote by rξ = (rξ 1,..., rξ n ) where r I, ξ j, j = 1,..., n, ξ = (ξ 1,..., ξ n ) and also r ξ = (r 1 ξ 1,..., r n ξ n ), where r I n, r = (r1,..., r n ), r j I, ξ j, j = 1,..., n. We formulate now a olydisk version of heorem 6. For < < and α >, let as above { ( B ( n ) = f H( n ) : su f( ) n } r ξ) dm n (ξ) (1 r k ) α <, r 1<1,...,r n<1 n k=1 { 1 1 ( B(,1 n ) = f H( n ) : f( ) n } r ξ) dm n (ξ) (1 r k ) α 1 dr 1 dr n <. n It is easy to see that B(,1 n ) B ( n ), < <, α >. We now define a new set on I n and then using its characteristic function we will give a new shar assertion concerning the distance function. For ε >, f H( n ), let L ε,α (f) = { r = (r 1,... r n ) I n : k=1 n (1 r k ) α f( r ξ) dm n (ξ) ε}. n k=1 heorem 11. Let f B, α >, 1 <. hen the following are equivalent: (a) ŝ 1 = dist B (n )(f, B,1 ( n )); (b) ŝ 2 = inf{ε > : 1 1 n k=1 (1 r k) 1 χ Lε,α(f)(r 1,... r n )dr 1 dr n < }. Proof. he roof is a reetition of arguments of the one dimensional case and we omit details. Now we formulate the olydisk version of heorem 2 the roof is quite similar to one dimensional case and will be also omitted.

13 On new estimates for distances in analytic function saces. 11 Let B α ( n ), α >, be the collection of the analytic functions on the olydisk satisfying f Bα ( n ) = su z 1,...,z n f(z 1,..., z n ) n (1 z k 2 ) α <. B α ( n ) is a Banach sace with the norm f Bα ( n ). For k > s, <, q, Bs q, ( n ) let be the class of analytic functions on the olydisk satisfying (see [8]) 1 f q Bs q, ( n ) = 1 ( k=1 n k f( r ξ) dm n (ξ) ) q n (1 r k ) (k s)q 1 dr 1 dr n <. k=1 It is known that B q,q s ( n ) B (s 1 q ) ( n ), s < (see [2]). heorem 12. Let < q 1, s <, t s 1 q, β > 1 sq q and β > 1 t. Let f B t ( n ). hen the following are equivalent: (a) l 1 = dist B t ( n )(f, Bq,q s ( n )); (b) l 2 = inf{ε > : ( n ) n k=1 (1 w k ) β+t q da 2n(w 1,...,w n) Ω n ε, t(f) k=1 1 z kw k 2+β n k=1 (1 z k ) sq 1 da 2n (z 1,..., z n ) < }. We will formulate now a shar theorem for analytic classes on the subframe. Let R s f(z) = k (k 1,...k n 1 + +k n +1) s a k1,...,k n z k1 1 zkn n and n = (, 1] n. It is obvious R s f H( n ) if f H( n ). It is easy to note that f B,s ( n ) = su r<1(1 r) α R s f(rξ) dm n n (ξ) ( 1 ( ) C R s f(rξ) dm n (ξ) (1 r) dr) 1 α 1 n = f B,1,s ( n ), where s R, α >, < <. he analytic classes on the subframe n were studied in [11], [12], [13]. For ε > and f H( n ), let K ε,α,s = {r I : (1 r) α n R s f(rξ) dm n (ξ) ε}. heorem 13. Let f B,s( n ), α >, 1 <, s R. hen the following are equivalent: (a) ν 1 = dist B,s ( n )(f, B,1,s( n )); (b) ν 2 = inf{ε > : 1 (1 r) 1 χ Kε,α,s(f)(r)dr < }. his aer only concerns with the situation when some Bergman (or mixed norm) sace is acting as a subsace of a larger analytic class where su can be seen somehow in quasinorm. It is also easy to notice that we systematically use the classical Bergman integral reresentation formula in all our roofs.

14 12 R. Shamoyan, O. Mihic We note that based on similar arguments we obtained corresonding shar results in cases when mentioned above Bergman classes are relaced by analytic (weighted) Hardy tye saces in the unit disk. he only visible difference is that in such cases roofs are based on the classical Causchy integral reresentation formula. We give an examle of such a result in the unit disk. heorem 14. Let α > 1, 1, f B, 1 (). hen the following are equivalent: (a) υ 1 = dist Bα ()(f, B, 1 ()); ( ) 1 (b) υ 2 = inf{ε > : su r<1 1 r χ Ω r,ε,α (ξ)dξ < }, where Ω r,ε,α = {ξ : f(rξ) (1 r) α ε}. In [15] the author rovided several obvious corollaries of his results. Similar corollaries can be obtained immediate from our theorems 6-9. As examle we note that from heorem 6 we have the following Proosition 1. Let α >, 1 1 < 2 <. hen dist B (f, B 1,1 ) = dist B (f, B 2,1 ). Proosition 2. Let α >, 1 1 < 2 <. hen the closures of B 1,1 and B 2,1 in B are the same and f is in closure of B 1,1 in B if and only if 1 (1 r) 1 χ Lε,(f)(r) <, for every ε >. Similar results obviously are true also in higher dimension. We omit details. References [1] L. V. Ahlfors:Bounded analytic functions, uke Math Journal, 14 (1947), [2] M. M. jrbashian, F. A. Shamoian:oics in the theory of A α classes, eubner exte zur Mathematics, 1988, v 15. [3] P. L. uren:heory of H saces, Academic Press, New York, 197. [4] P. G. Ghatage,. Zheng:Analytic functions of bounded mean oscillation and the Bloch sace, Integr. Equat. Oer. heory, 17 (1993), [5] M. Jevtić, M. Pavlović, R. F. Shamoyan:A note on diagonal maing theorem in saces of analytic functions in the unit olydisk, Publ. Math. ebrecen, 74/1-2, 29, 1-14.

15 On new estimates for distances in analytic function saces. 13 [6]. Khavinson, M. Stessin:Certain linear extremal roblems in Bergman saces of analityc function, Indiana Univ. Math. J., 3(46) (1997). [7] S. Ya Khavinson:On an extremal roblem in the theory of analityc function, Russian Math. Surv., 4(32) (1949), [8] J. Ortega, J. Fabrega:Hardy s inequality and embeddings in holomorhic riebel-lizorkin saces, Illinois J. Math. 43 (1999), [9] W. Rudin:Analytic functions of class H, rans AMS, 78 (1955), [1] W. Rudin:Function theory in olydisks, Benjamin, New York, [11] R. F. Shamoyan, O. Mihić:On some inequalities in holomorhic function theory in olydisk related to diagonal maing, Czechoslovak Mathematical Journal 6(2) (21), [12] R. F. Shamoyan, O. Mihić:Analytic classes on subframe and exanded disk and the R s differential oerator in olydisk, Journal of Inequalities and Alications,Volume 29, Article I 35381, 22 ages. [13] Songxiao Li, R. F. Shamoyan:On some roerties of a differential oerator on the olydisk, Banach Journal of Math. Analysis, v.3, n.1, (29), [14] W. Xu:istances from Bloch functions to some Möbius invariant function saces in the unit ball of C n, J. Funct. Saces Al., v.7, n.1, (29), [15] R. Zhao:istances from Bloch functions to some Möbius invariant saces, Ann. Acad. Sci. Fenn. Math. 33 (28), [16] K. Zhu:Saces of Holomorhic Functions in the Unit Ball, Graduate exts in Mathematics, 226. Sringer-Verlag, New York, 25. Romi Shamoyan Bryansk University, Bryansk, Russia. Olivera Mihic Fakultet organizacionih nauka, Jove Ilića 154, Belgrade, Serbia. rshamoyan@yahoo.com, oliveradj@fon.rs

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