ADJOINT OPERATOR OF BERGMAN PROJECTION AND BESOV SPACE B 1

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1 AJOINT OPERATOR OF BERGMAN PROJECTION AN BESOV SPACE B 1 AVI KALAJ and JORJIJE VUJAINOVIĆ The main result of this paper is related to finding two-sided bounds of norm for the adjoint operator P of the Bergman projection P, where P denotes the Bergman projection which maps L 1 (, dλ(z)) onto the Besov space B 1. Here, dλ(z) is the Möbius invariant measure (1 z 2 ) 2 da(z). It is shown that 2 P 4. AMS 21 Subject Classification: Primary 3H3, 32A25. Key words: Bergman projection, Besov space. 1. INTROUCTION The Bergman projections are some of the most important operators acting on the spaces of analytic functions over domains of the complex plane. Finding necessary and sufficient conditions for which Bergman projection is bounded is one of main subject of this area of research. Throughout the paper, is the open unit disc in complex plane C and da(z) = 1 π dxdy is the normalized Lebesgue area measure. H() is as usually the space of all analytic functions on the unit disc. Now, we recall some basic facts from the theory of Bergman spaces. The weighted Bergman space A p α, < p < and 1 < α < + is the set of analytic functions on which belong to the Lebesgue space L p (, da α ), where da α = (α + 1)(1 z 2 ) α da(z). In the case when α =, we have ordinary Bergman space denoted by A p. The weighted Bergman space A p α is a closed Banach subspace of L p (, da α ). The fact that A 2 α is a Hilbert subspace of L 2 (, da α ) gives natural way to define Bergman projection P α as an orthogonal projection from L 2 (, da α ) onto A 2 α and it is defined as P α f(z) = f(w)k α (z, w)da α (w), z, where K α is the Bergman reproducing kernel K α (z, w) = 1 (1 z w) 2+α, z, w. Since the previous formula is pointwise and A 2 α is dense in A 1 α, we have that for any f A 1 α MATH. REPORTS 15(65), 4 (213),

2 434 avid Kalaj and jordjije Vujadinović 2 f(z) = f(w) (1 z w) 2+α da α, z. In ([1], Theorem 1.1) there were given sufficient and necessary conditions for P β to be bounded, i.e. Proposition 1. Suppose 1 < α, β < + and 1 p < +. Then P β is a bounded projection from L p (, da α ) onto A p α if and only if α + 1 < (β + 1)p. The Besov space B p of, 1 < p < +, is the space of analytic functions f in such that ( ) 1 f Bp = (1 z 2 ) p f (z) p p dλ(z) < +, da(z) (1 z 2 ) 2 where dλ(z) = is a Möbius invariant measure on. By the previous relation, we define Bp which is a complete semi-norm on B p. The Besov space B p is a Banach space with the norm f = f() + f Bp. When p =, B = B is the Bloch space, and it is defined as B = {f H() : sup {(1 z 2 ) f (z) < + }. z The little Bloch space B is defined as B = {f H() : lim z 1 (1 z 2 ) f (z) = }. The Little Bloch space B is a closed subspace of B, i.e. more precisely B is the closure in B of the polynomials. The Besov space B 1 is defined in the different manner. B 1 is the space of analytic functions f in, which can be presented as f(z) = a n ϕ λn (z) for some sequence (a n ) l 1 and (λ n ). n=1 Here, ϕ λn denotes Möbius transform of, i.e. ϕ λn (z) = z λn, z. 1 z λ n B 1 is a Banach space with the norm { } f B1 = inf a n : f(z) = a n ϕ λn(z). n=1 The representation of the dual spaces of the Besov spaces are important in proving our results. So, let us recall some facts about dualities of Besov spaces ([6], Theorem 5.3.7). n=1

3 3 Adjoint operator of Bergman projection and Besov space B Proposition 2. Under the invariant paring f, g = f (z)g (z)da(z) we have the following dualities: (1) B p = B q if 1 p < + and 1 p + 1 q = 1; (2) B = B1. In this paper, by boundedness of Bergman projection on L 1 (, dλ), we mean that there exists a constant C > such that P f B1 C f L 1 (,dλ). Bergman projection connects the Besov B p space and L p (, dλ). This relation is expressed in the next theorem. Proposition 3. Suppose f H(), 1 p +. Then f B p f P L p (, dλ), where P is the Bergman projection. The proof of the previous theorem is given in [6], where it is shown that the Bergman projection maps boundedly L p (, dλ) onto B p, 1 < p < +. In [5] Perälä has found the exact norm of the following projection P : L (, da) B, i.e. he found that P = 8 π. For a generalization to several dimensional case see the work of Kalaj and Marković [2]. The extended Bergman projections from Lebesgue classes onto all Besov spaces on the unit ball in C n are defined and investigated in the work of H. Turgay Kaptanoglu [3]. In this paper, we consider the adjoint Bergman projection for the case p = 1, i.e. P : (B 1 ) (L 1 (, dλ)). By Proposition 1 the Bergman projection maps L 1 (, dλ) onto B 1. It is easy to show that Closed graph theorem implies that P is bounded in this case and therefore, P is also bounded. Our main results (Theorem 7) implies that its norm P is bounded by 4. It remains an open problem to find its exact value. 2. THE RESULT Assume that 2 < α 1 and define dλ α = (1 z 2 ) α da(z). As dλ α dλ, it follows that L 1 (, dλ) L 1 (, dλ α ). In the sequel, we consider the following sub-space of normed space L 1 (, dλ α ) { } L 1 α(, dλ) = f L 1 (, dλ α ) : f(z) dλ(z) <. In our first result, we prove that P : L 1 α(, dλ) B 1 is unbounded mapping. We need the following simple result related to the dual of L 1 α(, dλ).

4 436 avid Kalaj and jordjije Vujadinović 4 Lemma 4. The dual space (L 1 α(, dλ)) is isometrically isomorphic to the space L (, dλ α ). Proof. By Hahn-Banach theorem we can join to every bounded functional ϕ (L 1 α(, dλ)) its extension ψ on L 1 (, dλ α ), i.e. ψ (L 1 (, dλ α )) = L (, dλ α ), where ψ ϕ. On the other hand, the fact that L 1 α(, dλ) is dense in L 1 (, dλ α ) makes clear that ψ = ϕ, and that ψ is unique. Similarly, to every ψ (L 1 (, dλ α )) we can join its bounded restriction on L 1 α(, dλ α ) with the same properties as above. So, we conclude that appropriate correspondence is an isometric isomorphism. Theorem 5. Assume that dλ α = (1 z 2 ) α da(z), 2 < α 1 and let P be the Bergman projection P : L 1 (, dλ α ) B 1. Then P is an unbounded operator. Proof. According to Lemma 4, we are going to identify the dual space (L 1 α(, dλ α )) with L (, dλ α ). Now, we are going to consider adjoint operator P : B L (, dλ α ). Let us first determine the form of P. At the beginning, we can consider function g as a polynomial or some other function in Bloch space with bounded first derivative. Using classical definition of conjugate operator we have (1) P g(f) = g(p f), where g B, f L 1 (, dλ). (2) Identity in (1) is equivalent with f(z)p g(z)dλ α (z) = (P f) (z)g (z)da(z) 2f(w) w = (1 z w) 3 da(w)g (z)da(z). In order to apply Fubini s theorem on the right hand side of (2), we assume that f is a continuous function with a compact support in, i.e. f C c (). Now, we are in position to use Fubini s theorem, and we get (3) f(w)p g(w)dλ α (w) = 2 f(w)(1 w 2 ) α g w (z) (1 z w) 3 da(z)dλ α(w). On the left and right hand side of (3), we have two bounded functionals on L 1 (, dλ), which are identical on the set C c (), so we conclude that they are the same.

5 5 Adjoint operator of Bergman projection and Besov space B In other words, we have (4) P g(z) = 2(1 z 2 ) α z g (w) da(w), z. (1 z w) 3 So far, we have determined P on the subspace of bounded functions in B and in sequel, we shall prove that P is unbounded on this space w.r.t the norm of B. Let us observe functions g n z (w) = 1 C n n z k w k+1, w, for fixed z, n N and C n = 1 + ( ) k n k 2 k=1 k+1. It is easy to show that gz n B and gz n B 1. Also, we should notice that C n n, n +. The notation A(n) B(n), n +, means that there are constants C > and c > such that ca(n) = B(n) = CA(n), for all n large enough. On the other hand, we have (5) P gz n (z) = 2(1 z 2 ) α (gz n (w)) z (1 z w) 3 da(w), 1 for fixed z. By using Taylor expansion of the function, the formula (1 z w) 3 (gz n (w)) = 1 n C n (k + 1) zk w k and the orthogonality, for fixed z, we get (6) P g n z (z) = 2 C n (1 z 2 ) α z = 1 C n (1 z 2 ) α z = 1 C n (1 z 2 ) α z n (k + 1) z k w k n (k + 1) l= Γ(3 + l) l!γ(3) zl w l da(w) Γ(k + 3) z 2k w 2k da(w) k! n (k + 1)(k + 2) z 2k. Now, we see that lim n n + (k +1)(k +2) z 2k 2 =, z, i.e. (1 z 2 ) 3 more precisely, for z n (7) (k + 1)(k + 2) z 2k = 2 + z 2+2n ( 6 + 5n + n 2 2(1 + n)(3 + n) z 2 + (1 + n)(2 + n) z 4) ( z 2 1) 3. Choosing the sequence z n = 1 1 n, according to the relation (7), we estimate P gz n n (z n ) n 2+α, n +.

6 438 avid Kalaj and jordjije Vujadinović 6 Our next result is related to finding a positive constant C, such that P f C f, f B. In order to estimate the adjoint operator of the Bergman projection, when P : L 1 (, dλ) B 1, we need the following lemma. Lemma 6. Let B be the Bloch space. If f B then sup (1 z 2 ) 2 (z 2 f (z)) 4 f B. z Proof. By using Cauchy integral formula for an analytic function f, for fixed z and z < r < 1 we get (1 z 2 ) 2 (z 2 f (z)) = (1 z 2 ) 2 1 ξ 2 f (ξ) 2πi ξ =r (ξ z) 2 dξ = (1 z 2 ) 2 1 2π r 3 e 3it if (re it ) 2πi (re it z) 2 dt r3 f 2π B 2π(1 r 2 ) (1 z 2 ) 2 dt re it z 2 dt (8) 2π = r3 f B 2π(1 r 2 ) (1 z 2 ) 2 1 r 2 = r f B (1 r 2 ) (1 z 2 ) 2 z 2n r 2n n= = r 3 f B (1 z 2 ) 2 (1 r 2 )(r 2 z 2 ) f B (1 z 2 ) 2 (1 r 2 )(r 2 z 2 ) z r e it 1 z dt r eit On the other hand, for a fixed z we can choose r such that function ϕ(r) = (1 r 2 )(r 2 z 2 1+ z ) is maximal, i.e. for r = 2 2 we have maximal value of function ϕ max = (1 z 2 ) 2 4. Finally, sup (1 z 2 ) 2 (z 2 f (z)) ) 4 f B. z The following theorem is the main result of this paper. Theorem 7. Let P be the Bergman projection P : L 1 (, dλ) B 1. Then P is bounded operator and 2 P 4.

7 7 Adjoint operator of Bergman projection and Besov space B Proof. As we have already seen in the proof of Theorem 5, the adjoint operator P : B L acts in the following manner f(z)p g(z)dλ(z) = (P f) (z)g (z)da(z) (9) f(w) w = 2 (1 z w) 3 da(w)g (z)da(z), where f L 1 (, dλ) and g is Bloch function. By using elemental relation for norm of functional x on Banach space X, x = sup { x, x : x X, x 1}, we conclude that (1) P g = 2 sup f(w) w (1 z w) 3 da(w)g (z)da(z). f 1 It is clear that in the last relation we can take f to be a continuous function with compact support, i.e. f C c (). So, for fixed f C c (), f L 1 (,dλ) = 1, we have (11) f(w) w (1 z w) 3 da(w)g (z)da(z) = lim f(w) w r 1 r (1 z w) 3 da(w)g (z)da(z) = lim f(w) w(1 w 2 ) 2 g (z) r 1 (1 z w) 3 da(z)dλ(w). lim The duality argument (L 1 (, dλ)) = L (, dλ) implies lim f(w) w(1 w 2 ) 2 g (z) r 1 r (1 z w) 3 da(z)dadλ(w) lim sup (1 z 2 ) 2 g (w) z r 1 z r (1 z w) 3 da(w) (12) = lim sup (1 z 2 ) 2 g (rw) z r 1 (1 rz w) 3 rda(w) where ϕ(z) = (13) lim z sup r 1 z sup r 1 z (1 z 2 ) 2 z ϕ(rz), r g (rw) (1 z w) 3 da(w) is an analytic function of z. Moreover, (1 z 2 ) 2 z ϕ(rz) = sup(1 z 2 ) 2 z ϕ(z). Let us prove that (13). For ω = re it, t [, 2π), r [, 1), by using the fact that ϕ(ωz) is subharmonic for fixed z, we conclude that the function z

8 44 avid Kalaj and jordjije Vujadinović 8 sup z (1 z 2 ) 2 ωzϕ(ωz) is also subharmonic in ω. By the maximum principle for the subharmonic functions we obtain (14) lim sup (1 z 2 ) 2 rz ϕ(rz) sup sup(1 z 2 ) 2 ωz ϕ(ωz) r 1 z ω z sup t [,2π) z Further, if z n is a sequence in such that sup(1 z 2 ) 2 e it z ϕ(e it z) = sup(1 z 2 ) 2 z ϕ(z). z sup(1 z 2 ) 2 z ϕ(z) = lim (1 z n 2 ) 2 z n ϕ(z n ), z n then for r n = 1 (1 z n )/n, we have lim sup (1 z 2 ) 2 z ϕ(rz) (1 z n /r n 2 ) 2 z n ϕ(z n ) r 1 z = (1 y n 2 ) 2 r n y n ϕ(r n y n ). Here, y n = z n /(1 (1 z n )/n) < 1 for n > 1. Since it follows that lim n 1 z n 2 1 z n /r n 2 = 1, lim sup (1 z 2 ) 2 rz ϕ(rz) sup(1 z 2 ) 2 z ϕ(z). r 1 z Finally, we get (15) P g 2 sup(1 z 2 ) 2 z z z g (w) (1 z w) 3 da(w), for g B. Our goal is to determine a constant C such that (16) 2 sup(1 z 2 ) 2 g (w) z z (1 z w) 3 da(w) C g B. (17) If we write w in polar coordinate w = re it we get g (w) z (1 z w) 3 da(w) = z 1 1 2π g (re it ) rdr π (1 zre it ) 3 dt = z 1 1 g (ξ)ξ 2 rdr π (ξ zr 2 ) 3 dξ. ξ =r

9 9 Adjoint operator of Bergman projection and Besov space B Now, Cauchy formula applied to the last integral in (17) leads to z 1 1 g (ξ)ξ 2 1 rdr π ξ =r (ξ zr 2 ) 3 dξ = z (g (ξ)ξ 2 ) ξ=zr rdr 2 (18) (19) By Lemma 5, we get 2 sup(1 z 2 ) 2 z z = 1 2 z (g (ξ)ξ 2 ) dξ = 1 2 (z2 g (z)). g (w) (1 z w) 3 da(w) sup (1 z 2 ) 2 (z 2 g (z)) 4 g B. z Finally, we are going to give a lower bond for the P. Let g = 1 2 log[(1 + z)/(1 z)]. Then g = 1/(1 z 2 ). We make use of (11). We have P g, f = 2 lim f(w) w(1 w 2 ) 2 g (z) r 1 (1 z w) 3 da(z)dλ(w). Let us find We have Further, I r = I r = r r z 2 (1 z w) 3 = 1 2 Let z = ρe it. Then z 2 (1 z w) 3 = 1 2 r g (z) (1 z w) 3 da(z) z 2 (1 z w) 3 da(z). z 2k j=2 j(j 1)z j 2 w j 2. (2k + 2)(2k + 1)ρ 4k w 2k + l Z\{} e ilt A l (ρ, w), where the functions A l (ρ, w), l Z \ {} do not depend on t. Thus, I r = 1 2π = 2π dt r (k + 1)r 4k+2 w 2k = (2k + 2)(2k + 1)ρ 4k+1 w 2k dρ r 2 (1 r 4 w 2 ) 2.

10 442 avid Kalaj and jordjije Vujadinović 1 Hence, we have P g, f = 2 lim = 2 r 1 f(w) w(1 w 2 ) 2 r 2 (1 r 4 w 2 ) 2 dλ(w) f(w) w(1 w 2 ) 2 (1 w 2 ) 2 dλ(w). This implies that P g = sup P g, f = 2 sup f 1 w <1 w(1 w 2 ) 2 (1 w 2 ) 2 = 2. Since the Bloch norm of g is equal to 1, we obtain that P 2 as desired. Remark 8. It remains an open problem to find the exact value of P and, in view of the proof of Theorem 7, we believe that it is equivalent to the following extremal problem. Given the functional P(f) := sup z <1 (1 z 2 ) 2 (z 2 f(z)) find its supremum under the condition f B, f = 1. Acknowledgments. We are thankful to the referee for comments and suggestions that have improved this paper. REFERENCES [1] H. Hedenmalm, B. Korenblum and K. Zhu, Theory of Bergman spaces. Springer, New York, 2. [2]. Kalaj and M. Marković, Norm of the Bergman projection. arxiv: [3] H. Turgay Kaptanoglu, Bergman projections on Besov spaces on balls. Illinois J. Math. 49 (25). [4] M. Mateljević and M. Pavlović, An extension of the Forelli-Rudin projection theorem. Proc. Edinb. Math. Soc. (2) 36 (1993), [5] A. Perälä, On the optimal constant for the Bergman projection onto the Bloch space. Ann. Acad. Sci. Fenn. Math. 37 (212), [6] K. Zhu, Operator theory in function spaces. Marcel ekker, New York, 199. Received 15 ecember 212 University of Montenegro, Faculty of Natural Sciences and Mathematics, žorža Vašingtona b.b. 81 Podgorica, Montenegro davidkalaj@gmail.com University of Montenegro, Faculty of Natural Sciences and Mathematics, žorža Vašingtona b.b. 81 Podgorica, Montenegro djordjijevuj@t-com.me