RIEMANN-STIELTJES OPERATORS BETWEEN WEIGHTED BERGMAN SPACES

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1 RIEMANN-STIELTJES OPERATORS BETWEEN WEIGHTED BERGMAN SPACES JIE XIAO This aer is dedicated to the memory of Nikolaos Danikas ) Abstract. This note comletely describes the bounded or comact Riemann- Stieltes integral oerators T g acting between the weighted Bergman sace airs A, A β ) in terms of articular regularities of the holomorhic symbols g on the oen unit ball of C n. 1. Introduction Let = {z C n : z < 1} be the oen unit ball of C n. Set = {z C n : z = 1} be the comact unit shere of C n an n-dimensional Hilbert sace over the comlex field C under the inner roduct n z, w = z k w k, z = z 1, z 2,..., z n ), w = w 1, w 2,..., w n ) C n k=1 and the associated norm z = z, z 1/2, z C n. For two given holomorhic mas f, g : C, we, as in [5], define the Riemann- Stieltes integral of f with resect to g via 1.1) T g fz) = where Rgz) = n =1 1 0 ftz)rgtz)t 1 dt, z, z gz) z, z = z 1, z 2,..., z n ) stands for the radial derivative of g. In articular, if then 1.1) becomes 1 = 1, 1,..., 1) and gz) = log1 z, 1 ) T g fz) = 1 0 ftz) tz, 1 1 tz, 1 ) 1 t 1 dt, z, a higher dimensional version of the classical Cesáro oerator. We consider the roblem of determining the otimum on g such that 1.2) T g : A A β is bounded or comact Mathematics Subect Classification. Primary 30D55, 32H20, 47B38. Key words and hrases. Otimal estimates, boundedness, comactness, Riemann-Stieltes integrals, weighted Bergman saces, C n -ball. Suorted in art by NSERC Canada). 1

2 2 JIE XIAO Here and henceforth, for > 0 and > 1, A is the weighted Bergman sace of all holomorhic mas f : C satisfying ) 1/ f A = fz) 1 z 2 ) dvz) <, where dv denotes the Lebesgue volume measure on. It is known that if = 1 and = β > 1 then 1.2) holds if and only if g belongs to the Bloch sace or the little Bloch sace; see for examle [2] for n = 1) and [9] for n 1). But, in other cases even for n = 1), the otimal A, A β ) estimates have not yet been worked out; see also [1] and references therein) for the setting of H, H ) the 1-dimensional limit case of A, A β ) as = β 1. The goal of this note is to find out the otimal conditions of g such that 1.2) holds in all ossible cases on, ) 0, ) 0, ) and, β) 1, ) 1, ). Below is our result. Theorem 1.1. Let, β 1, ),, 0, ) and g : C be holomorhic. i) If > then T g : A A β is bounded or comact when and only when 1.3) Rgz) 1 z 2 ) 1 + ) β dvz) <. ii) If then T g : A A β is bounded when and only when 1.4) su Rgz) 1 z 2 ) 1 n+1+ + n+1+β <, z and T g : A A β is comact when and only when 1.5) lim Rgz) 1 z 2 ) 1 n+1+ + n+1+β = 0. z It is worth ointing out that the symbol g ensuring 1.3) or 1.4)/1.5) is a constant whenever β and > or n n β > 1 and. The rest of this note is organized as follows. In Section 2, we collect some reliminary but useful facts on the weighted Bergman saces and Khinchine s ineuality. In Section 3, we demonstrate Theorem 1.1 through these reliminary results and some of the ideas exosed in [6] and [8]. 2. Preliminaries When 1, the sace A is a Banach sace euied with the norm A, and when 0, 1), the sace A is a comlete metric sace with the distance df, g) = f g A. First of all, we need a growth roerty of holomorhic functions on. To do so, we denote by φ w the automorhism of sending 0 to w, i.e., φ w z) = { z, w = 0 w P wz 1 z 2 Q wz 1 z,w, w 0,

3 RIEMANN-STIELTJES OPERATORS BETWEEN WEIGHTED BERGMAN SPACES 3 where Q w = I P w, I is the identity ma and P w is the roection of C n onto the one-dimensional subsace sanned by w 0. For r > 0 and z ut { Dz, r) = w : 1 2 log 1 + φ } wz) 1 φ w z) < r which is called the oen Bergman metric ball with center z and radius r. Lemma 2.1. Let 0, ) and 1, ). If f : C is holomorhic then ) 1 2.1) fz) 1 z 2 ) n+1+ fw) 1 w 2 ) dvw), z. Dz,r) For a roof of 2.1), see also [10,. 59]. Next, we state two characterizations of the weighted Bergman saces. Lemma 2.2. Let 0, ), 1, ), and f : C be holomorhic. Then the following three statements are euivalent: i) f A. ii) f A = f0) + ) 1 <. Rfz) 1 z 2 ) + dvz) iii) For any η 0, 1], there exists a seuence {z } in such that a) = Dz, η); b) Dz, η 4 ) Dz k, η 4 ) = for k; c) Each oint z lies in at most N = Nη) of balls from {Dz, 2η)}; d) fz) = 1 z 2 ) b n 1 c 1 z, z ) b, z, where {c } is in the comlex seuence sace l and b is a constant greater than n max{1, 1 } Moreover, one has 2.2) f A f A {c } l. For a roof of Lemma 2.2 and its sources, see for examle [10, Chater 2] and [3] as well as [7]. Finally, we uote the following well-known form of Khinchine s ineuality. Lemma 2.3. Suose and r 0 t) = =1 { 1, 0 t [t] < 1/2 1, 1/2 t [t] < 1 r t) = r 0 2 t), = 1, 2,..., and let 0, ) and c 1,..., c m ) C m, m = 1, 2,... Then m 1 2.3) c 2 m c r t) dt. 0 =1

4 4 JIE XIAO Remark 2.4. In the above and below, the notation U V means that there are two constants κ 1, κ 2 > 0 such that κ 1 V U κ 2 V. Moreover, if U κ 2 V then we say U V or alternatively V U. 3. Proof To begin with, let us agree to two more conventions: dv β z) = 1 z 2 ) β dvz), z ; and T g A A = su{ T gf β A : f A β and f A = 1}. Meanwhile, for any two holomorhic mas f, g : C we always have 3.1) RT g fz) = fz)rgz), z, and conseuently 3.2) T g f fz)rgz) 1 z 2 ) dv A β z), β thanks to the lefthand comarability of 2.2). Proof of Theorem 1.1 i) Suose >. Let 1.3) be true. Alying Hölder s ineuality to the righthand integral of 3.2) we get that if f A then T g f f A β A Rgz) 1 z 2 ) 1 + β ) ) dvz), imlying the boundedness of T g : A A β. Conversely, suose T g : A A β is bounded. Then T g A A β each natural number let K z) = 1 z 2 ) b n 1 1 z, z ) b, is finite. For where {z } is the seuence in Lemma 2.2 iii). Let {c } l, and choose {r t)} as obeying Lemma 2.3. Then {c r t)} l with {c r t)} l = {c } l, and so c r t)k A with A c r t)k {c } l, due to the righthand comarability of 2.2). This fact derives T g c r t)k )z) dv β z) T g {c A A } l. β Furthermore, integrating this ineuality from 0 to 1 with resect to dt, as well as using 3.1), Fubini s theorem and 2.3) in Lemma 2.3, we get c K z) 2) 2 Rgz) 1 z 2 ) ) dvβ z) T g {c A } l. Noticing the estimate 1 z, z 1 z 2 when z Dz, 2η), A β

5 RIEMANN-STIELTJES OPERATORS BETWEEN WEIGHTED BERGMAN SPACES 5 emloying the condition c) in Lemma 2.2 iii), and letting 1 E be the characteristic function of a set E, we obtain = c Dz,2η) Rgz) 1 z 2 ) ) ) dvβ z) 1 z 2 n+1+) ) c 1 Dz,2η)z) Rgz) 1 z 2 ) ) dvβ z) 1 z 2 ) n+1+) c 2 1 Dz,2η)z) 1 z 2 ) 2n+1+) 2 Rgz) 1 z 2 ) ) dvβ z) c K z) 2) 2 Rgz) 1 z 2 ) ) dvβ z) Note that + { T g A A β {c } l T g A A β { c } l /. = 1. So the foregoing estimates actually indicate Dz,2η) Rgz) 1 z 2 ) ) } dvβ z) T 1 z 2 ) n+1+) g. A A β l Because Rg is holomorhic on, by the condition a) in Lemma 2.2 iii), the ineuality 2.1) and the last norm estimate, we achieve Rgz) 1 z 2 ) ) 1 z 2 ) β dvz) Rgz) 1 z 2 ) ) 1 z 2 ) β dvz) T g Dz,η) Dz,η) Dz,η) Rgw) 1 w 2 ) ) dvβ w) 1 z 2 ) +n+1) Dz,2η) Rgz) 1 z 2 ) ) dvβ z) 1 z 2 ) +n+1) Dz,2η) Rgz) 1 z 2 ) ) dvβ z), A A β 1 z 2 ) n+1+) ) ) ) dvz) 1 z 2 ) n+1 deriving 1.3). Regarding the comactness, it suffices to show that if 1.3) holds then T g : A A β is comact. Assuming 1.3), we obtain that T g is a bounded oerator from A A β and so g = g0) + T g1 A β. Additionally, we see that for any ɛ > 0

6 6 JIE XIAO there is a δ 0, 1) such that ) Rgz) 1 z 2 ) 1 z 2 ) β dvz) < ɛ. z >δ Since the weak convergence in A means the uniform convergence on comacta of, let {f } be a seuence in the closed unit ball of A that converges to 0 uniformly on comacta of. For the above ɛ > 0 there exists an integer 0 > 0 such that su z δ f z) < ɛ as 0. With the hel of 3.2) and Hölder s ineuality, we further obtain T g f A β + z δ z >δ ɛ g + f A β A ɛ g A β + ɛ. ) ) dvβ f z)rgz) 1 z 2 ) z) z >δ ) ) Rgz) 1 z 2 ) 1 z 2 ) β dvz) In other words, lim T g f A = 0 and so T g : A β A β is comact. Proof of Theorem 1.1 ii) Suose now. If 1.4) holds then g B γ = su Rgz) 1 z 2 ) γ < where γ = 1 n z + n β. From 3.2) and 2.1) it turns out that for f A, T g f A β fz) fz) Rgz) 1 z 2 ) dv β z) f A g B fz) 1 z 2 ) γ )n+1+) γ dv β z) f A g B. γ That is to say, T g : A A β is bounded. Conversely, if T g : A A β is bounded then T g A A β in mind, we handle two cases: > 1 and 1. Case 1: > 1. Define Kw, z) = 1 z, w ) n++1), z, w. A routine calculation see for examle [4]) yields 3.3) Kw, ) A 1 w 2 ) n+1+) 1). is finite. Keeing this

7 RIEMANN-STIELTJES OPERATORS BETWEEN WEIGHTED BERGMAN SPACES 7 Using 3.2), certain transformation roerties of φ w see also [10,. 5-8]) and 2.1), we get 1 w 2 ) + 1+)n+1+) β+ T g Kw, ) A β 1 w 2 ) + 1+)n+1+) β+ Kw, z) Rgz) 1 z 2 ) dv β z) u 1/2 Rgφ w 0)) Rgw). Rgφ w u)) 1 u 2 ) +β dvu) 1 u, w 2n+2 n+1+)+2β ) In brief, we have 3.4) Tg A A Kw, ) β A) Tg Kw, ) Rgw) A β 1 w 2 ). + 1+)n+1+) β+ This estimate, together with 3.3), roduces 1.4) right away. Case 2: 1. Select a ositive integer m > n and set K w, z) = 1 z, w ) m, z, w. Just like the case of > 1, it follows that 3.5) K w, ) A 1 w 2 ) n+1+ m and 3.6) T g K w, ) A β 1 w 2 m +n+1+β ) Rgw). A combination of 3.6), 3.5) and the boundedness of T g yields 1.4) too. To establish the corresonding comactness art, assume that g satisfies 1.5). Then g A β, and for any ɛ > 0 there is an δ 0, 1) such that as z δ, 1), Rgz) 1 z 2 ) γ < ɛ where γ = 1 n n β. In order to rove that T g : A A β is comact, let {f } be a seuence in the closed unit ball of A which converges to 0 uniformly on comacta of. Then, there exists an integer 0 > 0 such that su z δ f z) < ɛ when 0. Hence by 3.2) and 2.1), T g f A β f z)rgz) 1 z 2 ) dv β z) B n ɛ Rgz) 1 z 2 ) dv β z) + ɛ f A z δ ɛ ) g + f A β A ɛ ) g + 1. A β z >δ f z) 1 z 2 ) dvz) Namely, T g f A β 0 as. Thus, T g is a comact oerator from A to A β.

8 8 JIE XIAO On the other hand, if T g : A A β is comact, then, for z, w let Kw,z) Kw, ), > 1 k w, z) = A K w,z) K w, ), 1. A Obviously, k w, ) tends to 0 uniformly on comacta of as w. By the comactness of T g, we find lim w T g k w, ) A β = 0. This limit, along with 3.3), 3.4), 3.5) and 3.6), yields 1.5). Acknowledgment. The author is grateful to K. Zhu for his helful s, but also would like to thank the referee for several suggestions imroving the readability of the aer. References 1. A. Aleman and J. A. Cima, An integral oerator on H and Hardy s ineuality, J. Anal. Math ), A. Aleman and A. G. Siskakis, Integral oerators on Bergman saces, Indiana Univ. Math. J ), R. Coifman and R. Rochberg, Reresentation theorems for holomorhic and harmonic functions in L, Asterisue ), J. Faraut and A. Koranyi, Function saces and reroducing kernels on bounded symmetric domains, J. Funct. Anal ), Z. Hu, Extended Cesáro oerators on the Bloch sace in the unit ball of C n, Acta Math. Sci. Ser. B Engl. Ed ), D. H. Luecking, Embedding theorems for saces of analytic functions via Khinchine s ineuality, Michigan Math. J ), J. Shairo, Macey toologies, reroducing kernels, and diagonal mas on the Hardy and Bergman saces, Duke Math. J ), W. Smith and L. Yang, Comosition oerators that imrove integrability on weighted Bergman saces, Proc. Amer. Math. Soc ), J. Xiao, Riemann-Stieltes oerators on weighted Bloch and Bergman saces of the unit ball, J. London. Math. Soc. 2) ), K. Zhu, Saces of Holomorhic Functions in the Unit Ball, Graduate Texts in Mathematics 226. Sringer-Verlag, New York, Deartment of Mathematics and Statistics, Memorial University of Newfoundland, St. John s, NL A1C 5S7, Canada address: xiao@math.mun.ca