Weighted Composition Followed by Differentiation between Bergman Spaces

Transcription

1 International Mathematical Forum, 2, 2007, no. 33, Weighted Comosition Followed by ifferentiation between Bergman Saces Ajay K. Sharma 1 School of Alied Physics and Mathematics Shri Mata Vaishno evi University P/O Kakryal, Udhamur , India aksju 76@yahoo.com Som att Sharma and Sanjay Kumar eartment of Mathematics University of Jammu , India somdatt jammu@yahoo.co.in Abstract In this aer we consider linear oerators M ψ C ϕ and M ψ C ϕ acting between weighted Bergman saces, where M ψ,c ϕ and are multilication, comosition and differentation oerators resectively. Our goal is to characterize those holomorhic self mas ϕ of for which M ψ C ϕ and M ψ C ϕ acts boundedly and comactly between weighted Bergman saces. Primary 47B33, 46E10; Sec- Mathematics Subject Classification: ondary 3055 Keywords: comosition oerator, multilication oerator, weighted comosition oerator, differentation, generalized Nevanlinna counting function, vanishing Carleson measure, weighted Bergman saces 1.Introduction Thoughout this aer we denote by H(), the sace of holomorhic functions on, where is the oen unit disk in the comlex lane C. Let ψ and ϕ be 1 Suorted by CSIR grant ( F.No. 9/100(100)2002 EMR-1).

2 1648 Ajay K. Sharma, Som att Sharma and Sanjay Kumar holomorhic mas on such that ϕ(), we can define a linear oerator ψc ϕ on H(), called weighted comosition by ψc ϕ f = ψ(f ϕ). The oerator ψc ϕ can be regarded as a generalization of a multilication oerator and a comosition oerator, In case ψ 1orϕ(z) =z, ψc ϕ reduces to the comosition oerator C ϕ or the multilication oerator M ψ, resectively. For general back ground on comosition oerators, we refer [CoM 95] and [Sh 93] and references therein Weighted comosition oerators aear naturally in different contexts. For examle, Singh and Sharma [SiS 79] related the boundedness of comosition oerators on Hardy sace of the uer half-lane with the boundedness of weighted comosition oerators on the Hardy sace of the oen unit disk. Isometries in many Banach saces of analytic functions are just weighted comosition oerators, for examle see [Fo 64]. Recently, several authors have studied weighted comosition oerators on different saces of analytic functions. For examle, one can refer to [CoH 04] for study of these oerators on Hardy saces, [Ka 79] and [OhT 01] for disk algebra, [OhZ 01] and [OSZ 03] for Bloch-tye saces and [MiS 97] for study of these oerators on Bergman saces. In this aer we consider linear oerators M ψ C ϕ and M ψ C ϕ acting between weighted Bergman saces, where M ψ,c ϕ and are multilication, comosition and differentation oerators resectively. Our goal is to characterize those holomorhic self mas ϕ of for which M ψ C ϕ and M ψ C ϕ acts boundedly and comactly between weighted Bergman saces. 2. Preliminaries For > 1, the weighted Bergman sace A, is the set of analytic functions on the disk with f = f(z) dλ A (z) <, where dλ (z) =( + 1)(1 z 2 ) da(z) and da(z) =dxdy/π = rdrdθ/π, z = x+iy. The following shar estimate tells us how fast an arbitrary function from A grow near the boundary. Let f A. Then for every z in, we have f(z) f A (1 z 2 ) (2+)/ (2.1) with equality if and only if f is a constant multile of the function ( 1 z 2 ) 2+/ k a (z) =. (1 az) 2

3 Weighted comosition followed by differentiation 1649 It can be easily shown that k a A 1 with constant deending only on and [Sm 96, age 400]. For general background of weighted Bergman saces A and weighted Bloch saces, one may consult [Zh 90] and [HKZ 00] and the references therein. In what follows we make extensive use of Carleson measure techniques, so we give a short introduction to Carleson sets and Carleson measures. For a oint ζ on the boundary of we define the Carleson set S(ζ,δ)={z : ζ z <δ}. We use Carleson sets along with a more convinient choice of seudohyerbolic disks. For 0 <r<1and a, denote by (a, r), the disk whose seudohyerbolic center is a and whose seudohyerbolic radius is r : (a, r) = { z : a z } <r. 1 az The notation (a, r) A will denote the area of (a, r). For fixed 0 <r<1 the area of (a, r) has the estimation: (a, r) A (1 a 2 ) 2 (1 z 2 ) 2 (z, r) A, (2.2) for z (a, r), where means that the two quantities are bounded above and below by the constants indeendent of a. For fixed 0 <r<1, it is also known that for z (a, r), 1 az (1 a 2 ). (2.3) Also for each (a, r), there is a ζ so that (a, r) S(δ, ζ) for δ 1 a. A ositive Borel measure μ on is called -Carleson measure if su δ>0 su ζ μ(s(δ, ζ)) δ +2 <, and it will be called a vanishing Carleson measure if in addition lim su μ(s(ζ,δ)) =0. δ 0 ζ, δ δ Boundedness and Comactness of M ψ C ϕ In this section, we characterize those holomorhic self-mas of for which M ψ C ϕ mas A boundedly and comactly into A q β. To do so we need a generalized Nevanlinna counting function, which will be required for the change of variable.

4 1650 Ajay K. Sharma, Som att Sharma and Sanjay Kumar efinition 3.1. Let ϕ and ψ be holomorhic self-mas of such that ϕ(). Let q 1 and β> 1. For w, w 0, we define ( N q,β ψ,ϕ (w) = ψ(z) q ϕ (z) q 2 log 1 ) β, z where sum extends over all solutions of ϕ(z) =w and we name it generalized Nevanlinna counting function. We need a generalized change of variable formula. In the following formula, {z j (w)} denote the sequence of zeros of ϕ(z) w reeated according to multilicity. Theorem 3.2. [CoM 95] If g and W are non-negative measurable functions on, then g(ϕ(z)) ϕ (z) 2 W (z)da(z) = ϕ() ( ) g(ϕ(z)) W (z j (w)) da(w). We assume from now on that r (0, 1) is fixed. Theorem 3.3. Let 1 q, and, β > 1. Let ϕ and ψ be a holomorhic mas and such that ϕ() and ψϕ A q β. Let dμ(w) = N q,β ψ,ϕ (w)da(w). Then the following are equivalent: (1) M ψ C ϕ mas A boundedly into A q β. (2) μ((a, r)) = O((1 a 2 ) q(+2+)/ ) as a 1. In order to rove the Theorem 3.3 we need following result of Luecking [Lu 85 ] in which he characterized ositive measures μ with the roerty: f (n) L q (μ) C f A. The following result is a secial case of the Luecking s result [Lu 85, Theorem 2.2] for n = 1 in case 1 q. Theorem 3.4. Let 1 q, and, β > 1. Let μ be a finite ositive measure on. Then the following are equivalent: (1) f L q (μ) C f A for all f A. (2) μ((a, r)) = O((1 a 2 ) q(+2+)/ ) as a 1. For the case 1 q<,luecking used Khinchine s inequality and other estimates to obtain a version of Theorem 3.4 for f (n), where f A. We are interested in the case n = 1 and f A. The following result is a slight modification of Luecking s result (see [HiP 05 ] also). Theorem 3.5. Let 1 q<,and > 1. Let μ be a finite ositive measure on. Let Ω(z) =(1 z 2 ) (+2+q) μ((z, r)). Then the following are equivalent: (1) f L q (μ) C f A for all f A j 1

5 Weighted comosition followed by differentiation 1651 (2) Ω L / q (ν ). We are now ready to rove Theorem 3.3. Proof of Theorem 3.3. Since ψϕ A q β, change of variable formula 3.2 imlies that μ is a finite measure. Thus Theorem 3.4 alies. Note that M ψ C ϕ f q A q β = f (ϕ(z)) q ψ(z) q ϕ (z) q (log 1 z )β da(z) f (w) q N q,β ψ,ϕ (w)da(w) = f q L q (μ). Since M ψ C ϕ mas A boundedly into A q β, so we have f q L q (μ) = M ψc ϕ f q A q β C f A for all f A. Hence Theorem 3.4 imlies that μ((a, r)) = O((1 a 2 ) q(+2+) as a 1. Conversely, suose that (ii) holds. Then by Theorem 3.4, we have M ψ C ϕ f A q β = f L q (μ) C f A and hence M ψ C ϕ mas A boundedly into A q β. Theorem 3.6: Let 1 q, and, β > 1. Let ϕ and ψ be holomorhic mas on such that ϕ() and ψϕ A q q,β β. Let dμ(w) =Nψ,ϕ (w)da(w). Then the following are equivalent: (1) M ψ C ϕ mas A comactly into A q β. (2) μ((a, r)) = o((1 a 2 ) q(+2+)/ ) as a 1. Proof: First suose that M ψ C ϕ mas A comactly into A q β. Let a and consider the function f a (z) = (1 a 2 ) (+2)/ (1 az). 2(+2)/ Clearly f a A 1 and f a converges to zero uniformly on comact subsets of as a 1. Since M ψ C ϕ mas A comactly into Aq β, so for gives ɛ>0, we can find r 0, 0 <r 0 < 1 such that M ψ C ϕ f a A <ɛfor a >r 0. Thus ɛ> f a(z) q dμ(z) f a(z) q dμ(z) for a >r 0. Since for z (a, r), f a(z) (a,r) 1 (1 a 2 ) (+2+)/

6 1652 Ajay K. Sharma, Som att Sharma and Sanjay Kumar and so above estimate yields for all a with a >r 0. Hence μ((a, r)) <ɛ((1 a 2 ) q(+2+)/ μ((a, r)) = o((1 a 2 ) q(+2+)/) as a 1. Conversely, assume that (ii) holds and let {f n } be a sequence in A such that f n A q M and f n 0 uniformly on comact subsets of. To show that M ψ C ϕ mas A comactly into A q β, it is sufficient to show that M ψ C ϕ f n q f Aβ= q n q L q (μ) 0asn By a standard estimate of Luecking [Lu 93], age 338, we have M ψ C ϕ f n q 1 C f Aβ q (1 a 2 ) 2+q n (z) q da(z)dμ(a). (a,r) Note that χ (a,r) (z) =χ (z,r) (a) and 1 a 2 1 z 2 for a (z, r). Also by 2.1, we have f n f n (z) C A (1 z 2 ). (2+)/ By an alication of Fubini s theorem, we get M ψ C ϕ f n q C f A q n (z) q μ((z, r)) da(z) β (1 z 2 ) 2+q C f n q A q fn (z) μ((z, r)) (1 z 2 ) ( C M q f n (z) μ((z, r)) z 6r 0 (1 z 2 ) + f n (z) μ((z, r)) ) (1 z 2 ) da(z) (q+2q+q )/ = I + II. (q+2q q )/ da(z) (q+2q+q )/ da(z) Now (ii) imlies that for a given ɛ>0, we can find r 0, 0 <r 0 < 1 such that II = C M q f n (z) μ((z, r)) (1 z 2 ) (q+2q+q )/da(z) ɛc M q f n (z) (1 z 2 ) da(z) ɛc M q f n A ɛc M q.

7 Weighted comosition followed by differentiation 1653 Since f n 0 uniformly on comact subsets of I = C M q f n (z) μ((z, r)) z 6r 0 (1 z 2 ) C 1 C M q ɛ μ((z, r))da(z) C 1 C 2 C M q ɛ μ()da(z) = C 1 C 2 C 3 C M q ɛ for n large enough. Thus lim M ψc ϕ f n q =0, n A q β and hence M ψ C ϕ mas A comactly into A q β. (q+2q+q )/ da(z) Theorem 3.7 Let 1 <q,and, β > 1. Let dμ(z) =N q,β ψ,ϕda(z) and let ψϕ A q β. Let G(z) =(1 z 2 ) (+q+2) μ((a, r)).then the following are equivalent: (1) M ψ C ϕ mas A boundedly into A q β. (2) M ψ C ϕ mas A comactly into Aq β. (3) G L /( q) (ν ). Proof. (1) (3) Suose (1) holds. By change of variable formula as in Theorem 3.2, we have M ψ C ϕ f q A q β= f q L q (μ) Since M ψ C ϕ mas A boundedly into A q β, we can find a ositive constant C such that f L q (μ)= M ψ C ϕ f A q C f β A and so by Theorem 3.4, M ψ C ϕ mas A boundedly into A q β if and only if G L /( q) (ν ). It is clear that (2) imlies (1). It remains to verify that (3) imlies (2). Assume that f n A C and f n 0 uniformly on comact subsets of. It is sufficient to show that lim M ψc ϕ f n A q =0. n β As in the roof of the Theorem 3.5, we have M ψ C ϕ f n A q C f β n (z) q G(z)dν (z).

8 1654 Ajay K. Sharma, Som att Sharma and Sanjay Kumar Let ɛ>0. Then the hyothesis on G imlies that there exists r 0, 0 <r 0 < 1, with the roerty G / q (z)dν (z) <ɛ / q. It follows by Holder s inequality that f n (z) q G(z)dν (z) Thus we have ( ) q/ ( ) ( q)/ f n (z) dν G / q dν (z) ɛ f n q A Cɛ. f n (z) q G(z)dν (z) Cɛ. Since f n 0 uniformly on comact subsets of, so f n (z) <ɛfor all z such that z <r 0 and for all n n 0. Thus f n (z) q G(z)dν (z) ɛ q G(z)dν (z) z 6r 0 z 6r 0 for n n 0. Since ψϕ A q β, and thus Thus G(z) Cμ((z, r)) Cμ() < G(z)dν (z) C z 6r 0 z 6r μ((z, r))dν (z) C. f n (z) q G(z)dν (z) Cɛ for n n 0. Hence M ψ C ϕ mas A comactly into A q β. 4. Boundedness and Comactness of M ψ C ϕ Now we discuss boundedness and comactness of the oerator M ψ C ϕ acting between weighted Bergman saces. Proofs follow exactly on same lines, so we omit the details. Theorem 4.1. Let 1 q<, and, β > 1. Let ϕ and ψ be holomorhic mas on such that ϕ() and ψϕ A β. Then the following are equivalent: (1) M ψ C ϕ mas A boundedly into Aq β

9 Weighted comosition followed by differentiation 1655 (2) (μ β ϕ 1 )((a, r)) = O((1 a 2 ) q(a+2+)/ ) as a 1. where μ β ϕ 1 is the ull-back measure induced by ν β. Here dμ β (z) = ψ(z) q dν β (z). Proof: First suose that M ψ C ϕ mas A boundedly into Aq β. Since ψϕ A q β, change of variable formula of measure theory ( see [Ha 74], Theorem C, age 173) imlies that μ β ϕ 1 is a finite measure M ψ C ϕ f q = f (ϕ(z)) q ψ(z) q dν A q β (z) β = f (z) q d(μ β ϕ 1 )(z) = f q L q (μ β ϕ 1 ), Since M ψ C ϕ mas A boundedly into Aq β, so we have f q L q (μ) = M ψ C ϕ f q A q β C f q A q β for all f A q β. Hence Theorem imlies that (μ β ϕ 1 )((a, r)) = O((1 a 2 ) q(+2+) as a 1. Conversely suose that (ii) holds. Then by Theorem 3.4, we have M ψ C ϕ f A q β = f L q (μ β ϕ 1 ) C f A and hence M ψ C ϕ mas A boundedly into Aq β. Theorem 4.2. Let 1 q, and, β > 1. Let ϕ and ψ be holomorhic mas on such that ϕ() and ψϕ A q β. Then the following are equivalent. (1) M ψ C ϕ mas A comactly into A q β (2) (μ β ϕ 1 )((a, r)) = O((1 a 2 ) q(a+2+)/ ) as a 1. Theorem 4.3. Let 1 <,and, β > 1. Let ϕ and ψ be holomorhic mas on such that ϕ() and ψϕ A q β and let Φ(z) = (1 z 2 ) (+2+q) (μ β ϕ 1 )((z, r)). Then the following are equivalent. (1) M ψ C ϕ mas A boundedly into Aq β (2) M ψ C ϕ mas A comactly into A q β (3) Φ L / q (ν ) References [CoH 04] M.. Contreras and A. G. Hernandez-iaz, Weighted comosition oerators on saces of functions with derivative in a Hardy sace, J. Oerator Theory, 52 (2004),

10 1656 Ajay K. Sharma, Som att Sharma and Sanjay Kumar [CoM 95] C. C. Cowen and B.. MacCluer, Comosition oerators on saces of analytic functions, CRC Press Boca Raton, New York, [CuZ 04] Z. Cuckovic and R. Zhao, Weighted comosition oerators on the Bergman sace, J. London Math. Soc., 70 (2004), [Fo 64] F. Forelli, The isometries of H saces, Canad. J Math., 16 (1964) [Ha 74] P. R. Halmos, Measure theory, Sringer-Verlag, New York, [HiP 05] R.A. Hibschweiler and N. Portnoy, Comosition followed by differentiation between Bergman and Hardy saces Rocky Mountain Journal of Mathematics , [HKZ 00] H. Hedenmalm, B. Korenblum and K. Zhu, Theory of Bergman saces, Sringer, New York, Berlin, etc [Ka 79] H. Kamowitz, Comact oerators of the form uc ϕ, Pacific J. Math., 80 (1979), [Lu 85]. H. Luecking, Forward and reverse Carleson inequalities for functions in Bergman saces and their derivatives, Amer.. Math. J , [Lu 85]. H. Luecking, Embedding theorems for saces of analytic functions via Khinchine s inequality, Mich. Math. J , [MiS 97] G. Mirzakarimi and K. Seddighi, Weighted comosition oerators on Bergman and irichlet saces, Georgian. Math. J., 4 (1997), [OhT 01] S. Ohno and H. Takagi, Some roerties of weighted comosition oerators on algebras of analytic functions, J. Nonlinear Convex Anal., 2 (2001), [OhZ 01] S. Ohno and R. Zhao, Weighted comosition oerators on the Bloch sace, Bull. Austral. Math. Soc., 63 (2001), [OSZ 03] S. Ohno, K. Stroethoff and R. Zhao, Weighted comosition oerators between Bloch-tye saces, Rocky Mountain J. Math., 33 (2003), [Sh 93] J. H. Shairo, Comosition oerators and classical function theory, Sringer- Verlag, New York [SiS 79] R. K. Singh and S.. Sharma, Comosition oerators on a functional Hilbert sace,bull. Austral. Math. Soc. 20(1979), [Sm 96] W. Smith, Comosition oerators between Bergman and Hardy saces, Trans. Amer. Math. Soc, 348 (1996), [Zh 90] K. Zhu, Oerator theory in function saces, Marcel ekker, New York, Received: October 11, 2006