CESÁRO TYPE OPERATORS ON SPACES OF ANALYTIC FUNCTIONS. S. Naik

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1 Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: Filomat 25:4 2, DOI:.2298/FIL485N CESÁRO TYPE OPERATORS ON SPACES OF ANALYTIC FUNCTIONS S. Naik Abstract In this article, we consider a two parameter family of generalized Cesáro operators P b,c, Re b + > Re c >, on classical spaces of analytic functions such as Hardy H p, BMOA and a- Bloch space B a. We Prove that P b,c, Re b + > Re c > is bounded on H p if and only if p, and on B a if and only if a, and unbounded on H, BMOA and B a, a, ]. Also we prove that α- Cesáro operators C α is a bounded operator from the Hardy space H p to the Bergmann space A p for p,. Thus, we improve some well known results of the literature. Introduction Let D denote the unit disc in the complex plane C. A function f analytic in the unit disc D is belong to the Hardy space H p, < p <, if f H p where its integral mean M p r, f is given by M p r, f = = sup M p r, f <, <r< { fre iθ p dθ} /p. With this norm H p is a complete metric linear space or Fréchet space if < p < and a Banach space if p. It is a Hilbert space for p = 2. We let M r, f denote the maximum of fz on the circle z = r <. Thus for p = the Banach space H is the class of bounded analytic functions in D with the norm f H = sup fz. z D 2 Mathematics Subject Classifications. 33C5, 3D55, 47B38. Key words and Phrases. Gaussian hypergeometric functions, Cesáro operators. Received: November 3, 2 Communicated by Miodrag Mateljević

2 86 S. Naik The Bergman space A p < p consists of all analytic functions f in D for which /p f A p = Mp p r, fr dr <. and A = H. Thus A p is a Banach space if p, and Fréchet spaces if < p <. The space BMOA containing H is the space of functions in H, of which the radial limit functions have bounded mean oscillation on the unit circle D, more precisely a functions f H is said to be in the space f = sup I I fe iθ f I dθ <, where the supremum is taken over all subarcs I D, I = I f I = fe iθ dθ I. The proper inclusions H BMOA I p< H p dθ and and BMOA B are well known. Other information on BMOA can be found in the books [5] or [3]. For a >, we let a-bloch space denotes B a is the space of analytic functions f on D such that f B a = f + sup z a f z <. z D It is known that B a is a Banach space for each a > with the above norm. see for example [4]. When a >, the space can be identified with space of analytic functions f with f + sup z a fz <, z D see [4], Proposition 7. When < a < and a =, the space B a can be identified with analytic Lipschitz space see [4], Theorem B and Bloch space respectively. For any complex number a, b, c n, n =,, 2,..., the Gaussian hypergeometric function 2 F a, b; c; z is defined by power series expansion 2F a, b; c; z = F a, b; c; z = n= a, nb, n z n c, n n! z <, where a, n is the shifted factorial defined by Appel s symbol a, n = aa +... a + n = Γa + n, n N = {, 2,...} Γa and a, = for a, see [, ]. Obviously, F a, b; c; z is an analytic function in the unit disc D. We refer the reader to [, ] for a background on Gaussian

3 Cesáro type operators on spaces of analytic functions 87 hypergeometric functions. In [2], authors have used one of the contiguous relation differ by in the parameter values involving hypergeometric functions to introduce and study the generalized Cesáro operator. For b, c C, the generalized Cesáro operator P b,c, associates any analytic function fz = n= a nz n on the unit disc D with power series: P b,c fz := n= A b+;c n n b n k a k z n, k= where A b;c k = b, k c, k, b =, and for k b k = + b c A b;c k b, Notice that the α Cesáro operator introduce by see [] and Classical Cesáro operator are the special cases of generalized Cesáro operator P b,c. The boundedness of P b,c on H p for < p is proved in [2] and the case < p < is proved in [7, 9] for Re b + > Re c. Other cases, that is for Re c, remain open till date. In [7] also the boundedness of P b,c is proved on mixed norm space which include Bergman space. In this article, we study once again the action of the operator P b,c on the space H p, further on the spaces BMOA and B a. We show that P b,c is a bounded operator on H p if and only if p, for Re b + > Re c > by giving an alternate proof for p >. Thus our result improve the condition on the parameters b, c compare to the result given in [7]. In [2] it is proved that P b,c is bounded on B a for all a, whenever Re b + > Re c. In this article we study an equivalent condition for the boundedness of P b,c on B a, for a > with Re b + > Re c >. Also we give an alternate proof for unboundedness of P b,c on BMOA using a different technique. In Section 4, we consider the α- Cesáro operator from H p to A p for p, and prove their boundedness. Throughout the text the letter C will denote the constant depending only on related parameters such as p, b, c and so on, C may differ at different occurrences. The symbol a b means that a b is bounded from above and below by two positive constants. Also throughout the paper, an operator T is unbounded from X into Y means that, if for some f X we have T f / Y. 2 Preliminaries We start by stating a known result which associates an integral representation to the P b,c operator.

4 88 S. Naik Lemma. [2] For b, c C with Re b + > Re c >, we have P b,c fz = B = z b B z t c t b c ftz F c, c b ; c; tz dt tz b+ c ζ c z ζ b c fζ F c, c b ; c; ζ dζ, ζ b+ c where B = Bc, b + c is the usual beta function. The following lemmas will be used in the sequel. Lemma 2. [3] If < p < and f H p, then there exists a constant C = Cp independent of f such that sup r< fre iθ p dθ C f p H p. Lemma 3. [8] Let α >, < p and < q <. If r α+q M p p r, D f dr < then r α Mp q r, f dr <. We shall use an integral representation of P b,c which is obtained by choosing the path of integration between and z as tz γt = φ t z =, t [, ]. tz Using this path into the second equality of 2 and denoting the term F c, c b ; c; z by F z, we find P b,c fz = z b B = B φ t z c z φ t z b c fφ t z φ t z b+ c F φ tz φ tzdt t c t b c tz c fφ tzf φ t z dt. We define T t fz = ωt c zfφ t zf φ t z for t, ], where ω t z =, z D is the family of weight functions. It follows that tz where λt = t c t b c. P b,c fz = B 2 T t fzλt dt, 3

5 Cesáro type operators on spaces of analytic functions 89 3 Generalized Cesáro operators on H P, BMOA, B a spaces In this section we describe the boundedness of P b,c on the spaces H P, BMOA and B a. Though we provide the proofs only for reals b and c with b + > c >, these can easily be modified to see that all the results remain valid for b, c C and Re b + > Re c >. Time and again we use techniques developed in [6]. Theorem 4. Let < p and b, c C be numbers satisfying Re b + > Re c >. Then P b,c is a bounded operator from H p to H p if and only if < p <. Moreover, for < p < we have Bb +, p C P b,c H p Bc, b c + + p max cp, /p 2 c /p Γc C Bc, b + c Proof. Case a, Let < p, for this case see [2], Theorem 3.2. Case b, Let < p <. In this case our aim is to show that P b,c H p C f H p if p c, if p = c. for some constant C > depending on b, c and p. Suppose f H p, as H p norm is same as L p norm of the boundary function for < p <, 3 gives P b,c f H p = lim r M p r, P b,c f { = P b,c fe iθ p dθ B { } p p } ω te iθ c fφ t e iθ F φ t e iθ p λt dt dθ. Using Minkowski s inequality and the boundedness of hypergeometric function F c, c b ; c; z on z for b + > c in above calculations, we find P b,c f H p B C B Fix t, ] and define { { } ω t e iθ cp fφ t e iθ F φ t e iθ p p dθ λt dt } ω t e iθ cp fφ t e iθ p p dθ At = π π te iθ cp f te iθ p te iθ dθ. λt dt. 4

6 9 S. Naik te iθ Let e iφ te iθ = te iθ. A simple calculation shows that t dφ dθ C, < t. 5 t Suppose Mfe iφ = sup r< fre iφ is the radial maximal function. For 2 t, inequality 5 gives, At C C π π π π te iθ cp Mfe iφ p dθ dφ dφ Mfe iφ p dφ C f p Hp Lemma 2. 6 For < t < 2, let At = A t + A 2 t, where and A t = < θ t te iθ cp f te iθ p te iθ dθ A 2 t = t< θ π te iθ cp f te iθ p te iθ dθ. Using 5 in A t and since t < te iθ, we have A t C t cp Mfe iφ p dθ dφ dφ < θ t C t cp C t cp f p H te f iθ te C f te iθ H p as iθ te iθ. Hence for A 2 t, we have Therefore, for < t < 2, A 2 t C At < θ t Mfe iφ p dφ p Lemma 2. te iθ < and when t < θ < π, imply θ t< θ π θ cp f p H p dθ { C f p cp t if p cp c, H p log /t if p = c. { C f p H p cp t if p cp c, C + log /t f p H if p = p c. 7

7 Cesáro type operators on spaces of analytic functions 9 Finally, combining inequalities 6, 7, together with 4, we obtain for p = c, P b,c H p f H p /2 + log /t c λt dt + λt dt, /2 since b + > c >, convergence of the following integral imply /2 + log /t c t c t b c dt P b,c Γc H p C Bc, b + c f H p, and for p c /2 P b,c H p C f H p cp /p t/p c λt dt + λt dt /2 C max cp, f /p 2 c /p H p t /p t b c dt. If p =, it is rather simple to find P b,c is unbounded on H i.e. there exists f H such that P b,c f / H. Taking f H, from we have P b,c f z = n= + b c b + n zn = + b c F b, ; b + ; z, b which gives P b,c f / H, completes the proof. For our next result we consider the general version of P b,c which is defined as below. If µ be a finite positive Borel measure on, ], then P b,c µ fz = T t fz dµt, for each z D. Theorem 5. Let µ be a finite positive Borel measure on, ] and b, c C with Re b + > Re c >. Then Pµ b,c is a bounded operator on BMOA if and only if dµt t <. c Proof. Assume that dµt t <. It is known [5] that f BMOA if and only c if there exist f and f 2 analytic in the D with Re f, Re f 2 in L, f = f +if 2 + f, f = f 2 = and f BMOA Re f L + Re f 2 L + f. Now, let f BMOA posses such a decomposition. Define gz = fzf z. Since F z is bounded on z for Re b+ > Re c, gives gz BMOA. Let g = g +ig 2 with

8 92 S. Naik g j analytic in D such that g j = and g BMOA Re g L + Re g 2 L. Since tz t, We have P b,c µ fz C g tz dµt tz t c is well defined for each z D. Obviously, Re P b,c µ f = P b,c µ Re f. Hence for each j, Pµ b,c f j BMOA C{ Pµ b,c f + P b,c Re f j L } The above inequality gives C{ f + Pµ b,c f BMOA C f + C dµt t c g L }. dµt t c g BMOA dµt t c f BMOA. Conversely, Assume that Pµ b,c is bounded on BMOA. Let fz = log z and x [,. Then, Pµ b,c fx C log x Pb,c µ BMOA. 8 Since b + > c, F c, c b ; c; x is bounded, continuous and positive on [, ], we can quickly obtain P b,c µ fx = tx tx log F dµt tx c x tx x tx c F tx dµt tx 2 log x C log x Combining 8 and 9, we find dµt. 9 x tc dµt C Pb,c x tc µ BMOA. The desired conclusion follows if we take x in the above estimate. In particular, if we take µt = λtdt, then we have following result for P b,c.

9 Cesáro type operators on spaces of analytic functions 93 Corollary 6. Let b, c C be such that Re b + > Re c >. Then P b,c is unbounded operator from BM OA to BM OA i.e. there exists f BM OA but P b,c f / BMOA. Proof. Let f 2 z = log z. We see that f 2 BMOA see [2]. Our aim is to show that P b,c f 2 / BMOA. Since BMOA B, to proof our aim it is suffices to show that P b,c f 2 / B. From 3, we find where I = P b,c f 2 x = B I, tx log F tx c x tx tx λt dt and Since t < tx for < t <, F c, c b ; c; x is bounded, continuous and positive on [, ] for b + > c and using logarithm function is increasing, we have I C C Above inequality gives F log xi C tx tx log t x t t b c dt. t x t t b c dt log t t t b c dt. The integral log t t t b c dt is not finite, therefore it is clear that xp b,c f 2 x is also not finite, imply P b,c f 2 / B which completes the proof. Theorem 7. Let a, and b, c C. Then for Re b + > Re c >, P b,c is a bounded operator on B a, if and only if a,. Proof. Let f B a, a,. It is easy to see that fφ t z C f B a φ t z a. A simple calculation gives a tz fφ t z C f B a. z Using this in the integral representation of P b,c given in 3 and since F z is bounded on z, we find z a P b,c fz z a B C C f B a fφ t zf φ t z tz c λt dt f B a λt dt tz c a+ λt dt. tz c a+

10 94 S. Naik tz c a+ t c a+ if c a +, 2 c a+ if c a + <. As a >, above expression together with inequality, we have z a P b,c fz C f B a. Since P b,c f = f, last inequality gives P b,c f B a. Conversely, unboundedness of P b,c on B is contained in the proof of last corollary see also [2], Theorem 3.9i. Let a, and f 3. Then it is obivious that f B a. From the definition, we have P b,c f 3 z = + b c F b, ; b + ; z. b In view of the well-known Gauss identity [, ] F a, b; c; z = z c a b F c a, c b, c; z, and using derivative formula of hypergeometric function, we obtain Now P b,c f 3 z = + b c b + z F, b; b + 2; z. z a P b,c f 3 z = + b c b + z a z F, b; b + 2; z + b c b + z a F, b; b + 2; z. Since F, b; b + 2; z is bounded on z and a,, imply P b,c f 3 / B a, which completes the proof. 4 α-cesáro operator In this section we discuss the boundedness of the α- Cesáro operator C α from the Hardy space H p to the Bergman space A p for p, by rewriting C α as Hadamard or convolution product. Although the result follows from [2], Theorem 3.i here we give an alternate and simple proof. The Hadamard product f g is defined by f gz = n= a nb n z n, whenever fz = n= a nz n and gz = n= b nz n. If f and g are analytic in the unit disc D, then f gρz = fρe it gze it dt, < ρ <.

11 Cesáro type operators on spaces of analytic functions 95 For α C with Re α >, the α- Cesáro operator see [] or [2] is defined by C α, n n α +, n k fz := a k z n α + 2, n, n k n= We can rewrite the above formula of C α with the following expression. k= C α fz = G z G 2 z, fz where G z = z α+ and G 2z = F, ; α + 2; z. Therefore, C α fρz = = F ρe it Gze it dt Differentiation with respect to z in 2, gives C α fρz = α + fρe it ρe it α+ F, ; α + 2; ze it dt. 2 fρe it ρe it α+ F 2, 2; α + 3; ze it e it dt. Using the Gauss identity and since F α +, α + ; α + 3; z is bounded on z if Re α <, we have ρc α fρz fρe it F 2, 2; α + 3; ze it e it α + ρe it α+ dt C fρe it ρe it α+ ze it α dt C We will use above inequality to prove our next theorem. z α ρ α+ fρeit dt. 3 Theorem 8. Let α C with < Re α < and < q p <. If f H p, then C α f A q. Proof. i Let < q < p <. If f H p, then [3], Theorem 5.9 gives M r, f C. 4 r /p Assume that Re α <, taking z = r = ρ and z = re iθ in the last inequality of 3, a simple calculations show that ρ ρ q M q q ρ 2, C α f dρ ρ q M q ρ, f dρ ρ q p dρ by 4 < as q < p.

12 96 S. Naik Since ρ q M q q ρ, f dρ C ρ q ρm q q ρ 2, f dρ then ρ q Mq q ρ, C α f dρ <. Taking α = and p = q in Lemma 3, the above inequality gives That is C α f A q. ii If < q = p < and f H p, then M q q r, C α f dr <. r p M p r, f dr <, by Hardy-Littlewood theorem [3], Theorem 5.. Similar to above case, we have r r p M p p r 2, C α f dr < Desired result follows as before. This complete the proof. ρ p M p r, f dr <. References [] G. E. Andrews, R. Askey and R. Roy, Special functions, Cambridge University Press, 999. [2] M. R. Agrawal, P. G. Howlett, S. K. Lucas, S. Naik and S. Ponnusamy, Boundedness of generalized Cesáro averaging operator on certain function spaces, J. Comput. Appl. Math. 8 25, [3] P. L. Duren, Theory of H p spaces, Academic Press, New York, 97. [4] P. L. Duren, B. Romberg and A. L. shields, Linear functionals on H p spaces with < p <, J. Reine Angew. Math , [5] J. B. Garnett, Bounded analytic functions, Academic Press, New York, 98. [6] P. Galanopoules and M. Papadimitrakis, Hausdorff and quasi-hausdroff matrices on the spaces of analytic funcions, Canad. J. Math , [7] Songxiao Li, A note on the boundeness of generalized Cesáro averaging operator on certain function spaces, Indian Journal of Mathematics, 48 26, 53 6.

13 Cesáro type operators on spaces of analytic functions 97 [8] M. Mateljevic and M. Pavlovic, Multiplier of H p and BMOA, Pacific Journal of Mathematics, 46 99, [9] S. Naik, Generalized Cesáro operators on certain function spaces, Annales Polonici Mathematici, 982 2, [] K. Stempak, Cesáro averaging operators, Proc. Royal Soc. of Edinburgh, 24A 994, [] N. M. Temme, Special functions: An introduction to the classical functions of mathematical physics, John Wiley, New York, 996. [2] J. Xiao, Cesáro type operators on Hardy, BMOA and Bloch spaces, Arch. Math , [3] K. Zhu, Operators on Banach spaces of analytic functions, Marcel Dekker, 99. [4] K. Zhu, Bloch type spaces of analytic functions, Rocky Mountain J. of Maths , Sunanda Naik: Institute of Science and Technology, Gauhati University, Guwahati, 78 4, India spn2@yahoo.com