Multipliers on Spaces of Analytic Functions

Blasco, Oscar

Canadian journal of mathematics, Tome 47 (1995) no. 1, pp. 44-64

Résumé

In the paper we find, for certain values of the parameters, the spaces of multipliers (H(p, q, α), H(s, t, β) and (H(p, q, α), ls), where H(p, q, α) denotes the space of analytic functions on the unit disc such that . As corollaries we recover some new results about multipliers on Bergman spaces and Hardy spaces.

Détail

DOI : 10.4153/CJM-1995-003-5

Mots-clés : 42A45, multipliers, spaces of analytic functions

Bibliographie

Anderson, J.M., Clunie, J. and Pommerenke, Ch., On Block functions and normal functions, J. Reine Angew. Math. 270(1974), 12–37. Google Scholar

[AS] Anderson, J.M. and Shields, A.L., Coefficient multipliers on Block functions, Trans. Amer. Math. Soc. 224(1976), 256–265. Google Scholar

[A] Axler, S., Bergman spaces and their operators, Surveys on some recent results on Operator Theory ‘eds. Conway, J.B. and Morrel, B.B.), Pitman Res. Notes Math. 171, 1988, 1–50. Google Scholar

Bennett, G., Stegenga, D.A. and Timoney, R.M., Coefficients of Block and Lipschitz functions, Illinois J. Math. 25(1981), 520–531. Google Scholar

Blasco, O., Operators on weighted Bergman spaces and applications, Duke Math. J. 66(1992), 443–167. Google Scholar

Blasco, O., Multipliers on weighted Besov spaces of analytic functions, Contemp. Math. 144(1993), 23–33. Google Scholar

Duren, P., Theory ofHp-spaces, Academic Press, New York, 1970. Google Scholar

Duren, P. and Shields, A.L., Coefficient multipliers of W and B? spaces, Pacific J. Math. 32(1970), 69–78. Google Scholar

Duren, P., Properties of Hp (0 < p < , 1) and its containing Banach space, Trans. Amer. Math. Soc. 141(1969), 255–262. Google Scholar

Duren, P.L., Romberg, B.W. and Shields, A.L., Linear funetionals on Hp-spaces with 0 < p <, 1, J. Reine Angew. Math. 238(1969), 32–60. Google Scholar

Flett, T.M., The dual of an inequality of Hardy and Littlewood and some related inequalities, J. Math. Anal. Appl. 38(1972), 746–765. Google Scholar

Flett, T.M., Lipschitz spaces of functions on the circle and the disc, J. Math. Anal. Appl. 39(1972), 125–158. Google Scholar

Flett, T.M., On the rate of growth of mean valeus of holomorphic and harmonic functions, Proc. London Math. Soc. 20(1970), 749–768. Google Scholar

Hardy, G. H. and Littlewood, J. E., Some properties of fractional integrals II, Math. Z. 34(1932), 403-439. Google Scholar

Hardy, G. H., Theorems concerning mean values of analytic or harmonic functions, Quart. J. Math. 12(1941), 221-256. Google Scholar

Jevtic, M. and Pavlovic, M., On multipliers from HP to Iq, 0 < q < p < 1, Arch. Math. 56(1991), 174-180. Google Scholar

Katnetzon, Y., An introduction to Harmonic Analysis, John Wiley and Sons, New York, 1968. Google Scholar

Kellogg, C. N., An extension of the Hausdorff-Young Theorem, Michigan Math. J. 18(1971), 121-127. Google Scholar

Kisliakov, S. V., Fourier coefficients of boundary values of analytic functions on the disc and the bidisc, Trudy Mat. Inst. Steklov. 155(1981), 77-91. Google Scholar

Littlewood, J. E. and Paley, R. E. A. C., Theorems on Fourier series and power series, J. London Math. Soc. 42(1931), 52-89. Google Scholar

MacGregor, T. and Zhu, K., Coefficient multipliers between Bergman and Hardy spaces, preprint. Google Scholar

Marzuq, M., Linear Junctionals on some weighted Bergman spaces, Bull. Austral. Math. Soc. 42(1990), 417-425. Google Scholar

Mateljevic, M. and Pavlovic, M., LP behaviour of power series with positive coefficients and Hardy spaces, Proc. Amer. Math. Soc. 87(1983), 309-316. Google Scholar

Mateljevic, M., LP behaviour of the integral means of analytic functions, Studia Math. 77(1984), 219-237. Google Scholar

Mateljevic, M. , Multipliers of HP and BMO, Pacific J. Math. 146(1990), 71-84. Google Scholar

Matheson, A., A Multipliers theorem for analytic functions of slow mean growth, Proc. Amer. Math. Soc. 77(1979), 53-57. Google Scholar

Sledd, W., Some results about spaces of analytic functions introduced by Hardy and Littlewood, J. London Math. Soc. 2(1974), 328-336. Google Scholar

Sledd, W., On multipliers of HP spaces, Indiana Univ. Math. J. 27(1978), 797-803. Google Scholar

Shapiro, J. H., Mackey topologies, reproducing kernels and diagonal maps on the Hardy and Bergman spaces, Duke Math. J. 43(1976), 187-202. Google Scholar

Shields, A. L. and D. L. Williams, Bounded projections, duality and multipliers in spaces of analytic functions, Trans. Amer. Math. Soc. 162(1971), 287-302. Google Scholar

Stegenga, D., Multipliers of the Dirichlet space, Illinois J. Math. 24(1980), 113-139. Google Scholar

Vukotic, D., On the coefficient multipliers of Bergman spaces, preprint. Google Scholar

Wojtaszcyk, P., Multipliers into Bergman spaces and Nevalinna class, Canad. Math. Bull. 33(1990), 151-161. Google Scholar

Zhu, K., Operator theory in function spaces, Marcel Dekker, Inc., New York, 1990. Google Scholar

Zygmund, A., Trigonometric series, Cambrigde Univ. Press., New York, 1959. Google Scholar