Revista Matemática Iberoamericana

Coefficient multipliers on Banach spaces of analytic functions

Óscar Blasco and Miroslav Pavlović

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Motivated by an old paper of Wells [J. London Math. Soc. {\bf 2} (1970), 549-556] we define the space $X\otimes Y$, where $X$ and $Y$ are "homogeneous" Banach spaces of analytic functions on the unit disk $\mathbb{D}$, by the requirement that $f$ can be represented as $f=\sum_{j=0}^\infty g_n * h_n$, with $g_n\in X$, $h_n\in Y$ and $\sum_{n=1}^\infty \|g_n\|_X \|h_n\|_Y < \infty$. We show that this construction is closely related to coefficient multipliers. For example, we prove the formula $((X\otimes Y),Z)=(X,(Y,Z))$, where $(U,V)$ denotes the space of multipliers from $U$ to $V$, and as a special case $(X\otimes Y)^*=(X,Y^*)$, where $U^*=(U,H^\infty)$. We determine $H^1\otimes X$ for a class of spaces that contains $H^p$ and $\ell^p$ $(1\le p\le 2)$, and use this together with the above formulas to give quick proofs of some important results on multipliers due to Hardy and Littlewood, Zygmund and Stein, and others.

Article information

Rev. Mat. Iberoamericana, Volume 27, Number 2 (2011), 415-447.

First available in Project Euclid: 10 June 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42A45: Multipliers 30A99: None of the above, but in this section
Secondary: 30D55 46E15: Banach spaces of continuous, differentiable or analytic functions 46A45: Sequence spaces (including Köthe sequence spaces) [See also 46B45]

Banach spaces analytic functions coefficient multipliers tensor products Hardy spaces


Blasco, Óscar; Pavlović, Miroslav. Coefficient multipliers on Banach spaces of analytic functions. Rev. Mat. Iberoamericana 27 (2011), no. 2, 415--447.

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  • Ahern, P. and Jevtić, M.: Duality and multipliers for mixed norm spaces. Michigan Math. J. 30 (1983), no. 1, 53-64.
  • Anderson, J.M. and Shields, A.L.: Coefficient multipliers of Bloch functions. Trans. Amer. Math. Soc. 224 (1976), no. 2, 255-265.
  • Arazy, J., Fisher, S. and Peetre, J.: Möbius invariant function spaces. J. Reine Angew. Math. 363 (1985), 110-145.
  • Blasco, O.: Multipliers on spaces of analytic functions. Canad. J. Math. 47 (1995), no. 1, 44-64.
  • Blasco, O.: Vector-valued analytic functions of bounded mean oscillation and geometry of Banach spaces. Illinois J. Math. 41 (1997), 532-558.
  • Blasco, O.: Composition operators on the minimal space invariant under Moebious transformations. In Complex and harmonic analysis, 157-166. DEStech Publ., Lancaster, PA, 2007.
  • Buckley, S.M.: Relative solidity for spaces of holomorphic functions. Math. Proc. R. Ir. Acad. 104A (2004), no. 1, 83-97 (electronic).
  • Bukhvalov, A.V.: On the analytic Radon-Nikodým property. In Function Spaces (Poznań 1989), 211-228. Teubner-Texte Math 120. Teubner, Stuttgart, 1991.
  • Bukhvalov, A.V. and Danilevich, A.A.: Boundary properties of analytic functions with values in Banach spaces. Mat. Zametki 31 (1982), 203-214 (russian).
  • Duren, P.L.: Theory of $H^p$-spaces. Pure and Applied Mathematics 38. Academic Press, New York-London, 1970.
  • Figà-Talamanca, A.: Translation invariant operators in $L^p$. Duke Math. J. 32 (1965), 495-501.
  • Fisher, S.: The Möbius group and invariant spaces of analytic functions. Amer. Math. Monthly 95 (1988), 514-527.
  • Flett, T.M.: Lipschitz spaces of functions on the circle and the disc. J. Math. Anal. Appl. 39 (1972), 125-158.
  • Girela, D., Pavlović, M. and Peláez, J.A.: Spaces of analytic functions of Hardy-Bloch type. J. Anal. Math. 100 (2006), 53-81.
  • Hardy, G.H. and Littlewood, J.E.: Notes on the theory of series. XX: Generalizations of a theorem of Paley. Q. J. Math., Oxford Ser. 8 (1937), 161-171.
  • Hardy, G.H. and Littlewood, J.E.: Theorems concerning mean values of analytic or harmonic functions Quart. J. Math., Oxford Ser. 12 (1941), 221-256.
  • Jevtić, M. and Pavlović, M.: Coefficient multipliers on spaces of analytic functions. Acta Sci. Math. (Szeged) 64 (1998), no. 3-4, 531-545.
  • Kellogg, C.N.: An extension of the Hausdorff-Young theorem. Michigan Math. J. 18 (1971), 121-127.
  • Lengfield, M.: A nested embedding theorem for Hardy-Lorentz spaces with applications to coefficient multiplier problems. Rocky Mountain J. Math. 38 (2008), no. 4, 1215-1251.
  • Mateljević, M. and Pavlović, M.: $L^p$-behaviour of power series with positive coefficients and Hardy spaces. Proc. Amer. Math. Soc. 87 (1983), 309-316.
  • Mateljević, M. and Pavlović, M.: $L^p$-behaviour of the integral means of analytic functions. Studia Math. 77 (1984), no. 3, 219-237.
  • Mateljević, M. and Pavlović, M.: Multipliers of $H^p$ and BMOA. Pac. J. Math. 146 (1990), no. 1, 71-84.
  • Mateljević, M. and Pavlović, M.: The best approximation and composition with inner functions. Michigan Math. J. 42 (1995), no. 2, 367-378.
  • Nowak, M.: A note on coefficient multipliers $(H\sp p,\mathfrak B)$ and $(H\sp p,\rm BMOA)$. In Topics in complex analysis (Warsaw, 1992), 299-302. Banach Center Publ. 31. Polish Acad. Sci., Warsaw, 1995.
  • Pavlović, M.: Mixed norm spaces of analytic and harmonic functions. I. Publ. Inst. Math. (Beograd) (N.S.) 40(54) (1986), 117-141.
  • Rubel, L.A. and Timoney, R.M.: An extremal property of the Bloch space. Proc. Amer. Math. Soc. 75 (1979), 45-49.
  • Sledd, W.T.: On multipliers of $H^p$ spaces. Indiana Univ. Math. J. 27 (1978), no. 5, 797-803.
  • Stein, E.M. and Zygmund, A.: Boundedness of translation invariant operators on Hölder spaces and $L^p$-spaces. Ann. of Math. (2) 85 (1967), 337-349.
  • Taylor, A.E.: Banach spaces of functions analytic in the unit circle. I. Studia Math. 11 (1950), 145-170.
  • Taylor, A.E.: Banach spaces of functions analytic in the unit circle. II. Studia Math. 12 (1951), 25-50.
  • Timoney, R.M.: Natural function spaces. J. London Math. Soc. (2) 41 (1990), 78-88.
  • Timoney, R.M.: Maximal invariant spaces of analytic functions. Indiana Univ. Math. J. 31 (1982), 651-663.
  • Torchinsky, A.: Real-variable methods in harmonic analysis. (Reprint of the 1986 original). Dover Publications, Mineola, NY, 2004.
  • Wells, J.H.: Some results concerning multipliers of $H^p$. J. London Math. Soc. (2) 2 (1970), 549-556.