DOI QR코드

DOI QR Code

MULTIPLIERS OF DIRICHLET-TYPE SUBSPACES OF BLOCH SPACE

  • Received : 2019.03.18
  • Accepted : 2019.10.16
  • Published : 2020.03.31

Abstract

Let M(X, Y) denote the space of multipliers from X to Y, where X and Y are analytic function spaces. As we known, for Dirichlet-type spaces 𝓓αp, M(𝓓p-1p, 𝓓q-1q) = {0}, if p ≠ q, 0 < p, q < ∞. If 0 < p, q < ∞, p ≠ q, 0 < s < 1 such that p + s, q + s > 1, then M(𝓓p-2+sp, 𝓓q-2+sq) = {0}. However, X ∩ 𝓓p-1p ⊆ X ∩ 𝓓q-1q and X ∩ 𝓓p-2+sp ⊆ X ∩ 𝓓q-2+sp whenever X is a subspace of the Bloch space 𝓑 and 0 < p ≤ q < ∞. This says that the set of multipliers M(X ∩ 𝓓 p-2+sp, X∩𝓓q-2+sq) is nontrivial. In this paper, we study the multipliers M(X ∩ 𝓓p-2+sp, X ∩ 𝓓q-2+sq) for distinct classical subspaces X of the Bloch space 𝓑, where X = 𝓑, BMOA or 𝓗.

Acknowledgement

Supported by : National Natural Science Foundation of China

References

  1. J. M. Anderson, J. Clunie, and Ch. Pommerenke, On Bloch functions and normal functions, J. Reine Angew. Math. 270 (1974), 12-37.
  2. J. Arazy, Multipliers of Bloch functions, University of Haifa Mathem. Public. Series, 54, 1982.
  3. L. Brown and A. L. Shields, Multipliers and cyclic vectors in the Bloch space, Michigan Math. J. 38 (1991), no. 1, 141-146. https://doi.org/10.1307/mmj/1029004269
  4. S. M. Buckley, P. Koskela, and D. Vukotic, Fractional integration, differentiation, and weighted Bergman spaces, Math. Proc. Cambridge Philos. Soc. 126 (1999), no. 2, 369- 385. https://doi.org/10.1017/S030500419800334X
  5. C. Chatzifountas, D. Girela, and J. Pelaez, Multipliers of Dirichlet subspaces of the Bloch space, J. Operator Theory 72 (2014), no. 1, 159-191. https://doi.org/10.7900/jot.2012nov20.1979
  6. P. Duren, Theory of Hp Spaces, Academic Press, New York-London 1970. Reprint: Dover, Mineola, New York, 2000.
  7. T. M. Flett, The dual of an inequality of Hardy and Littlewood and some related inequalities, J. Math. Anal. Appl. 38 (1972), 746-765. https://doi.org/10.1016/0022-247X(72)90081-9
  8. P. Galanopoulos, D. Girela, and M. J. Martin, Besov spaces, multipliers and univalent functions, Complex Anal. Oper. Theory 7 (2013), no. 4, 1081-1116. https://doi.org/10.1007/s11785-011-0160-3
  9. P. Galanopoulos, D. Girela, and J. Pelaez, Multipliers and integration operators on Dirichlet spaces, Trans. Amer. Math. Soc. 363 (2011), no. 4, 1855-1886. https://doi.org/10.1090/S0002-9947-2010-05137-2
  10. D. Girela, Analytic functions of bounded mean oscillation, in Complex function spaces (Mekrijarvi, 1999), 61-170, Univ. Joensuu Dept. Math. Rep. Ser., 4, Univ. Joensuu, Joensuu, 2001.
  11. D. Girela and J. Pelaez, Carleson measures for spaces of Dirichlet type, Integral Equations Operator Theory 55 (2006), no. 3, 415-427. https://doi.org/10.1007/s00020-005-1391-3
  12. D. Girela and J. Pelaez, Carleson measures, multipliers and integration operators for spaces of Dirichlet type, J. Funct. Anal. 241 (2006), no. 1, 334-358. https://doi.org/10.1016/j.jfa.2006.04.025
  13. D. Gnuschke, Relations between certain sums and integrals concerning power series with Hadamard gaps, Complex Variables Theory Appl. 4 (1984), no. 1, 89-100. https://doi.org/10.1080/17476938408814094
  14. J. E. Littlewood and R. E. A. C. Paley, Theorems on Fourier series and power series (II), Proc. London Math. Soc. (2) 42 (1936), no. 1, 52-89. https://doi.org/10.1112/plms/s2-42.1.52
  15. J. M. Ortega and J. Fabrega, Pointwise multipliers and corona type decomposition in BMOA, Ann. Inst. Fourier (Grenoble) 46 (1996), no. 1, 111-137.
  16. J. Pau and R. Zhao, Carleson measures, Riemann-Stieltjes and multiplication operators on a general family of function spaces, Integral Equations Operator Theory 78 (2014), no. 4, 483-514. https://doi.org/10.1007/s00020-014-2124-2
  17. R. Zhao, On logarithmic Carleson measures, Acta Sci. Math. (Szeged) 69 (2003), no. 3-4, 605-618.
  18. K. Zhu, Operator Theory in Function Spaces, second edition, Mathematical Surveys and Monographs, 138, American Mathematical Society, Providence, RI, 2007. https://doi.org/10.1090/surv/138
  19. A. Zygmund, Trigonometric Series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959.