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Q p Spaces and Dirichlet Type Spaces

  • Guanlong Bao (a1), Nihat Gökhan Gögüs (a2) and Stamatis Pouliasis (a2)


In this paper, we show that the Möbius invariant function space ${{\mathcal{Q}}_{p}}$ can be generated by variant Dirichlet type spaces ${{\mathcal{D}}_{\mu ,p}}$ induced by finite positive Borel measures $\mu $ on the open unit disk. A criterion for the equality between the space ${{\mathcal{D}}_{\mu ,p}}$ and the usual Dirichlet type space ${{\mathcal{D}}_{p}}$ is given. We obtain a sufficient condition to construct different ${{\mathcal{D}}_{\mu ,p}}$ spaces and provide examples. We establish decomposition theorems for ${{\mathcal{D}}_{\mu ,p}}$ spaces and prove that the non-Hilbert space ${{\mathcal{Q}}_{p}}$ is equal to the intersection of Hilbert spaces ${{\mathcal{D}}_{\mu ,p}}$ . As an application of the relation between ${{\mathcal{Q}}_{p}}$ and ${{\mathcal{D}}_{\mu ,p}}$ spaces, we also obtain that there exist different ${{\mathcal{D}}_{\mu ,p}}$ spaces; this is a trick to prove the existence without constructing examples.



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[1] Aleman, A., The multiplication operator on Hilbert spaces of analytic functions. Habilitationsschrift, Fern Universitat, Hagen, 1993.
[2] Aleman, A. and A. Simbotin, Estimates in Mobius invariant spaces of analytic functions. Complex Var. Theory Appl. 49(2004), 487510. 731 657
[3] Arazy, J. and Fisher, S. D., The uniqueness of the Dirichlet space among Mb'bius-invariant Hilbert spaces. Illinois J. Math. 29(1985), 449462.
[4] Arazy, J., Fisher, S. D., and J. Peetre, Mobius invariant function spaces. J. Reine Angew. Math. 363(1985), 110145.
[5] Aulaskari, R. and P. Lappan, Criteria for an analytic function to be Bloch and a harmonic or meromorphic function to be normal. In: Complex analysis and its applications, Pitman Res. Notes Math. Ser., 305, Longman Sci. Tec, Harlow, 1994, pp. 136146.
[6] Aulaskari, R., J. Xiao, and R. Zhao, On subspaces and subsets ofBMOA and UBC. Analysis 15(1995), 101121.
[7] Axler, S., The Bergman space, the Bloch Space, and commutators of multiplication operators. Duke Math. J. 53(1986), 315332.
[8] Baernstein, A., Analytic functions of bounded mean oscillation. In: Aspects of contemporary complex analysis (Proc. NATO Adv. Study Inst., Univ. Durham, Durham), Academic Press, 1980, pp. 336.
[9] Bao, G., N. Gogiis, and S. Pouliasis, On Dirichlet spaces with a class of superharmonic weights.
[10] El-Fallah, O., K. Kellay, J. Mashreghi, and T. Ransford, A primer on the Dirichlet space. Cambridge Tracts in Mathematics, 203, Cambridge University Press, Cambridge, 2014.
[11] Essen, M. and H. Wulan, On analytic and meromorphic function and spaces ofQ^-type. Illionis. J. Math. 46(2002), no. 4, 12331258.
[12] Girela, D., Analytic functions of bounded mean oscillation. In: Complex function spaces, Mekrijarvi 1999 Univ. Joensuu Dept. Math. Rep. Ser., 4, Univ. Joensuu, Joensuu, 2001, pp. 61170.
[13] Hedenmalm, H., B. Korenblum, and K. Zhu, Theory of Bergman spaces. Graduate Texts in Mathematics, Springer-Verlag, New York, 2000.
[14] Liu, X., G. Chacon, and Z. Lou, Characterizations of the Dirichlet-type space. Complex Anal. Oper. Theory 9(2015), 12691286.
[15] Ortega, J. and J. Fabrega, Pointwise multipliers and corona type decomposition in BMOA. Ann. Inst. Fourier (Grenoble) 46(1996), 111137.
[16] Pau, J. and P. Perez, Composition operators acting on weighted Dirichlet spaces. J. Math. Anal. Appl. 401(2013), 682694.
[17] Richter, S., A representation theorem for cyclic analytic two-isometries. Trans. Amer. Math. Soc. 328(1991), 325349.
[18] Rochberg, R., Decomposition theorems for Bergman spaces and their applications. In: Operators and function theory (Lancaster, 1984), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 153, Reidel, Dordrecht, 1985, pp. 225277.
[19] Rochberg, R. and S. Semmes, A decomposition theorem for BMO and applications. J. Funct. Anal. 67(1986), 228263. 6/0022-1236(86)90038-8
[20] Rubel, L. and R. Timoney, An extremal property of the Bloch space. Proc. Amer. Math. Soc. 75(1979), 4549.
[21] Wirths, K. J. and J. Xiao, Recognizing Qp,a functions per Dirichlet space structure. Bull. Belg. Math. Soc. Simon Stevin 8(2001), 4759.
[22] Wu, Z. and C. Xie, Decomposition theorems for Qp spaces. Ark. Mat. 40(2002), 383401. 0.1007/BF02384542
[23] Wulan, H., D. Zheng, and K. Zhu, Compact composition operators on BMOA and the Bloch space. Proc. Amer. Math. Soc. 137(2009), 38613868.
[24] Xiao, J., Holomorphic Q classes. Lecture Notes in Mathematics, 1767, Springer-Verlag, Berlin, 2001.
[25] Xiao, J., Geometric Qp functions. Frontiers in Mathematics, Birkhauser Verlag, Basel, 2006.
[26] Zhao, R., On a general family of function spaces. Ann. Acad. Sci. Fenn. Math. Diss. 105(1996).
[27] Zhao, R., Distances from Bloch functions to some Mobius invariant spaces. Ann. Acad. Sci. Fenn. Math. 33(2008), 303313.
[28] Zhu, K., A class of Mobius invariant function spaces. Illinois J. Math. 51(2007), 9771002.
[29] Zhu, K., Operator theory in function spaces. Mathematical Surveys and Monographs, 138, American Mathematical Society, Providence, RI, 2007.! 0.1090/surv/138
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Q p Spaces and Dirichlet Type Spaces

  • Guanlong Bao (a1), Nihat Gökhan Gögüs (a2) and Stamatis Pouliasis (a2)


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