Qp Spaces and Dirichlet Type Spaces

Bao, Guanlong ; Gögüs, Nihat Gökhan ; Pouliasis, Stamatis

Canadian mathematical bulletin, Tome 60 (2017) no. 4, pp. 690-704

Résumé

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.

Détail

DOI : 10.4153/CMB-2017-006-1

Mots-clés : 30H25, 31C25, 46E15, Qp space, Dirichlet type space, Möbius invariant function space

Bibliographie

[1] [1] Aleman, A., The multiplication operator on Hilbert spaces of analytic functions. Habilitationsschrift, Fern Universitat, Hagen, 1993. Google Scholar

[2] [2] Aleman, A. and A. Simbotin, Estimates in Mobius invariant spaces of analytic functions. Complex Var. Theory Appl. 49(2004), 487–510. Google Scholar | DOI

[3] [3] Arazy, J. and Fisher, S. D., The uniqueness of the Dirichlet space among Mb'bius-invariant Hilbert spaces. Illinois J. Math. 29(1985), 449–462. Google Scholar

[4] [4] Arazy, J., Fisher, S. D., and J. Peetre, Mobius invariant function spaces. J. Reine Angew. Math. 363(1985), 110–145. Google Scholar | DOI

[5] [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. 136–146. Google Scholar

[6] [6] Aulaskari, R., J. Xiao, and R. Zhao, On subspaces and subsets ofBMOA and UBC. Analysis 15(1995), 101–121. Google Scholar | DOI

[7] [7] Axler, S., The Bergman space, the Bloch Space, and commutators of multiplication operators. Duke Math. J. 53(1986), 315–332. Google Scholar | DOI

[8] [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. 3–36. Google Scholar

[9] [9] Bao, G., N. Gogiis, and S. Pouliasis, On Dirichlet spaces with a class of superharmonic weights. http://goo.gl/4t8clA Google Scholar

[10] [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. Google Scholar

[11] [11] Essen, M. and H. Wulan, On analytic and meromorphic function and spaces ofQ^-type. Illionis. J. Math. 46(2002), no. 4, 1233–1258. Google Scholar

[12] [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. 61–170. Google Scholar

[13] [13] Hedenmalm, H., B. Korenblum, and K. Zhu, Theory of Bergman spaces. Graduate Texts in Mathematics, Springer-Verlag, New York, 2000. Google Scholar | DOI

[14] [14] Liu, X., G. Chacon, and Z. Lou, Characterizations of the Dirichlet-type space. Complex Anal. Oper. Theory 9(2015), 1269–1286. Google Scholar | DOI

[15] [15] Ortega, J. and J. Fabrega, Pointwise multipliers and corona type decomposition in BMOA. Ann. Inst. Fourier (Grenoble) 46(1996), 111–137. Google Scholar | DOI

[16] [16] Pau, J. and P. Perez, Composition operators acting on weighted Dirichlet spaces. J. Math. Anal. Appl. 401(2013), 682–694. Google Scholar | DOI

[17] [17] Richter, S., A representation theorem for cyclic analytic two-isometries. Trans. Amer. Math. Soc. 328(1991), 325–349. Google Scholar | DOI

[18] [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. 225–277. Google Scholar

[19] [19] Rochberg, R. and S. Semmes, A decomposition theorem for BMO and applications. J. Funct. Anal. 67(1986), 228–263. Google Scholar | DOI

[20] [20] Rubel, L. and R. Timoney, An extremal property of the Bloch space. Proc. Amer. Math. Soc. 75(1979), 45–49. Google Scholar | DOI

[21] [21] Wirths, K. J. and J. Xiao, Recognizing Qp,a functions per Dirichlet space structure. Bull. Belg. Math. Soc. Simon Stevin 8(2001), 47–59. Google Scholar

[22] [22] Wu, Z. and C. Xie, Decomposition theorems for Q spaces. Ark. Mat. 40(2002), 383–401. Google Scholar | DOI

[23] [23] Wulan, H., D. Zheng, and K. Zhu, Compact composition operators on BMOA and the Bloch space. Proc. Amer. Math. Soc. 137(2009), 3861–3868. Google Scholar | DOI

[24] [24] Xiao, J., Holomorphic Q classes. Lecture Notes in Mathematics, 1767, Springer-Verlag, Berlin, 2001. Google Scholar | DOI

[25] [25] Xiao, J., Geometric Qp functions. Frontiers in Mathematics, Birkhauser Verlag, Basel, 2006. Google Scholar

[26] [26] Zhao, R., On a general family of function spaces. Ann. Acad. Sci. Fenn. Math. Diss. 105(1996). Google Scholar

[27] [27] Zhao, R., Distances from Bloch functions to some Mobius invariant spaces. Ann. Acad. Sci. Fenn. Math. 33(2008), 303–313. Google Scholar

[28] [28] Zhu, K., A class of Mobius invariant function spaces. Illinois J. Math. 51(2007), 977–1002. Google Scholar

[29] [29] Zhu, K., Operator theory in function spaces. Mathematical Surveys and Monographs, 138, American Mathematical Society, Providence, RI, 2007. Google Scholar | DOI