Comptes Rendus
Mathematical analysis/Complex analysis
Intersection of harmonically weighted Dirichlet spaces
Comptes Rendus. Mathématique, Volume 355 (2017) no. 8, pp. 859-865.

In 1991, S. Richter introduced harmonically weighted Dirichlet spaces D(μ), motivated by his study of cyclic analytic two-isometries. In this paper, we consider μPD(μ), the intersection of D(μ) spaces, where P is the family of Borel probability measures. Several function-theoretic characterizations of the Banach space μPD(μ) are given. We also show that μPD(μ) is located strictly between some classical analytic Lipschitz spaces and μPD(μ) can be regarded as the endpoint case of analytic Morrey spaces in some sense.

En 1991, S. Richter a introduit les espaces de Dirichlet D(μ) à poids harmonique, motivé par l'étude des 2-isométries analytiques. Dans cet article, on considère une intersection μPD(μ) d'espaces D(μ), où P est l'espace des mesures de probabilité boréliennes. On donne plusieurs caractérisations de μPD(μ) en termes de théorie des fonctions. On montre également que μPD(μ) se compare dans les deux sens par des relations d'inclusion strictes avec certains espaces de fonctions analytiques Lipschitz et que μPD(μ) peut être considéré comme le cas extrême des espaces de Morrey analytiques en un certain sens.

Published online:
DOI: 10.1016/j.crma.2017.07.013

Guanlong Bao 1; Nihat Gökhan Göğüş 2; Stamatis Pouliasis 2

1 Department of Mathematics, Shantou University, Shantou, Guangdong 515063, China
2 Faculty of Engineering and Natural Sciences, Sabanci University, Tuzla, Istanbul 34956, Turkey
     author = {Guanlong Bao and Nihat G\"okhan G\"o\u{g}\"u\c{s} and Stamatis Pouliasis},
     title = {Intersection of harmonically weighted {Dirichlet} spaces},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {859--865},
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Guanlong Bao; Nihat Gökhan Göğüş; Stamatis Pouliasis. Intersection of harmonically weighted Dirichlet spaces. Comptes Rendus. Mathématique, Volume 355 (2017) no. 8, pp. 859-865. doi : 10.1016/j.crma.2017.07.013.

[1] A. Baernstein Analytic functions of bounded mean oscillation (D.A. Brannan; J.G. Clunie, eds.), Aspects of Contemporary Complex Analysis, Academic Press, London, New York, 1980, pp. 3-36

[2] P. Duren Theory of Hp Spaces, Academic Press, New York, 1970

[3] O. El-Fallah; K. Kellay; J. Mashreghi; T. Ransford A Primer on the Dirichlet Space, Cambridge Tracts in Mathematics, vol. 203, Cambridge University Press, 2014

[4] G. Hardy; J. Littlewood Some properties of fractional integrals II, Math. Z., Volume 34 (1932), pp. 403-439

[5] E.A. Poletsky Projective limits of Poletsky–Stessin Hardy spaces, Complex Anal. Oper. Theory, Volume 10 (2016) no. 5, pp. 1001-1016

[6] E.A. Poletsky; K.R. Shrestha (Banach Cent. Publ.), Volume vol. 107, Institute of Mathematics of the Polish Academy of Sciences, Warsaw (2015), pp. 195-204

[7] S. Richter A representation theorem for cyclic analytic two-isometries, Trans. Amer. Math. Soc., Volume 328 (1991), pp. 325-349

[8] S. Richter; C. Sundberg A formula for the local Dirichlet integral, Mich. Math. J., Volume 38 (1991), pp. 355-379

[9] S. Richter; C. Sundberg Multipliers and invariant subspaces in the Dirichlet space, J. Oper. Theory, Volume 28 (1992), pp. 167-186

[10] Z. Wu; C. Xie Q spaces and Morrey spaces, J. Funct. Anal., Volume 201 (2003), pp. 282-297

[11] H. Wulan On the coefficients of α-Bloch functions, Math. Montisnigri, Volume 3 (1994), pp. 45-57

[12] J. Xiao Geometric Qp Functions, Birkhäuser Verlag, Basel–Boston–Berlin, 2006

[13] J. Xiao; W. Xu Composition operators between analytic Campanato spaces, J. Geom. Anal., Volume 24 (2014), pp. 649-666

[14] K. Zhu Operator Theory in Function Spaces, American Mathematical Society, Providence, RI, 2007

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