Abstract
We give an overview of some results on the class of functions with subharmonic behaviour and their invariance properties under conformal and quasiconformal mappings. While many of the results we present will be related to author’s own work, we shall present also some other results and examples about this class of functions.
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Notes
For a set A, by \(A^c\) we denote its complement.
References
Ahern, P., Bruna, J.: Maximal and area integral characterization of Hardy–Sobolev spaces in the unit ball of \({\mathbb{C}}^n\). Rev. Mat. Iberoam. 4(1), 123–153 (1988)
Astala, K.: A remark on quasiconformal mappings and BMO-functions. Mich. Math. J. 30(2), 209–212 (1983)
Coifman, R.R., Fefferman, C.: Weighted norm inequalities for maximal functions and singular integrals. Stud. Math. 51, 241–250 (1974)
Cruz-Uribe, D.: The minimal operator and the geometric maximal operator in \({\mathbb{R}}^n\). Stud. Math. 144(1), 1–37 (2001)
Dovgoshey, O., Riihentaus, J.: Bi-Lipschitz mappings and quasinearly subharmonic functions. Int. J. Math. Math. Sci. 2010, 8 (2010). https://doi.org/10.1155/2010/382179
Duren, P.: Univalent Functions, Fundamental Principles of Mathematical Sciences 259. Springer, New York (1983)
Fefferman, C., Stein, E.M.: \({\mathbb{H}}^p\)-spaces of several variables. Acta Math. 129(3–4), 137–193 (1972)
Garnett, J.B.: Bounded Analytic Functions. Academic Press, New York (1981)
Hardy, G.H., Littlewood, J.E.: Some properties of conjugate functions. J. Reine Angew. Math. 167, 405–423 (1933)
Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford Mathematical Monographs. Oxford University Press, New York (1993)
Heinonen, J., Koskela, P.: Weighted Sobolev and Poincaré inequalities and quasiregular mappings of polynomial type. Math. Scand. 77(2), 251–271 (1995)
Heinonen, J., Koskela, P.: Definitions of quasiconformality. Invent. Math. 120(1), 61–79 (1995)
Hervé, M.: Analytic and plurisubharmonic functions in finite and infinte dimensional spaces. Lecture Notes in Mathematics, vol. 198. Springer, Berlin (1971)
Iwaniec, T.: p-harmonic tensors and quasiregular mappings. Ann. Math. (2) 136(3), 589–624 (1992)
Koskela, P.: Lectures on Quasiconformal and Quasisymmetric Mappings. University of Jyväskylä, Jyväskylä (2009)
Koskela, P., Manojlović, V.: Quasi-nearly subharmonic functions and quasiconformal mappings. Potential Anal. 37(2), 187–196 (2012)
Pavlović, M.: On subharmonic behaviour and oscillation of functions on balls in \({\mathbb{R}}^n\). Publ. Inst. Math. (Belgrade) 55, 18–22 (1994)
Pavlović, M., Riihentaus, J.: Classes of quasi-nearly subharmonic functions. Potential Anal. 29(1), 89–104 (2008)
Reimann, H.M.: Functions of bounded mean oscillation and quasiconformal mappings. Comment. Math. Helv. 49, 260–276 (1974)
Rickman, S.: Quasiregular Mappings. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 26. Springer, Berlin (1993)
Riihentaus, J.: Subharmonic functions: Non-tangential and tangental boundary behavior. In: Function Spaces, Differential Operators and Non-linear Analysis, Acad. Sci. Czech. Repub. Inst. Math. Prague, pp. 229–238 (2000)
Riihentaus, J.: A generalized mean value inequality for subharmonic functions. Expo. Math. 19(2), 187–190 (2001)
Staples, S.G.: Maximal functions, \(A_{\infty }\) measures and quasiconformal maps. Proc. Am. Math. Soc. 113(3), 689–700 (1991)
Staples, S.G.: Doubling measures and quasiconformal maps. Comment. Math. Helv. 67, 119–128 (1992)
Todorčević, V.: Quasi-nearly subharmonic functions and conformal mappings. Filomat (Niš) 21(2), 243–249 (2007)
Uchiyama, A.: Weight functions of the class \((A_{\infty })\) and quasi-conformal mappings. Proc. Jpn. Acad. 51, 811–814 (1975). suppl
Ziemer, W.P.: Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation. Graduate Texts in Mathematics, 120. Springer, New York (1989)
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Todorčević, V. Subharmonic behavior and quasiconformal mappings. Anal.Math.Phys. 9, 1211–1225 (2019). https://doi.org/10.1007/s13324-019-00308-8
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DOI: https://doi.org/10.1007/s13324-019-00308-8