## Classes of quasi-nearly subharmonic functions.(English)Zbl 1158.31002

It is known that if $$u\geq 0$$ is subharmonic on a domain $$\Omega\subset\mathbb{R}^n$$, $$n\geq 2$$ and $$p>0$$, then there is a constant $$C=C(n,p)\geq\frac{1}{m(B(0,1))}$$ such that $u(x)^p\leq\frac{c}{r^n}\int_{B(x,r)}u^p\,dm,$ for any ball $$B(x,r)\subset\Omega$$. In an attempt to generalize this mean value inequality, the authors define a new class of functions $$QNS$$: a nonnegative, locally integrable function $$u$$ on $$\Omega$$ is called quasi-nearly subharmonic if there is a constant $$K=K(n,u,\Omega)>0$$ such that $u(x)\leq\frac{K}{r^n}\int_{B(x,r)}u\,dm$ for any $$B(x,r)\subset\Omega$$. The authors show that the $$QNS$$ class is large. For example, if $$h$$ is a real valued and harmonic on $$\Omega$$, then the function $$|\nabla h|^p$$ is $$QNS$$ for every $$p>0$$. The authors prove some of the basic properties of these functions and give a characterization of the $$QNS$$ functions with the aid of the quasihyperbolic metric.

### MSC:

 31C05 Harmonic, subharmonic, superharmonic functions on other spaces 31B25 Boundary behavior of harmonic functions in higher dimensions 31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
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### References:

 [1] Ahern, P., Bruna, J.: Maximal and area integral characterizations of Hardy-Sobolev spaces in the unit ball of $$\mathbb{C}$$ n . Rev. Mat. Iberoam. 4, 123–153 (1988) · Zbl 0685.42008 [2] Di Benedetto, E., Trudinger, N.S.: Harnack inequalities for quasi-minima of variational integrals. Ann. Inst. Henri Poincare Anal. Non Lineaire 1, 295–308 (1984) · Zbl 0565.35012 [3] Domar, Y.: Uniform boundedness in families related to subharmonic functions. J. Lond. Math. Soc. 38(2), 485–491 (1988) · Zbl 0631.31002 [4] Dyakonov, K.M.: Equivalent norms on Lipschitz-type spaces of holomorphic functions. Acta Math. 178, 143–167 (1997) · Zbl 0898.30040 [5] Dyakonov, K.M.: Holomorphic functions and quasiconformal mappings with smooth moduli. Adv. Math. 187, 146–172 (2004) · Zbl 1056.30018 [6] Fefferman, C., Stein, E.M.: H p spaces of several variables. Acta Math. 129, 137–193 (1972) · Zbl 0257.46078 [7] Garnett, J.B.: Bounded Analytic Functions. Academic, New York (1981) · Zbl 0469.30024 [8] Hallenbeck, D.J.: Radial growth of subharmonic functions. Pitman Res. Notes 262, 113–121 (1992) · Zbl 0803.31002 [9] Hervé, M.: Analytic and Plurisubharmonic Functions in Finite and Infinite Dimensional Spaces. Lecture Notes in Mathematics 198. Springer, Berlin (1971) · Zbl 0214.36404 [10] Jevtić, M., Pavlović, M.: $${\mathcal{M}}$$ -Besov p-classes and Hankel operators in the Bergman spaces on the unit ball. Arch. Math. 61, 367–376 (1993) · Zbl 0785.32003 [11] Jevtić, M., Pavlović, M.: Subharmonic behavior of generalized ({$$\alpha$$},{$$\beta$$})-harmonic functions and their derivatives. Indian J. Pure Appl. Math. 30, 407–418 (1999) · Zbl 0931.32003 [12] Krasnoselskii, M.A., Rutickii, Ja.B.: Convex functions and Orlicz spaces. Translated from the first Russian edition by Leo F. Boron, P. Noordhoff Ltd., Groningen (1961) [13] Kuran, Ü.: Subharmonic behavior of |h| p (p > 0, h harmonic). J. Lond. Math. Soc. 8(2), 529–538 (1974) · Zbl 0289.31007 [14] Mateljević, M., Pavlović, M.: $${\mathcal{L}}^p$$ -behavior of power series with positive coefficients and Hardy spaces. Proc. Am. Math. Soc. 87, 309–316 (1983) [15] Mizuta, Y.: Potential theory in euclidean spaces. Gaguto International Series, Mathematical Sciences and Applications, 6. Gakkōtosho Co., Tokyo (1996) · Zbl 0938.76539 [16] Mizuta, Y.: Boundary limits of functions in weighted Lebesgue or Sobolev classes. Rev. Roum. Math. Pures Appl. 46, 67–75 (2001) · Zbl 1049.31007 [17] Pavlović, M.: Mean values of harmonic congugates in the unit disc. Complex Var. Theory Appl. 10, 53–65 (1988) · Zbl 0648.30030 [18] Pavlović, M.: Inequalities for the gradient of eigenfunctions of the invariant Laplacian on the unit ball. Indag. Math. (N.S.) 2, 89–98 (1991) · Zbl 0731.32003 [19] Pavlović, M.: A proof of the Hardy-Littlewood theorem on fractional integration and a generalization. Publ. Inst. Math. (Belgrade) 59, 31–38 (1996) · Zbl 0944.31007 [20] Pavlović, M.: On subharmonic behavior and oscillation of functions on balls in $$\mathbb{R}$$ n . Publ. Inst. Math. (Beograd) 55(69), 18–22 (1994) · Zbl 0824.31003 [21] Pavlović, M.: Subharmonic behavior of smooth functions. Mat. Vesn. 48, 15–21 (1996) · Zbl 0944.31003 [22] Pavlović, M.: Decompositions of $${\mathcal{L}}^p$$ and Hardy spaces of polyharmonic functions. J. Math. Anal. Appl. 216, 499–509 (1997) · Zbl 0912.46032 [23] Pavlović, M.: On Dyakonov’s paper ’Equivalent norms on Lipschitz-type spaces of holomorphic functions’. Acta Math. 183, 141–143 (1999) · Zbl 0989.46011 [24] Pavlović, M.: Introduction to Function Spaces on the Disk. Posebna Izdanja 20, Matematicki Institut SANU (2004) · Zbl 1107.30001 [25] Pavlović, M.: Lipschitz conditions on the modulus of a harmonic function. Rev. Mat. Iberoam. 23, 831–845 (2007) · Zbl 1148.31003 [26] Riihentaus, J.: On a theorem of Avanissian–Arsove. Expo. Math. 7, 69–72 (1989) · Zbl 0677.31004 [27] Riihentaus, J.: Subharmonic functions: Non-tangential and tangential boundary behavior. In: Mustonen, V., Rákosnik, J. (eds.) Function Spaces, Differential Operators and Nonlinear Analysis (FSDONA’99), Proceedings of the Syöte Conference 1999, pp. 229–238. Math. Inst., Czech Acad. Science, Praha, (2000) (ISBN 80-85823-42-X) [28] Riihentaus, J.: A generalized mean value inequality for subharmonic functions. Expo. Math. 19, 187–190 (2001) · Zbl 0979.31002 [29] Riihentaus, J.: A generalized mean value inequality for subharmonic functions and applications. arXiv:math.CA/0302261 v1, 21 Feb (2003) [30] Riihentaus, J.: A weighted boundary behavior result for subharmonic functions. Adv. Algebra Anal. 1, 27–38 (2006) · Zbl 1170.31300 [31] Rudin, W.: Function Theory in the Unit Ball of $$\mathbb{C}$$ n . Springer, New York (1980) · Zbl 0495.32001 [32] Stein, E.M., Weiss, G.: On the theory of harmonic functions of several variables I. The theory of H p spaces. Acta Math. 103, 25–62 (1960) · Zbl 0097.28501 [33] Stoll, M.: Invariant Potential Theory in the Unit Ball of $$\mathbb{C}$$ n . London Mathematical Society Lecture Notes Series, Cambridge (1994) · Zbl 0797.31001 [34] Stoll, M.: Boundary limits and non-integrability of $${\mathcal{M}}$$ -subharmonic functions in the unit ball of $$\mathbb{C}$$ n (n 1). Trans. Am. Math. Soc. 349, 3773–3785 (1997) · Zbl 0880.31003 [35] Stoll, M.: Weighted tangential boundary limits of subharmonic functions on domains in $$\mathbb{R}$$ n (n 2). Math. Scand. 83, 300–308 (1998) · Zbl 0935.31002 [36] Suzuki, N.: Nonintegrability of harmonic functions in a domain. Jpn. J. Math. 16, 269–278 (1990) · Zbl 0728.31002 [37] Suzuki, N.: Nonintegrability of superharmonic functions. Proc. Am. Math. Soc. 113, 113–115 (1991) · Zbl 0734.31002 [38] Torchinsky, A.: Real-Variable Methods in Harmonic Analysis. Academic, London (1986) · Zbl 0621.42001 [39] Vuorinen, M.: Conformal Geometry and Quasiregular Mappings. Lecture Notes in Mathematics 1319, Springer, Berlin (1988) · Zbl 0646.30025
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