Abstract
In this paper, we are going to characterize the space BMO(ℝn) through variable Lebesgue spaces and Morrey spaces. There have been many attempts to characterize the space BMO(ℝn) by using various function spaces. For example, Ho obtained a characterization of BMO(ℝn) with respect to rearrangement invariant spaces. However, variable Lebesgue spaces and Morrey spaces do not appear in the characterization. One of the reasons is that these spaces are not rearrangement invariant. We also obtain an analogue of the well-known John-Nirenberg inequality which can be seen as an extension to the variable Lebesgue spaces.
Similar content being viewed by others
References
D. Cruz-Uribe, L. Diening, A. Fiorenza: A new proof of the boundedness of maximal operators on variable Lebesgue spaces. Boll. Unione Mat. Ital. (9) 2 (2009), 151–173.
D. Cruz-Uribe, A. Fiorenza, C. J. Neugebauer: The maximal function on variable L p spaces. Ann. Acad. Sci. Fenn. Math. 28 (2003), 223–238; 29 (2004), 247–249.
L. Diening: Maximal function on generalized Lebesgue spaces L p(·). Math. Inequal. Appl. 7 (2004), 245–253.
L. Diening: Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces. Bull. Sci. Math. 129 (2005), 657–700.
L. Diening, P. Harjulehto, P. Hästö, Y. Mizuta, T. Shimomura: Maximal functions in variable exponent spaces: limiting cases of the exponent. Ann. Acad. Sci. Fenn. Math. 34 (2009), 503–522.
L. Diening, P. Harjulehto, P. Hästö, M. Růžička: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Math. 2017, Springer, Berlin, 2011.
K. Ho: Characterization of BMO in terms of rearrangement-invariant Banach function spaces. Expo. Math. 27 (2009), 363–372.
K. Ho: Characterizations of BMO by A p weights and p-convexity. Hiroshima Math. J. 41 (2011), 153–165.
M. Izuki: Boundedness of commutators on Herz spaces with variable exponent. Rend. Circ. Mat. Palermo 59 (2010), 199–213.
F. John, L. Nirenberg: On functions of bounded mean oscillation. Comm. Pure Appl. Math. 14 (1961), 415–426.
O. Kováčik, J. Rákosník: On spaces L p(x) and W k,p(x). Czech. Math. J. 41 (1991), 592–618.
A.K. Lerner: On some questions related to the maximal operator on variable L p spaces. Trans. Amer. Math. Soc. 362 (2010), 4229–4242.
W. A. J. Luxenberg: Banach Function Spaces. Technische Hogeschool te Delft, Assen, 1955.
H. Nakano: Modulared Semi-Ordered Linear Spaces. Maruzen Co., Ltd., Tokyo, 1950.
H. Nakano: Topology of Linear Topological Spaces. Maruzen Co., Ltd., Tokyo, 1951.
Y. Sawano, S. Sugano, H. Tanaka: Orlicz-Morrey spaces and fractional operators. Potential Anal. 36 (2012), 517–556.
Y. Sawano, S. Sugano, H. Tanaka: Olsen’s inequality and its applications to Schrödinger equations. RIMS Kôkyûroku Bessatsu B26 (2011), 51–80
Author information
Authors and Affiliations
Corresponding author
Additional information
The research has been supported by Osaka City University Advanced Mathematical Institute. This work is partially supported by Grant-in-Aid for Young Scientists (B) No. 21740104, Japan Society for the Promotion of Science.
Rights and permissions
About this article
Cite this article
Izuki, M., Sawano, Y. Variable Lebesgue norm estimates for BMO functions. Czech Math J 62, 717–727 (2012). https://doi.org/10.1007/s10587-012-0042-5
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10587-012-0042-5