Variable Lebesgue norm estimates for BMO functions

Mitsuo Izuki; Yoshihiro Sawano

Czechoslovak Mathematical Journal (2012)

  • Volume: 62, Issue: 3, page 717-727
  • ISSN: 0011-4642

Abstract

top
In this paper, we are going to characterize the space through variable Lebesgue spaces and Morrey spaces. There have been many attempts to characterize the space by using various function spaces. For example, Ho obtained a characterization of 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.

How to cite

top

Izuki, Mitsuo, and Sawano, Yoshihiro. "Variable Lebesgue norm estimates for BMO functions." Czechoslovak Mathematical Journal 62.3 (2012): 717-727. <http://eudml.org/doc/246501>.

@article{Izuki2012,
abstract = {In this paper, we are going to characterize the space $\{\rm BMO\}(\{\mathbb \{R\}\}^n)$ through variable Lebesgue spaces and Morrey spaces. There have been many attempts to characterize the space $\{\rm BMO\}(\{\mathbb \{R\}\}^n)$ by using various function spaces. For example, Ho obtained a characterization of $\{\rm BMO\}(\{\mathbb \{R\}\}^n)$ 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.},
author = {Izuki, Mitsuo, Sawano, Yoshihiro},
journal = {Czechoslovak Mathematical Journal},
keywords = {variable exponent; Morrey space; BMO; variable exponent; Morrey space; BMO},
language = {eng},
number = {3},
pages = {717-727},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Variable Lebesgue norm estimates for BMO functions},
url = {http://eudml.org/doc/246501},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Izuki, Mitsuo
AU - Sawano, Yoshihiro
TI - Variable Lebesgue norm estimates for BMO functions
JO - Czechoslovak Mathematical Journal
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 3
SP - 717
EP - 727
AB - In this paper, we are going to characterize the space ${\rm BMO}({\mathbb {R}}^n)$ through variable Lebesgue spaces and Morrey spaces. There have been many attempts to characterize the space ${\rm BMO}({\mathbb {R}}^n)$ by using various function spaces. For example, Ho obtained a characterization of ${\rm BMO}({\mathbb {R}}^n)$ 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.
LA - eng
KW - variable exponent; Morrey space; BMO; variable exponent; Morrey space; BMO
UR - http://eudml.org/doc/246501
ER -

References

top
  1. Cruz-Uribe, D., Diening, L., Fiorenza, A., A new proof of the boundedness of maximal operators on variable Lebesgue spaces, Boll. Unione Mat. Ital. (9) 2 (2009), 151-173. (2009) Zbl1207.42011MR2493649
  2. Cruz-Uribe, D., Fiorenza, A., Neugebauer, C. J., The maximal function on variable spaces, Ann. Acad. Sci. Fenn. Math. 28 (2003), 223-238 29 (2004), 247-249. (2004) MR2041952
  3. Diening, L., Maximal function on generalized Lebesgue spaces , Math. Inequal. Appl. 7 (2004), 245-253. (2004) MR2057643
  4. Diening, L., 10.1016/j.bulsci.2003.10.003, Bull. Sci. Math. 129 (2005), 657-700. (2005) MR2166733DOI10.1016/j.bulsci.2003.10.003
  5. Diening, L., Harjulehto, P., Hästö, P., Mizuta, Y., Shimomura, T., Maximal functions in variable exponent spaces: limiting cases of the exponent, Ann. Acad. Sci. Fenn. Math. 34 (2009), 503-522. (2009) MR2553809
  6. Diening, L., Harjulehto, P., Hästö, P., Růžička, M., Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Math. {2017} Springer, Berlin (2011). (2011) Zbl1222.46002MR2790542
  7. Ho, K., 10.1016/j.exmath.2009.02.007, Expo. Math. 27 (2009), 363-372. (2009) Zbl1174.42025MR2567029DOI10.1016/j.exmath.2009.02.007
  8. Ho, K., 10.32917/hmj/1314204559, Hiroshima Math. J. 41 (2011), 153-165. (2011) Zbl1227.42024MR2849152DOI10.32917/hmj/1314204559
  9. Izuki, M., 10.1007/s12215-010-0015-1, Rend. Circ. Mat. Palermo 59 (2010), 199-213. (2010) Zbl1202.42029MR2670690DOI10.1007/s12215-010-0015-1
  10. John, F., Nirenberg, L., 10.1002/cpa.3160140317, Comm. Pure Appl. Math. 14 (1961), 415-426. (1961) Zbl0102.04302MR0131498DOI10.1002/cpa.3160140317
  11. Kováčik, O., Rákosník, J., On spaces and , Czech. Math. J. 41 (1991), 592-618. (1991) MR1134951
  12. Lerner, A. K., 10.1090/S0002-9947-10-05066-X, Trans. Amer. Math. Soc. 362 (2010), 4229-4242. (2010) MR2608404DOI10.1090/S0002-9947-10-05066-X
  13. Luxenberg, W. A. J., Banach Function Spaces, Technische Hogeschool te Delft Assen (1955) 0072440. 
  14. Nakano, H., Modulared Semi-Ordered Linear Spaces, Maruzen Co., Ltd. Tokyo (1950) 0038565. Zbl0041.23401MR0038565
  15. Nakano, H., Topology of Linear Topological Spaces, Maruzen Co., Ltd. Tokyo (1951) 0046560. MR0046560
  16. Sawano, Y., Sugano, S., Tanaka, H., 10.1007/s11118-011-9239-8, Potential Anal. 36 (2012), 517-556. (2012) Zbl1242.42017MR2904632DOI10.1007/s11118-011-9239-8
  17. Sawano, Y., Sugano, S., Tanaka, H., Olsen's inequality and its applications to Schrödinger equations, RIMS Kôkyûroku Bessatsu B26 (2011), 51-80. (2011) Zbl1236.42018MR2883846

NotesEmbed ?

top

You must be logged in to post comments.