A New Proof of the Boundedness of Maximal Operators on Variable Lebesgue Spaces

D. Cruz-Uribe; L. Diening; A. Fiorenza

Bollettino dell'Unione Matematica Italiana (2009)

  • Volume: 2, Issue: 1, page 151-173
  • ISSN: 0392-4041

Abstract

top
We give a new proof using the classic Calderón-Zygmund decomposition that the Hardy-Littlewood maximal operator is bounded on the variable Lebesgue space L p ⁢ ( ⋅ ) whenever the exponent function p ⁢ ( ⋅ ) satisfies log-Hölder continuity conditions. We include the case where p ⁢ ( ⋅ ) assumes the value infinity. The same proof also shows that the fractional maximal operator M a , 0 < a < n , maps L p ⁢ ( ⋅ ) into L q ⁢ ( ⋅ ) , where 1 / p ⁢ ( ⋅ ) - 1 / q ⁢ ( ⋅ ) = a / n .

How to cite

top

Cruz-Uribe, D., Diening, L., and Fiorenza, A.. "A New Proof of the Boundedness of Maximal Operators on Variable Lebesgue Spaces." Bollettino dell'Unione Matematica Italiana 2.1 (2009): 151-173. <http://eudml.org/doc/290576>.

@article{Cruz2009,
abstract = {We give a new proof using the classic Calderón-Zygmund decomposition that the Hardy-Littlewood maximal operator is bounded on the variable Lebesgue space $L^\{p(\cdot)\}$ whenever the exponent function $p(\cdot)$ satisfies log-Hölder continuity conditions. We include the case where $p(\cdot)$ assumes the value infinity. The same proof also shows that the fractional maximal operator $M_\{a\}$, $0 < a < n$, maps $L^\{p(\cdot)\}$ into $L^\{q(\cdot)\}$, where $1/p(\cdot) - 1/q(\cdot) = a/n$.},
author = {Cruz-Uribe, D., Diening, L., Fiorenza, A.},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {151-173},
publisher = {Unione Matematica Italiana},
title = {A New Proof of the Boundedness of Maximal Operators on Variable Lebesgue Spaces},
url = {http://eudml.org/doc/290576},
volume = {2},
year = {2009},
}

TY - JOUR
AU - Cruz-Uribe, D.
AU - Diening, L.
AU - Fiorenza, A.
TI - A New Proof of the Boundedness of Maximal Operators on Variable Lebesgue Spaces
JO - Bollettino dell'Unione Matematica Italiana
DA - 2009/2//
PB - Unione Matematica Italiana
VL - 2
IS - 1
SP - 151
EP - 173
AB - We give a new proof using the classic Calderón-Zygmund decomposition that the Hardy-Littlewood maximal operator is bounded on the variable Lebesgue space $L^{p(\cdot)}$ whenever the exponent function $p(\cdot)$ satisfies log-Hölder continuity conditions. We include the case where $p(\cdot)$ assumes the value infinity. The same proof also shows that the fractional maximal operator $M_{a}$, $0 < a < n$, maps $L^{p(\cdot)}$ into $L^{q(\cdot)}$, where $1/p(\cdot) - 1/q(\cdot) = a/n$.
LA - eng
UR - http://eudml.org/doc/290576
ER -

References

top
  1. BENNETT, C. - SHARPLEY, R., Interpolation of operators, volume 129 of Pure and Applied Mathematics. Academic Press Inc. (Boston, MA, 1988). Zbl0647.46057MR928802
  2. CAPONE, C. - CRUZ-URIBE, D. - FIORENZA, A., The fractional maximal operator and fractional integrals on variable L p spaces. Revista Math. Iberoamericana, 23, 3 (2007), 743-770. Zbl1213.42063MR2414490DOI10.4171/RMI/511
  3. CRUZ-URIBE, D., New proofs of two-weight norm inequalities for the maximal operator. Georgian Math. J., 7, 1 (2000), 33-42. Zbl0987.42019MR1768043
  4. CRUZ-URIBE, D. - FIORENZA, A. - NEUGEBAUER, C. J., The maximal function on variable L p spaces. Ann. Acad. Sci. Fenn. Math., 28, 1 (2003), 223-238. See errata [5]. Zbl1037.42023MR1976842
  5. CRUZ-URIBE, D. - FIORENZA, A. - NEUGEBAUER, C. J., Corrections to: "The maximal function on variable L p spaces" [Ann. Acad. Sci. Fenn. Math., 28, no. 1 (2003), 223- 238]. Ann. Acad. Sci. Fenn. Math., 29, 1 (2004), 247-249. Zbl1037.42023MR2041952
  6. DIENING, L., Maximal function on generalized Lebesgue spaces L p ⁢ ( ⋅ ) . Math. Inequal. Appl., 7, 2 (2004), 245-253. Zbl1071.42014MR2057643DOI10.7153/mia-07-27
  7. DIENING, L., Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces. Bull. Sci. Math., 129, 8 (2005), 657-700. Zbl1096.46013MR2166733DOI10.1016/j.bulsci.2003.10.003
  8. DIENING, L., Habilitation, Universität Freiburg, 2007. 
  9. DIENING, L. - HARJULEHTO, P. - HÄSTO, P. - MIZUTA, Y. - SHIMOMURA, T., Maximal functions in variable exponent spaces: limiting cases of the exponent. Preprint, 2007. MR2553809
  10. DIENING, L. - HÄSTO, P. - NEKVINDA, A., Open problems in variable exponent Lebesgue and Sobolev spaces. In FSDONA04 Proceedings ( Drabek and Rakosnik (eds.); Milovy, Czech Republic, pages 38-58. Academy of Sciences of the Czech Republic (Prague, 2005). 
  11. DUOANDIKOETXEA, J., Fourier analysis, volume 29 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2001. MR1800316
  12. FAN, X. - ZHAO, D., On the spaces L p ⁢ ( x ) ⁢ ( Ω ) and W m , p ⁢ ( x ) ⁢ ( Ω ) . J. Math. Anal. Appl., 263, 2 (2001), 424-446. Zbl1028.46041MR1866056DOI10.1006/jmaa.2000.7617
  13. GARCÍA-CUERVA, J. - RUBIO DE FRANCIA, J. L., Weighted norm inequalities and related topics, volume 116 of North-Holland Mathematics Studies (North-Holland Publishing Co., Amsterdam, 1985). Zbl0578.46046MR807149
  14. KOKILASHVILI, V. - SAMKO, S., On Sobolev theorem for Riesz-type potentials in Lebesgue spaces with variable exponent. Z. Anal. Anwendungen, 22, 4 (2003), 899-910. Zbl1040.42013MR2036935DOI10.4171/ZAA/1178
  15. KOVÁČIK, O. - RÁKOSNÍK, J., On spaces L p ⁢ ( x ) and W k , p ⁢ ( x ) . Czechoslovak Math. J., 41, 116 (4) (1991), 592-618. MR1134951
  16. LERNER, A. K., Some remarks on the Hardy-Littlewood maximal function on variable L p ⁢ ( x ) spaces. Math. Z., 251, 3 (2005), 509-521. Zbl1092.42009MR2190341DOI10.1007/s00209-005-0818-5
  17. MUCKENHOUPT, B. - WHEEDEN, R., Weighted norm inequalities for fractional integrals. Trans. Amer. Math. Soc., 192 (1974), 261-274. Zbl0289.26010MR340523DOI10.2307/1996833
  18. MUSIELAK, J., Orlicz spaces and modular spaces, volume 1034 of Lecture Notes in Mathematics (Springer-Verlag, Berlin, 1983). Zbl0557.46020MR724434DOI10.1007/BFb0072210
  19. NEKVINDA, A., Hardy-Littlewood maximal operator on L p ⁢ ( x ) ⁢ ( ℝ n ) . Math. Inequal. Appl., 7, 2 (2004), 255-265. Zbl1059.42016MR2057644DOI10.7153/mia-07-28
  20. NEKVINDA, A., A note on maximal operator on l p n and L p ⁢ ( x ) ⁢ ( ℝ ) . J. Funct. Spaces Appl., 5, 1 (2007), 49-88. Zbl1143.46011MR2296013DOI10.1155/2007/294367
  21. PICK, L. - RŮŽIČKA, M., An example of a space L p ⁢ ( x ) on which the Hardy-Littlewood maximal operator is not bounded. Expo. Math., 19, 4 (2001), 369-371. Zbl1003.42013MR1876258DOI10.1016/S0723-0869(01)80023-2
  22. SAMKO, S., On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators. Integral Transforms Spec. Funct., 16, 5-6 (2005), 461-482. Zbl1069.47056MR2138062DOI10.1080/10652460412331320322

Citations in EuDML Documents

top
  1. Mitsuo Izuki, Yoshihiro Sawano, Variable Lebesgue norm estimates for BMO functions
  2. Yan Lu, Yue Ping Zhu, Boundedness of some sublinear operators and commutators on Morrey-Herz spaces with variable exponents
  3. Ka Luen Cheung, Kwok-Pun Ho, Boundedness of Hardy-Littlewood maximal operator on block spaces with variable exponent
  4. Kristóf Szarvas, Ferenc Weisz, Weak- and strong-type inequality for the cone-like maximal operator in variable Lebesgue spaces
  5. Alberto Fiorenza, Categories of results in variable Lebesgue spaces theory

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.