Weak L ∞ and BMO in Metric Spaces

Daniel Aalto

Bollettino dell'Unione Matematica Italiana (2012)

  • Volume: 5, Issue: 2, page 369-385
  • ISSN: 0392-4041

Abstract

top
Bennett, DeVore and Sharpley introduced the space weak L ∞ in 1981 and studied its relationship with functions of bounded mean oscillation. Here we characterize the weak L ∞ in measure spaces without using the decreasing rearrangement of a function. Instead, we use exponential estimates for the distribution function. In addition, we consider a localized version of the characterization that leads to a new characterization of BMO.

How to cite

top

Aalto, Daniel. "Weak $L^\infty$ and BMO in Metric Spaces." Bollettino dell'Unione Matematica Italiana 5.2 (2012): 369-385. <http://eudml.org/doc/290888>.

@article{Aalto2012,
abstract = {Bennett, DeVore and Sharpley introduced the space weak $L^\{\infty\}$ in 1981 and studied its relationship with functions of bounded mean oscillation. Here we characterize the weak $L^\{\infty\}$ in measure spaces without using the decreasing rearrangement of a function. Instead, we use exponential estimates for the distribution function. In addition, we consider a localized version of the characterization that leads to a new characterization of BMO.},
author = {Aalto, Daniel},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {369-385},
publisher = {Unione Matematica Italiana},
title = {Weak $L^\infty$ and BMO in Metric Spaces},
url = {http://eudml.org/doc/290888},
volume = {5},
year = {2012},
}

TY - JOUR
AU - Aalto, Daniel
TI - Weak $L^\infty$ and BMO in Metric Spaces
JO - Bollettino dell'Unione Matematica Italiana
DA - 2012/6//
PB - Unione Matematica Italiana
VL - 5
IS - 2
SP - 369
EP - 385
AB - Bennett, DeVore and Sharpley introduced the space weak $L^{\infty}$ in 1981 and studied its relationship with functions of bounded mean oscillation. Here we characterize the weak $L^{\infty}$ in measure spaces without using the decreasing rearrangement of a function. Instead, we use exponential estimates for the distribution function. In addition, we consider a localized version of the characterization that leads to a new characterization of BMO.
LA - eng
UR - http://eudml.org/doc/290888
ER -

References

top
  1. AALTO, D. - KINNUNEN, J., The discrete maximal operator in metric spaces, J. Anal. Math., 111 (2010). Zbl1210.42029MR2747071DOI10.1007/s11854-010-0022-3
  2. BASTERO, J. - MILMAN, M. - RUIZ, B. F. J., A note on L ⁢ ( ∞ , q ) spaces and Sobolev embeddings, Indiana Univ. Math. J., 52, 5 (2003), 1215-1230. Zbl1098.46023MR2010324DOI10.1512/iumj.2003.52.2364
  3. BENNETT, C. - DEVORE, R. A. - SHARPLEY, R., Weak- L ∞ and BMO, Ann. of Math. (2), 113 (1981), 601-611. MR621018DOI10.2307/2006999
  4. BENNETT, C. - SHARPLEY, R., Interpolation of operators, vol. 129 of Pure and Applied Mathematics, Academic Press Inc. (Boston, MA, 1988). Zbl0647.46057MR928802
  5. CHIARENZA, F. - FRASCA, M., Morrey spaces and Hardy-Littlewood maximal function, Rend. Mat. Appl. (7), 7 (1987), 3-4 (1988), 273-279. Zbl0717.42023MR985999
  6. COIFMAN, R. R. - WEISS, G., Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Mathematics, 242 (Springer-Verlag, Berlin, 1971), Étude de certaines intégrales singulières. Zbl0224.43006MR499948
  7. HEINONEN, J., Lectures on analysis on metric spaces, Universitext (Springer-Verlag, New York, 2001). Zbl0985.46008MR1800917DOI10.1007/978-1-4613-0131-8
  8. KORENOVSKII, A., Mean oscillations and equimeasurable rearrangements of functions, vol. 4 of Lecture Notes of the Unione Matematica Italiana, (Springer, Berlin, 2007). Zbl1133.42035MR2363526DOI10.1007/978-3-540-74709-3
  9. MACMANUS, P. - PÉREZ, C., Generalized Poincaré inequalities: sharp self-improving properties, Internat. Math. Res. Notices, 2 (1998), 101-116. MR1604816DOI10.1155/S1073792898000099
  10. MALÝ, J. - PICK, L., An elementary proof of sharp Sobolev embeddings, Proc. Amer. Math. Soc., 130 (2002), 555-563. Zbl0990.46022MR1862137DOI10.1090/S0002-9939-01-06060-9
  11. MATEU, J. - MATTILA, P. - NICOLAU, A. - OROBITG, J., BMO for nondoubling measures, Duke Math. J., 102, 3 (2000), 533-565. MR1756109DOI10.1215/S0012-7094-00-10238-4
  12. MILMAN, M. - PUSTYLNIK, E., On sharp higher order Sobolev embeddings, Commun. Contemp. Math., 6, 3 (2004), 495-511. Zbl1108.46029MR2068850DOI10.1142/S0219199704001380
  13. RIVIÈRE, N. M., Interpolation à la Marcinkiewicz, Rev. Un. Mat. Argentina, 25 (1970/71), 363-377. Collection of articles dedicated to Alberto González Domínguez on his sixty-fifth birthday. MR370169

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.