Hostname: page-component-7b7567c8c-wxvwp Total loading time: 0 Render date: 2026-11-15T15:19:34.465Z Has data issue: false hasContentIssue false
  • English
  • Français

On the domain and range of the maximal operator

Published online by Cambridge University Press:  22 January 2016

Alberto Fiorenza and
Miroslav Krbec
Alberto Fiorenza
Affiliation:
Dipartimento di Costruzioni e Metodi Matematici in Architettura, via Monteoliveto, 3, 80134, Napoli, Italy, fiorenza@cds.unina.it
Miroslav Krbec
Affiliation:
Institute of Mathematics, Academy of Sciences of the Czech Republic, Žitá 25, 115 67 Prague 1, Czech Republic, krbecm@matsrv.math.cas.cz
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We give a detailed survey, known and new results on the domain and the range of the maximal operator. In particular we employ the grand Lp spaces and logarithmic Lebesgue spaces.

Keywords

42B2546E30

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 158 , December 2000, pp. 43 - 61
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2000

References

[B] Bandle, C., Isoperimetric Inequalities and Applications, Monographs and Studies in Math. 7, Pitman, London, 1980.Google Scholar
[BI] Bojarski, B. and Iwaniec, T., Analytical foundations of the theory of quasiconformal mappings in R N , Ann. Ac. Sc. Fen., 8 (1983), 257–324.Google Scholar
[BR] Bennett, C. and Rudnick, K., On Lorentz-Zygmund spaces, Dissertationes Math. (Rozprawy Mat.), 175 (1980), 1–72.Google Scholar
[BS] Bennett, C. and Sharpley, R., Interpolation of Operators, Academic Press, Boston, 1988.Google Scholar
[CP] Caffarelli, L. A. and Peral, I., On W1,p estimates of elliptic equations in divergence form, Comm. Pure Appl. Math., 51 (1998), 1–21.3.0.CO;2-G>CrossRefGoogle Scholar
[DS] DeVore, R. A. and Sharpley, R. C., Maximal functions measuring smoothness, American Mathematical Society, Memoirs of the AMS vol. 47, 293 (1984).Google Scholar
[ET] Edmunds, D. E. and Triebel, H., Logarithmic Sobolev spaces and their applications to spectral theory, Proc. London Math. Soc. (3), 71 (1995), 333–371.Google Scholar
[F1] Fiorenza, A., A note on the spherical maximal function, Rend. Accad. Sci. Fis. Mat. Napoli, 65 (1987), 77–83.Google Scholar
[F2] Fiorenza, A., Some remarks on Stein’s Llog L result, Diff. and Int. Equations, 5 (6) (1992), 1355–1362.CrossRefGoogle Scholar
[FS1] Fusco, N. and Sbordone, C., Higher integrability from reverse Jensen inequalities with different supports, Partial Differential Equations and the Calculus of Variations: Essays in Honor of Ennio De Giorgi, Progress in Non-linear Differential Equations and their Applications, Birkhaüser Boston Inc. (1989), pp. 541–561.Google Scholar
[FS2] Fusco, N. and Sbordone, C., Higher integrability of the gradient of minimizers of functionals with nonstandard growth conditions, Comm. Pure Appl. Math., 43 (1990), 673–683.Google Scholar
[GCRF] Garcéıa-Cuerva, J. and de Francia, J. L. Rubio, Weighted Norm Inequalities and Related Topics, North Holland, Amsterdam, 1985.Google Scholar
[GIM] Greco, L., Iwaniec, T. and Moscariello, G., Limits of the improved integrability of the volume forms, Indiana Univ. Math. J., 44 (1995), 305–339.Google Scholar
[Gr] Greco, L., A Remark on the Equality det Du = DetDu, Diff. and Int. Equations, 6 (1993), 1089–1100.Google Scholar
[Gu] Guzméan, M. de, Differentiation of Integrals in R n , Lecture Notes in Math. vol. 481, Springer-Verlag, Berlin-Heidelberg-New York, 1975.Google Scholar
[IS] Iwaniec, T. and Sbordone, C., On the integrability of the Jacobian under minimal hypothesis, Arch. Rat. Mech. Anal., 119 (1992), 129–143.Google Scholar
[K] Kawohl, B., Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Math. vol. 1150, Springer Verlag, Berlin-New York, 1985.CrossRefGoogle Scholar
[KK] Kokilashvili, V. and Krbec, M., Weighted Inequalities in Lorentz and Orlicz Spaces, World Scientific, Singapore, 1991.Google Scholar
[LZ] Li, C. and Zhang, K., Higher integrability of certain bilinear forms on Orlicz Spaces, Macquarie Math. Reports, Sydney (1994), pp. 94–157.Google Scholar
[Ma] Maz’ya, V. G., Sobolev Spaces, Springer-Verlag, Berlin, 1985.Google Scholar
[Mi] Milman, M., Extrapolation and Optimal Decompositions, Springer-Verlag, Berlin, 1994.Google Scholar
[Mo] Mossino, J., Inégalités Isopérimétriques et Applications en Physique, Collection Travaux en Cours, Hermann, Paris (1984).Google Scholar
[Sa] Sadosky, C., Interpolation of Operators and Singular Integrals, M. Dekker Inc., New York, 1979.Google Scholar
[Sc] Schwartz, L., Théorie des Distributions. Tome II, Publications de l’Institut de Mathématique de l’Université de Strasbourg, vol X, Hermann, Paris, 1957.Google Scholar
[S1] Stein, E., Note on the class Llog L, Studia Math., 31 (1969), 305–310.CrossRefGoogle Scholar
[S2] Stein, E. M., Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, New Jersey, 1970.CrossRefGoogle Scholar
[S3] Stein, E. M., Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, Princeton, New Jersey, 1993.Google Scholar
[T] Torchinsky, A., Real Variable Methods in Harmonic Analysis, Academic Press, San Diego, 1986.Google Scholar
[Wik] Wik, I., A comparison of the integrability of f and Mf with that of f#, Preprint No. 2, University of Umeå (1983), pp. 1–30.Google Scholar
[Wie] Wiener, N., The ergodic theorem, Duke Math. J., 5 (1939), 1–18.Google Scholar