Tohoku Mathematical Journal

Continuity properties of Riesz potentials of Orlicz functions

Yoshihiro Mizuta and Tetsu Shimomura

Full-text: Open access

Abstract

In this paper we are concerned with Sobolev type inequalities for Riesz potentials of functions in Orlicz classes. As an application, we study continuity properties of Riesz potentials.

Article information

Source
Tohoku Math. J. (2), Volume 61, Number 2 (2009), 225-240.

Dates
First available in Project Euclid: 24 June 2009

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1245849445

Digital Object Identifier
doi:10.2748/tmj/1245849445

Mathematical Reviews number (MathSciNet)
MR2541407

Zentralblatt MATH identifier
1181.46026

Subjects
Primary: 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems
Secondary: 31B15: Potentials and capacities, extremal length

Citation

Mizuta, Yoshihiro; Shimomura, Tetsu. Continuity properties of Riesz potentials of Orlicz functions. Tohoku Math. J. (2) 61 (2009), no. 2, 225--240. doi:10.2748/tmj/1245849445. https://projecteuclid.org/euclid.tmj/1245849445


Export citation

References

  • D. R. Adams and R. Hurri-Syrjänen, Vanishing exponential integrability for functions whose gradients belong to $L^n (\log (e+L))^\alpha$, J. Funct. Anal. 197 (2003), 162--178.
  • A. Cianchi, Strong and weak type inequalities for some classical operators in Orlicz spaces, J. London Math. Soc. 60 (1999), 187--202.
  • A. Alberico and A. Cianchi, Differentiability properties of Orlicz-Sobolev functions, Ark. Mat. 43 (2005), 1--28.
  • D. E. Edmunds and W. D. Evans, Hardy operators, function spaces and embeddings, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2004.
  • D. E. Edmunds, P. Gurka and B. Opic, Double exponential integrability, Bessel potentials and embedding theorems, Studia Math. 115 (1995), 151--181.
  • D. E. Edmunds, P. Gurka and B. Opic, Sharpness of embeddings in logarithmic Bessel-potential spaces, Proc. Royal Soc. Edinburgh. 126 (1996), 995--1009.
  • L. I. Hedberg, On certain convolution inequalities, Proc. Amer. Math. Soc. 36 (1972), 505--510.
  • N. G. Meyers, A theory of capacities for potentials in Lebesgue classes, Math. Scand. 8 (1970), 255--292.
  • Y. Mizuta, Continuity properties of Riesz potentials and boundary limits of Beppo Levi functions, Math. Scand. 63 (1988), 238--260.
  • Y. Mizuta, Continuity properties of potentials and Beppo-Levi-Deny functions, Hiroshima Math. J. 23 (1993), 79--153.
  • Y. Mizuta, Potential theory in Euclidean spaces, Gakkōtosyo, Tokyo, 1996.
  • Y. Mizuta and T. Shimomura, Exponential integrability for Riesz potentials of functions in Orlicz classes, Hiroshima Math. J. 28 (1998), 355--371.
  • Y. Mizuta and T. Shimomura, Differentiability and Hölder continuity of Riesz potentials of functions in Orlicz classes, Analysis 20 (2000), 201--223.
  • Y. Mizuta and T. Shimomura, Vanishing exponential integrability for Riesz potentials of functions in Orlicz classes, Illinois J. Math. 51 (2007), 1039--1060.
  • E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, 1970.