Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Equivalent quasi-norms on Lorentz spaces

Authors: David E. Edmunds and Bohumír Opic
Journal: Proc. Amer. Math. Soc. 131 (2003), 745-754
MSC (2000): Primary 46E30, 26D10, 47B38, 47G10
Published electronically: October 15, 2002
MathSciNet review: 1937412
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We give new characterizations of Lorentz spaces by means of certain quasi-norms which are shown to be equivalent to the classical ones.

References [Enhancements On Off] (What's this?)

  • [AH] D. R. Adams and L. I. Hedberg, Function spaces and potential theory, Springer, Berlin, 1996. MR 97j:46024
  • [BR] C. Bennett and K. Rudnick, On Lorentz-Zygmund spaces, Dissert. Math. 175 (1980), 1-72. MR 81i:42020
  • [BS] C. Bennett and R. Sharpley, Interpolation of operators, Pure and Appl. Math. 129, Academic Press, New York, 1988. MR 89e:46001
  • [C] A. P. Calderón, Spaces between $L^{1}$ and $L^{\infty }$ and the theorem of Marcinkiewicz, Studia Math. 26 (1966), 273-299. MR 34:3295
  • [CKOP] A. Cianchi, R. Kerman, B. Opic and L. Pick, Sharp rearrangement inequality for the fractional maximal operator, Studia Math. 138 (2000), 277-284. MR 2001h:42029
  • [EGO] D. E. Edmunds P. Gurka and B. Opic, Double exponential integrability of convolution operators in generalized Lorentz-Zygmund spaces, Indiana Univ. Math. J. 44 (1995), 19-43. MR 96f:47048
  • [EO] D. E. Edmunds and B. Opic, Boundedness of fractional maximal operators between classical and weak-type Lorentz spaces, Research Report No: 2000-15, CMAIA, University of Sussex at Brighton, 2000, 40 pp. (to appear in Dissert. Math. (2000)).
  • [La] S. Lai, Weighted inequalities for general operators on monotone functions, Trans. Amer. Math. Soc. 340 (1993), 811-836. MR 94b:42005
  • [Lo1] G. G. Lorentz, Some new function spaces, Ann. of Math. 51 (1950), 37-55. MR 11:442d
  • [Lo2] G. G. Lorentz, On the theory of spaces $\Lambda $, Pacific J. Math. 1 (1951), 411-429. MR 13:470c
  • [MW] B. Muckenhoupt and R. L. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc. 192 (1974), 261-274. MR 49:5275
  • [O] B. Opic, New characterizations of Lorentz spaces (to appear in Proc. Royal Soc. Edinburgh, Section A).
  • [OK] B. Opic and A. Kufner, Hardy-type inequalities, Pitman Research Notes in Math., Series 219, Longman Sci. & Tech., Harlow, 1990. MR 92b:26028
  • [OP] B. Opic and L. Pick, On generalized Lorentz-Zygmund spaces, Math. Inequal. 2 (1999), 391-467. MR 2000m:46067
  • [S] E. T. Sawyer, Boundedness of classical operators on classical Lorentz spaces, Studia Math. 96 (1990), 145-158. MR 91d:26026

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46E30, 26D10, 47B38, 47G10

Retrieve articles in all journals with MSC (2000): 46E30, 26D10, 47B38, 47G10

Additional Information

David E. Edmunds
Affiliation: Centre for Mathematical Analysis and Its Applications, University of Sussex, Falmer, Brighton BN1 9QH, England

Bohumír Opic
Affiliation: Mathematical Institute, Academy of Sciences of the Czech Republic, Žitná 25, 115 67 Praha 1, Czech Republic

Keywords: Lorentz spaces, equivalent quasi-norms, weighted norm inequalities, fractional maximal operators, Riesz potentials, Hilbert transform
Received by editor(s): July 1, 2001
Published electronically: October 15, 2002
Additional Notes: This research was supported by NATO Collaborative Research Grant PST.CLG 970071 and by grant no.201/01/0333 of the Grant Agency of the Czech Republic
Communicated by: Andreas Seeger
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society