Double exponential integrability, Bessel potentials and embedding theorems

David Edmunds; Petr Gurka; Bohumír Opic

Studia Mathematica (1995)

  • Volume: 115, Issue: 2, page 151-181
  • ISSN: 0039-3223

Abstract

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This paper is a continuation of [5] and provides necessary and sufficient conditions for double exponential integrability of the Bessel potential of functions from suitable (generalized) Lorentz-Zygmund spaces. These results are used to establish embedding theorems for Bessel potential spaces which extend Trudinger's result.

How to cite

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Edmunds, David, Gurka, Petr, and Opic, Bohumír. "Double exponential integrability, Bessel potentials and embedding theorems." Studia Mathematica 115.2 (1995): 151-181. <http://eudml.org/doc/216205>.

@article{Edmunds1995,
abstract = {This paper is a continuation of [5] and provides necessary and sufficient conditions for double exponential integrability of the Bessel potential of functions from suitable (generalized) Lorentz-Zygmund spaces. These results are used to establish embedding theorems for Bessel potential spaces which extend Trudinger's result.},
author = {Edmunds, David, Gurka, Petr, Opic, Bohumír},
journal = {Studia Mathematica},
keywords = {Bessel potential; Riesz potential, generalized Lorentz-Zygmund spaces; exponential integrability; Hardy inequality; Orlicz spaces; Bessel potential spaces; double exponential integrability of the Bessel potential; Lorentz-Zygmund spaces},
language = {eng},
number = {2},
pages = {151-181},
title = {Double exponential integrability, Bessel potentials and embedding theorems},
url = {http://eudml.org/doc/216205},
volume = {115},
year = {1995},
}

TY - JOUR
AU - Edmunds, David
AU - Gurka, Petr
AU - Opic, Bohumír
TI - Double exponential integrability, Bessel potentials and embedding theorems
JO - Studia Mathematica
PY - 1995
VL - 115
IS - 2
SP - 151
EP - 181
AB - This paper is a continuation of [5] and provides necessary and sufficient conditions for double exponential integrability of the Bessel potential of functions from suitable (generalized) Lorentz-Zygmund spaces. These results are used to establish embedding theorems for Bessel potential spaces which extend Trudinger's result.
LA - eng
KW - Bessel potential; Riesz potential, generalized Lorentz-Zygmund spaces; exponential integrability; Hardy inequality; Orlicz spaces; Bessel potential spaces; double exponential integrability of the Bessel potential; Lorentz-Zygmund spaces
UR - http://eudml.org/doc/216205
ER -

References

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  1. [1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. 
  2. [2] C. Bennett and K. Rudnick, On Lorentz-Zygmund spaces, Dissertationes Math. 175 (1980). Zbl0456.46028
  3. [3] C. Bennett and R. Sharpley, Interpolation of Operators, Pure Appl. Math. 129, Academic Press, 1988. Zbl0647.46057
  4. [4] H. Brézis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differential Equations 5 (1980), 773-789. Zbl0437.35071
  5. [5] D. E. Edmunds, P. Gurka and B. Opic, Double exponential integrability of convolution operators in generalized Lorentz-Zygmund spaces, Indiana Univ. Math. J. (1995), to appear. Zbl0826.47021
  6. [6] D. E. Edmunds and M. Krbec, Two limiting cases of Sobolev imbeddings, preprint no. 89, Math. Inst. Czech Acad. Sci., Prague, 1994, 8 pp. Zbl0835.46027
  7. [7] W. D. Evans, B. Opic and L. Pick, Interpolation of operators on scales of generalized Lorentz-Zygmund spaces, preprint no. 99, Math. Inst. Czech Acad. Sci., Prague, 1995, 57 pp. Zbl0865.46016
  8. [8] N. Fusco, P. L. Lions and C. Sbordone, Some remarks on Sobolev embeddings in borderline cases, preprint no. 25, Università degli Studi di Napoli "Federico II", 1993, 7 pp. 
  9. [9] A. Kufner, O. John and S. Fučík, Function Spaces, Academia, Prague, 1977. 
  10. [10] G. Lorentz, On the theory of spaces Λ, Pacific J. Math. 1 (1951), 411-429. Zbl0043.11302
  11. [11] F. J. Martín-Reyes and E. T. Sawyer, Weighted inequalities for Riemann-Liouville fractional integrals of order one and greater, Proc. Amer. Math. Soc. 106 (1989) 727-733. Zbl0704.42018
  12. [12] R. O'Neil, Convolution operators and L(p,q) spaces, Duke Math. J. 30 (1963), 129-142. 
  13. [13] E. T. Sawyer, Boundedness of classical operators on classical Lorentz spaces, Studia Math. 96 (1990), 145-158. Zbl0705.42014
  14. [14] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, N.J., 1970. Zbl0207.13501
  15. [15] V. D. Stepanov, Weighted inequalities for a class of Volterra convolution operators, J. London Math. Soc. 45 (1992), 232-242. Zbl0703.42011
  16. [16] V. D. Stepanov, The weighted Hardy's inequality for non-increasing functions, Trans. Amer. Math. Soc. 338 (1993), 173-186. Zbl0786.26015
  17. [17] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978. Zbl0387.46033
  18. [18] N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473-484. Zbl0163.36402
  19. [19] W. P. Ziemer, Weakly Differentiable Functions, Graduate Texts in Math. 120, Springer, Berlin, 1989. 

Citations in EuDML Documents

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  1. Robert Černý, Generalized n-Laplacian: boundedness of weak solutions to the Dirichlet problem with nonlinearity in the critical growth range
  2. Robert Černý, Silvie Mašková, A sharp form of an embedding into multiple exponential spaces
  3. Robert Černý, On generalized Moser-Trudinger inequalities without boundary condition
  4. Robert Černý, Sharp constants for Moser-type inequalities concerning embeddings into Zygmund spaces
  5. Robert Černý, Note on the concentration-compactness principle for generalized Moser-Trudinger inequalities
  6. Robert Černý, Sharp generalized Trudinger inequalities via truncation for embedding into multiple exponential spaces
  7. Robert Černý, Concentration-Compactness Principle for embedding into multiple exponential spaces on unbounded domains
  8. Robert Černý, Generalized n -Laplacian: semilinear Neumann problem with the critical growth
  9. Takao Ohno, Tetsu Shimomura, Sobolev embeddings for Riesz potentials of functions in grand Morrey spaces of variable exponents over non-doubling measure spaces

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