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. Takao Ohno, Tetsu Shimomura, Sobolev embeddings for Riesz potentials of functions in grand Morrey spaces of variable exponents over non-doubling measure spaces
  8. Robert Černý, Generalized n -Laplacian: semilinear Neumann problem with the critical growth
  9. Robert Černý, Concentration-Compactness Principle for embedding into multiple exponential spaces on unbounded domains

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