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Alternative characterisations of Lorentz-Karamata spaces
Czechoslovak Mathematical Journal, Tome 58 (2008) no. 2, pp. 517-540 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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We present new formulae providing equivalent quasi-norms on Lorentz-Karamata spaces. Our results are based on properties of certain averaging operators on the cone of non-negative and non-increasing functions in convenient weighted Lebesgue spaces. We also illustrate connections between our results and mapping properties of such classical operators as the fractional maximal operator and the Riesz potential (and their variants) on the Lorentz-Karamata spaces.
We present new formulae providing equivalent quasi-norms on Lorentz-Karamata spaces. Our results are based on properties of certain averaging operators on the cone of non-negative and non-increasing functions in convenient weighted Lebesgue spaces. We also illustrate connections between our results and mapping properties of such classical operators as the fractional maximal operator and the Riesz potential (and their variants) on the Lorentz-Karamata spaces.
Classification : 26D10, 42B35, 46E30, 47B38, 47G10
Keywords: Lorentz-Karamata spaces; equivalent quasi-norms; weighted norm inequalities; fractional maximal operators; Riesz potentials 
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Edmunds, D. E.; Opic, B. Alternative characterisations of Lorentz-Karamata spaces. Czechoslovak Mathematical Journal, Tome 58 (2008) no. 2, pp. 517-540. http://geodesic.mathdoc.fr/item/CMJ_2008_58_2_a16/

[1] C. Bennett and K. Rudnick: On Lorentz-Zygmund spaces. Dissertationes Math. 175 (1980), 1–72. | MR

[2] C. Bennett and R. Sharpley: Interpolation of operators. Academic Press, New York, 1988. | MR

[3] N. H. Bingham, C. M. Goldie and J. L. Teugels: Regular variation. Cambridge Univ. Press, Cambridge, 1987. | MR

[4] A. P. Calderón: Spaces between $L^1$ and $L^{\infty }$ and the theorem of Marcinkiewicz. Studia Math. 26 (1966), 273–299. | MR

[5] D. E. Edmunds and W. D. Evans: Hardy operators, function spaces and embeddings. Springer-Verlag, Berlin-Heidelberg, 2004. | MR

[6] D. E. Edmunds, P. Gurka and B. Opic: Double exponential integrability of convolution operators in generalised Lorentz-Zygmund spaces. Indiana Univ. Math. J. 44 (1995), 19–43. | MR

[7] D. E. Edmunds, P. Gurka and B. Opic: On embeddings of logarithmic Bessel potential spaces. J. Functional Anal. 146 (1997), 116–150. | DOI | MR

[8] D. E. Edmunds and B. Opic: Boundedness of fractional maximal operators between classical and weak-type Lorentz spaces. Dissertationes Math. 410 (2002), 1–53. | DOI | MR

[9] D. E. Edmunds and B. Opic: Equivalent quasi-norms on Lorentz spaces. Proc. Amer. Math. Soc. 131 (2003), 745–754. | DOI | MR

[10] W. D. Evans and B. Opic: Real interpolation with logarithmic functors and reiteration. Canad. J. Math. 52 (2000), 920–960. | DOI | MR

[11] A. Gogatishvili, B. Opic and W. Trebels: Limiting reiteration for real interpolation with slowly varying functions. Math. Nachr. 278 (2005), 86–107. | DOI | MR

[12] S. Lai: Weighted norm inequalities for general operators on monotone functions. Trans. Amer. Math. Soc. 340 (1993), 811–836. | DOI | MR | Zbl

[13] J. S. Neves: Lorentz-Karamata spaces, Bessel and Riesz potentials and embeddings. Dissertationes Math. 405 (2002), 1–46. | DOI | MR | Zbl

[14] B. Opic: New characterizations of Lorentz spaces. Proc. Royal Soc. Edinburgh 133A (2003), 439–448. | MR | Zbl

[15] B. Opic: On equivalent quasi-norms on Lorentz spaces. In Function Spaces, Differential Operators and Nonlinear Analysis. The Hans Triebel Aniversary Volume, D. Haroske, T. Runst, H.-J. Schmeisser (eds.), Birkhäuser Verlag, Basel/Switzerland, 2003, pp. 415–426. | MR | Zbl

[16] B. Opic and W. Trebels: Sharp embeddings of Bessel potential spaces with logarithmic smoothness. Math. Proc. Camb. Phil. Soc. 134 (2003), 347–384. | DOI | MR