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Lorentz-Karamata spaces; equivalent quasi-norms; weighted norm inequalities; fractional maximal operators; Riesz potentials
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.
[1] C. Bennett and K. Rudnick: On Lorentz-Zygmund spaces. Dissertationes Math. 175 (1980), 1–72. MR 0576995
[2] C. Bennett and R. Sharpley: Interpolation of operators. Academic Press, New York, 1988. MR 0928802
[3] N. H. Bingham, C. M. Goldie and J. L. Teugels: Regular variation. Cambridge Univ. Press, Cambridge, 1987. MR 0898871
[4] A. P. Calderón: Spaces between $L^1$ and $L^{\infty }$ and the theorem of Marcinkiewicz. Studia Math. 26 (1966), 273–299. MR 0203444
[5] D. E. Edmunds and W. D. Evans: Hardy operators, function spaces and embeddings. Springer-Verlag, Berlin-Heidelberg, 2004. MR 2091115
[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 1336431
[7] D. E. Edmunds, P. Gurka and B. Opic: On embeddings of logarithmic Bessel potential spaces. J. Functional Anal. 146 (1997), 116–150. DOI 10.1006/jfan.1996.3037 | MR 1446377
[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 10.4064/dm410-0-1 | MR 1952673
[9] D. E. Edmunds and B. Opic: Equivalent quasi-norms on Lorentz spaces. Proc. Amer. Math. Soc. 131 (2003), 745–754. DOI 10.1090/S0002-9939-02-06870-3 | MR 1937412
[10] W. D. Evans and B. Opic: Real interpolation with logarithmic functors and reiteration. Canad. J. Math. 52 (2000), 920–960. DOI 10.4153/CJM-2000-039-2 | MR 1782334
[11] A. Gogatishvili, B. Opic and W. Trebels: Limiting reiteration for real interpolation with slowly varying functions. Math. Nachr. 278 (2005), 86–107. DOI 10.1002/mana.200310228 | MR 2111802
[12] S. Lai: Weighted norm inequalities for general operators on monotone functions. Trans. Amer. Math. Soc. 340 (1993), 811–836. DOI 10.1090/S0002-9947-1993-1132877-X | MR 1132877 | Zbl 0819.47044
[13] J. S. Neves: Lorentz-Karamata spaces, Bessel and Riesz potentials and embeddings. Dissertationes Math. 405 (2002), 1–46. DOI 10.4064/dm405-0-1 | MR 1927106 | Zbl 1017.46021
[14] B. Opic: New characterizations of Lorentz spaces. Proc. Royal Soc. Edinburgh 133A (2003), 439–448. MR 1969821 | Zbl 1037.46026
[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 1984189 | Zbl 1036.46022
[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 10.1017/S0305004102006321 | MR 1972143
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