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1994 | 109 | 1 | 73-90
Tytuł artykułu

Compactness of Hardy-type integral operators in weighted Banach function spaces

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EN
Abstrakty
EN
We consider a generalized Hardy operator $Tf(x) = ϕ(x) ʃ_{0}^{x} ψfv$. For T to be bounded from a weighted Banach function space (X,v) into another, (Y,w), it is always necessary that the Muckenhoupt-type condition $ℬ = sup_{R>0} ∥ϕχ_{(R,∞)}∥_{Y}∥ψχ_{(0,R)}∥_{X'} < ∞$ be satisfied. We say that (X,Y) belongs to the category M(T) if this Muckenhoupt condition is also sufficient. We prove a general criterion for compactness of T from X to Y when (X,Y) ∈ M(T) and give an estimate for the distance of T from the finite rank operators. We apply the results to Lorentz spaces and characterize pairs of Lorentz spaces which fall into M (T).
Twórcy
  • Centre for Mathematical Analysis and its Applications, The University of Sussex, Falmer, Brighton BN1 9QH, England , mmfb7@syma.sussex.ac.uk
autor
  • Department of Mathematics, University of Agriculture, 160 21 Praha 6, Czech Republic , gurka@csearn.bitnet
autor
  • School of Mathematics, University of Wales College of Cardiff, Senghenydd Road, Cardiff CF2 4AG, UK , pick@csearn.bitnet
  • Mathematical Institute, Academy of Sciences of the Czech Republic, Žitná 25, 115 67 Praha 1, Czech Republic
Bibliografia
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  • [EE] D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Oxford Univ. Press, Oxford, 1987.
  • [EEH] D. E. Edmunds, W. D. Evans and D. J. Harris, Approximation numbers of certain Volterra integral operators, J. London Math. Soc. 38 (1988), 471-489.
  • [EH] W. D. Evans and D. J. Harris, Sobolev embeddings for generalized ridged domains, Proc. London Math. Soc. 54 (1987), 141-175.
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  • [K] V. M. Kokilashvili, On Hardy's inequalities in weighted spaces, Soobshch. Akad. Nauk. Gruzin. SSR 96 (1979), 37-40 (in Russian).
  • [LP] Q. Lai and L. Pick, The Hardy operator, $L_∞$, and BMO, J. London Math. Soc. (2) 48 (1993), 167-177.
  • [LUX] W. A. J. Luxemburg, Banach Function Spaces, thesis, Delft, 1955.
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  • [M] V. G. Maz'ya, Sobolev Spaces, Springer, Berlin, 1985.
  • [MU] B. Muckenhoupt, Hardy's inequality with weights, Studia Math. 44 (1972), 31-38.
  • [OK] B. Opic and A. Kufner, Hardy-type Inequalities, Pitman Res. Notes Math. Ser. 219, Longman Sci. & Tech., Harlow, 1990.
  • [S] E. T. Sawyer, Weighted Lebesgue and Lorentz norm inequalities for the Hardy operator, Trans. Amer. Math. Soc. 281 (1984), 329-337.
  • [T] G. Talenti, Osservazioni sopra una classe di disuguaglianze, Rend. Sem. Mat. Fis. Milano 39 (1969), 171-185.
  • [TO] G. Tomaselli, A class of inequalities, Boll. Un. Mat. Ital. 21 (1969), 622-631.
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