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1998 | 129 | 2 | 157-177
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Mapping properties of integral averaging operators

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EN
Abstrakty
EN
Characterizations are obtained for those pairs of weight functions u and v for which the operators $Tf(x) = ʃ_{a(x)}^{b(x)} f(t)dt$ with a and b certain non-negative functions are bounded from $L^p_u(0,∞)$ to $L^q_v(0,∞)$, 0 < p,q < ∞, p≥ 1. Sufficient conditions are given for T to be bounded on the cones of monotone functions. The results are applied to give a weighted inequality comparing differences and derivatives as well as a weight characterization for the Steklov operator.
Twórcy
autor
  • Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, L8S 4K1, Canada, heinig@mcmaster.ca
autor
  • Department of Mathematics, University of Western Ontario, London, Ontario, N6A 5B7, Canada, sinnamon@uwo.ca
Bibliografia
  • [1] M. Ariño and B. Muckenhoupt, Maximal functions on classical Lorentz spaces and Hardy's inequality with weights for non-increasing functions, Trans. Amer. Math. Soc. 320 (1990), 727-735.
  • [2] E. N. Batuev and V. D. Stepanov, Weighted inequalities of Hardy type, Siberian Math. J. 30 (1989), 8-16.
  • [3] J. S. Bradley, Hardy inequalities with mixed norms, Canad. Math. Bull. 21 (1978), 405-408.
  • [4] V. Burenkov and W. D. Evans, Hardy inequalities for differences and the extension problem for spaces with generalized smoothness, to appear.
  • [5] M. J. Carro and J. Soria, Boundedness of some integral operators, Canad. J. Math. 45 (1993), 1155-1166.
  • [6] P. Grisvard, Espaces intermédiaires entre espaces de Sobolev avec poids, Ann. Scoula Norm. Sup. Pisa 23 (1969), 373-386.
  • [7] H. P. Heinig, A. Kufner and L.-E. Persson, On some fractional order Hardy inequalities, J. Inequalities Appl. 1 (1997), 25-46.
  • [8] G. N. Jakovlev, Boundary properties of functions from the space $W_p^(l)$ on domains with angular points, Dokl. Akad. Nauk SSSR 140 (1961), 73-76 (in Russian).
  • [9] L. V. Kantorovitch and G. P. Akilov, Functional Analysis, 2nd ed., Pergamon Press, Oxford, 1982.
  • [10] B. Opic and A. Kufner, Hardy-Type Inequalities, Longman Sci. Tech., Harlow, 1990.
  • [11] E. T. Sawyer, Boundedness of classical operators on classical Lorentz spaces, Studia Math. 96 (1990), 145-158.
  • [12] E. T. Sawyer, personal communication, ~1985.
  • [13] G. Sinnamon and V. Stepanov, The weighted Hardy inequality: New proofs and the case p=1, J. London Math. Soc. (2) 54 (1996), 89-101.
  • [14] V. D. Stepanov, Integral operators on the cone of monotone functions, ibid. 48 (1993), 465-487.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-smv129i2p157bwm
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