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1994 | 110 | 1 | 35-52
Tytuł artykułu

Weighted $L_{Φ}$ integral inequalities for operators of Hardy type

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EN
Abstrakty
EN
Necessary and sufficient conditions are given on the weights t, u, v, and w, in order for $Φ_2^{-1} (ʃΦ_2(w(x)|Tf(x)|)t(x)dx) ≤ Φ_{1}^{-1}(ʃΦ_{1}(Cu(x)|f(x)|)v(x)dx)$ to hold when $Φ_1$ and $Φ_2$ are N-functions with $Φ_2∘Φ_{1}^{-1}$ convex, and T is the Hardy operator or a generalized Hardy operator. Weak-type characterizations are given for monotone operators and the connection between weak-type and strong-type inequalities is explored.
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Twórcy
autor
  • Department of Mathematics, Siena College, Loudonville, New York 12211, U.S.A.
autor
  • Department of Mathematics, Brock University, St. Catharines, Ontario, Canada LS2A1
Bibliografia
  • [1] M. Artola, untitled and unpublished manuscript.
  • [2] S. Bloom and R. Kerman, Weighted norm inequalities for operators of Hardy type, Proc. Amer. Math. Soc. 113 (1991), 135-141.
  • [3] J. S. Bradley, Hardy inequalities with mixed norms, Canad. Math. Bull. 21 (1978), 405-408.
  • [4] H. Heinig and L. Maligranda, Interpolation with weights in Orlicz spaces, Luleå University of Technology, Department of Applied Mathematics Research Report 03 (1992).
  • [5] S. S. Kazarian, Integral inequalities in Orlicz reflexive weighted spaces for the conjugate function, Dokl. Akad. Nauk Armyan. SSR 25 (3) (1990), 261-273 (in Russian).
  • [6] R. Kerman and A. Torchinsky, Integral inequalities with weights for the Hardy maximal function, Studia Math. 71 (1981/82), 277-284.
  • [7] M. A. Krasnosel'skii [M. A. Krasnosel'skiĭ] and Ya. B. Rutickii [Ya. B. Rutitskiĭ], Convex Functions and Orlicz Spaces, Noordhoff, Groningen, 1961.
  • [8] F. Martín-Reyes and E. Sawyer, Weighted norm inequalities for the Riemann-Liouville fractional integral operators, Proc. Amer. Math. Soc. 106 (1989), 727-733.
  • [9] B. Muckenhoupt, Hardy's inequality with weights, Studia Math. 34 (1972), 31-38.
  • [10] L. Quinsheng, Two weight Φ-inequalities for the Hardy operator, Hardy-Littlewood maximal operator and fractional integrals, Proc. Amer. Math. Soc., to appear.
  • [11] M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Marcel Dekker, New York, 1991.
  • [12] E. Sawyer, A characterization of a two-weight norm inequality for maximal operators, Studia Math. 75 (1982), 1-11.
  • [13] E. Sawyer, A characterization of two weight norm inequalities for fractional and Poisson integrals, Trans. Amer. Math. Soc. 302 (1988), 533-545.
  • [14] V. Stepanov, Two-weighted estimates for Riemann-Liouville integrals, preprint no. 39, Czech. Acad. Sci., 1988.
  • [15] G. Talenti, Osservazioni sopra una classe di disuguaglianze, Rend. Sem. Mat. Fis. Milano 39 (1969), 171-185.
  • [16] G. Tomaselli, A class of inequalities, Boll. Un. Mat. Ital. 21 (1969), 622-631.
  • [17] A. Zygmund, Trigonometric Series, Vol. I, 2nd ed., Cambridge Univ. Press, Cambridge, 1959.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-smv110i1p35bwm
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