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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Hardy inequalities in Orlicz spaces

Author: Andrea Cianchi
Journal: Trans. Amer. Math. Soc. 351 (1999), 2459-2478
MSC (1991): Primary 46E35; Secondary 46E30
Published electronically: January 27, 1999
MathSciNet review: 1433113
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Abstract: We establish a sharp extension, in the framework of Orlicz spaces, of the ($n$-dimensional) Hardy inequality, involving functions defined on a domain $G$, their gradients and the distance function from the boundary of $G$.

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  • [A] Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR 0450957
  • [BaS] N. K. Bari and S. B. Stečkin, Best approximations and differential properties of two conjugate functions, Trudy Moskov. Mat. Obšč. 5 (1956), 483–522 (Russian). MR 0080797
  • [BS] Colin Bennett and Robert Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press, Inc., Boston, MA, 1988. MR 928802
  • [BK] Steven Bloom and Ron Kerman, Weighted 𝐿_{Φ} integral inequalities for operators of Hardy type, Studia Math. 110 (1994), no. 1, 35–52. MR 1279373
  • [B] David W. Boyd, Indices for the Orlicz spaces, Pacific J. Math. 38 (1971), 315–323. MR 0306887
  • [BF] P. L. Butzer and F. Fehér, Generalized Hardy and Hardy-Littlewood inequalities in rearrangement-invariant spaces, Comment. Math. Special Issue 1 (1978), 41–64. Special issue dedicated to Władysław Orlicz on the occasion of his seventy-fifth birthday. MR 504152
  • [C] Andrea Cianchi, A sharp embedding theorem for Orlicz-Sobolev spaces, Indiana Univ. Math. J. 45 (1996), no. 1, 39–65. MR 1406683, 10.1512/iumj.1996.45.1958
  • [EGP] David E. Edmunds, Petr Gurka, and Luboš Pick, Compactness of Hardy-type integral operators in weighted Banach function spaces, Studia Math. 109 (1994), no. 1, 73–90. MR 1267713
  • [H] G. H. Hardy, Note on a theorem of Hilbert, Math. Z. 6 (1920), 314-317.
  • [K] Alois Kufner, Weighted Sobolev spaces, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], vol. 31, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1980. With German, French and Russian summaries. MR 664599
  • [L] Qinsheng Lai, Weighted integral inequalities for the Hardy type operator and the fractional maximal operator, J. London Math. Soc. (2) 49 (1994), no. 2, 244–266. MR 1260111, 10.1112/jlms/49.2.244
  • [LT] Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin-New York, 1979. Function spaces. MR 540367
  • [M] Benjamin Muckenhoupt, Hardy’s inequality with weights, Studia Math. 44 (1972), 31–38. Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, I. MR 0311856
  • [OK] B. Opic and A. Kufner, Hardy-type inequalities, Pitman Research Notes in Mathematics Series, vol. 219, Longman Scientific & Technical, Harlow, 1990. MR 1069756
  • [P] Giuliana Palmieri, An approach to the theory of some trace spaces related to the Orlicz-Sobolev spaces, Boll. Un. Mat. Ital. B (5) 16 (1979), no. 1, 100–119 (Italian, with English summary). MR 536530
  • [Stp] Vladimir D. Stepanov, Weighted norm inequalities for integral operators and related topics, Nonlinear analysis, function spaces and applications, Vol. 5 (Prague, 1994), Prometheus, Prague, 1994, pp. 139–175. MR 1322312
  • [Str] Jan-Olov Strömberg, Bounded mean oscillation with Orlicz norms and duality of Hardy spaces, Indiana Univ. Math. J. 28 (1979), no. 3, 511–544. MR 529683, 10.1512/iumj.1979.28.28037
  • [RR] M. M. Rao and Z. D. Ren, Theory of Orlicz spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 146, Marcel Dekker, Inc., New York, 1991. MR 1113700
  • [Ta1] Giorgio Talenti, An embedding theorem, Partial differential equations and the calculus of variations, Vol. II, Progr. Nonlinear Differential Equations Appl., vol. 2, Birkhäuser Boston, Boston, MA, 1989, pp. 919–924. MR 1034035
  • [Ta2] Giorgio Talenti, Boundedness of minimizers, Hokkaido Math. J. 19 (1990), no. 2, 259–279. MR 1059170, 10.14492/hokmj/1381517360

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Additional Information

Andrea Cianchi
Affiliation: Istituto di Matematica, Facoltà di Architettura, Università di Firenze, Via dell’ Agnolo 14, 50122 Firenze, Italy

Received by editor(s): May 15, 1996
Received by editor(s) in revised form: November 15, 1996
Published electronically: January 27, 1999
Article copyright: © Copyright 1999 American Mathematical Society