Hardy inequalities in Orlicz spaces
Author:
Andrea Cianchi
Journal:
Trans. Amer. Math. Soc. 351 (1999), 24592478
MSC (1991):
Primary 46E35; Secondary 46E30
Published electronically:
January 27, 1999
MathSciNet review:
1433113
Fulltext PDF Free Access
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Additional Information
Abstract: We establish a sharp extension, in the framework of Orlicz spaces, of the (dimensional) Hardy inequality, involving functions defined on a domain , their gradients and the distance function from the boundary of .
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 [A]
 R. A. Adams, Sobolev spaces, Academic Press, New York, 1975. MR 56:9247
 [BaS]
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 [BS]
 C. Bennett and R. Sharpley, Interpolation of operators, Academic Press, Boston, 1988. MR 89e:46001
 [BK]
 S. Bloom and R. Kerman, Weighted integral inequalities for operators of Hardy type, Studia Math. 110 (1994), 3552. MR 95f:42031
 [B]
 D. W. Boyd, Indices for the Orlicz spaces, Pacific J. Math. 38 (1971), 315323. MR 46:6008
 [BF]
 P. L. Butzer and F. Fehér, Generalized Hardy and HardyLittlewood inequalities in rearrangementinvariant spaces, Comment. Math. Prace Mat. Tomus Specialis in Honorum Ladislai Orlicz 1 (1978), 4164. MR 80c:46037
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 A. Cianchi, A sharp embedding theorem for OrliczSobolev spaces, Indiana Univ. Math. J. 45 (1996), 3965. MR 97h:46044
 [EGP]
 D. E. Edmunds, P. Gurka and L. Pick, Compactness of Hardytype operators in weighted Banach function spaces, Studia Math. 109 (1994), 7390. MR 95c:47033
 [H]
 G. H. Hardy, Note on a theorem of Hilbert, Math. Z. 6 (1920), 314317.
 [K]
 A. Kufner, Weighted Sobolev spaces, Teubner, Leipzig, 1980. MR 84e:46029
 [L]
 Lai Qin Sheng, Weighted integral inequalities for the Hardy type operator and the fractional maximal operator, J. London Math. Soc. 49 (1994), 244266. MR 95g:26029
 [LT]
 J. Lindenstrauss and L. Tzafriri, Classical Banach spaces II, SpringerVerlag, Berlin, 1979. MR 81c:46001
 [M]
 B. Muckenhoupt, Hardy's inequality with weights, Studia Math. 44 (1972), 3138. MR 47:418
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 B. Opic and A. Kufner, Hardytype inequalities, Longman Scientific and Technical, Harlow, 1990. MR 92b:26028
 [P]
 G. Palmieri, An approach to the theory of some trace spaces related to the OrliczSobolev spaces (Italian), Boll. Un. Mat. Ital. 16 (1979), 100119. MR 80f:46037
 [Stp]
 V. D. Stepanov, Weighted norm inequalities and related topics, in Nonlinear analysis, function spaces and applications, Vol. 5, Proceedings of the spring school in Prague, Prometheus, 1994. MR 96m:26019
 [Str]
 J. Strömberg, Bounded mean oscillation with Orlicz norms and duality of Hardy spaces, Indiana Univ. Math. J. 28 (1979), 511544. MR 81f:42021
 [RR]
 M. M. Rao and Z. D. Ren, Theory of Orlicz spaces, Marcel Dekker, Inc., New York, 1991. MR 92e:46059
 [Ta1]
 G. Talenti, An embedding theorem, in ``Essays of Math. Analysis in honour of E. De Giorgi'', Birkhäuser Verlag, Boston, 1989. MR 91d:46040
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 , Boundedness of minimizers, Hokkaido Math. J. 19 (1990), 259279. MR 91g:58054
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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
Email:
cianchi@cesit1.unifi.it
DOI:
http://dx.doi.org/10.1090/S0002994799019856
PII:
S 00029947(99)019856
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
