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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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 .
 [A]
Robert
A. Adams, Sobolev spaces, Academic Press [A subsidiary of
Harcourt Brace Jovanovich, Publishers], New YorkLondon, 1975. Pure and
Applied Mathematics, Vol. 65. MR 0450957
(56 #9247)
 [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
(18,303e)
 [BS]
Colin
Bennett and Robert
Sharpley, Interpolation of operators, Pure and Applied
Mathematics, vol. 129, Academic Press Inc., Boston, MA, 1988. MR 928802
(89e:46001)
 [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
(95f:42031)
 [B]
David
W. Boyd, Indices for the Orlicz spaces, Pacific J. Math.
38 (1971), 315–323. MR 0306887
(46 #6008)
 [BF]
P.
L. Butzer and F.
Fehér, Generalized Hardy and HardyLittlewood inequalities
in rearrangementinvariant spaces, Comment. Math. Special Issue
1 (1978), 41–64. Special issue dedicated to
Władysław Orlicz on the occasion of his seventyfifth
birthday. MR
504152 (80c:46037)
 [C]
Andrea
Cianchi, A sharp embedding theorem for OrliczSobolev spaces,
Indiana Univ. Math. J. 45 (1996), no. 1, 39–65.
MR
1406683 (97h:46044), http://dx.doi.org/10.1512/iumj.1996.45.1958
 [EGP]
David
E. Edmunds, Petr
Gurka, and Luboš
Pick, Compactness of Hardytype integral operators in weighted
Banach function spaces, Studia Math. 109 (1994),
no. 1, 73–90. MR 1267713
(95c:47033)
 [H]
G. H. Hardy, Note on a theorem of Hilbert, Math. Z. 6 (1920), 314317.
 [K]
Alois
Kufner, Weighted Sobolev spaces, TeubnerTexte zur Mathematik
[Teubner Texts in Mathematics], vol. 31, BSB B. G. Teubner
Verlagsgesellschaft, Leipzig, 1980. With German, French and Russian
summaries. MR
664599 (84e:46029)
 [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
(95g:26029), http://dx.doi.org/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, SpringerVerlag, Berlin, 1979. Function spaces. MR 540367
(81c:46001)
 [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 (47 #418)
 [OK]
B.
Opic and A.
Kufner, Hardytype inequalities, Pitman Research Notes in
Mathematics Series, vol. 219, Longman Scientific & Technical,
Harlow, 1990. MR
1069756 (92b:26028)
 [P]
Giuliana
Palmieri, An approach to the theory of some trace spaces related to
the OrliczSobolev spaces, Boll. Un. Mat. Ital. B (5)
16 (1979), no. 1, 100–119 (Italian, with
English summary). MR 536530
(80f:46037)
 [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 (96m:26019)
 [Str]
JanOlov
Strömberg, Bounded mean oscillation with Orlicz norms and
duality of Hardy spaces, Indiana Univ. Math. J. 28
(1979), no. 3, 511–544. MR 529683
(81f:42021), http://dx.doi.org/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 (92e:46059)
 [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
(91d:46040)
 [Ta2]
Giorgio
Talenti, Boundedness of minimizers, Hokkaido Math. J.
19 (1990), no. 2, 259–279. MR 1059170
(91g:58054)
 [A]
 R. A. Adams, Sobolev spaces, Academic Press, New York, 1975. MR 56:9247
 [BaS]
 N. K. Bari and S. B. Stechkin, Best approximation and differential properties of two conjugate functions, Trudy Moskov Mat. Obshch. 5 (1956), 483522. MR 18:303e
 [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
 [C]
 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
 [OK]
 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
 [Ta2]
 , 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
