The weighted Hardy’s inequality for nonincreasing functions
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- by Vladimir D. Stepanov PDF
- Trans. Amer. Math. Soc. 338 (1993), 173-186 Request permission
Abstract:
The purpose of this paper is to give an alternative proof of recent results of M. Arino and B. Muckenhoupt [1] and E. Sawyer [8], concerning Hardy’s inequality for nonincreasing functions and related applications to the boundedness of some classical operators on general Lorentz spaces. Our approach will extend the results of [1,8] to the values of the parameters which are inaccessible by the methods of these papers.References
- Miguel A. Ariño and Benjamin Muckenhoupt, Maximal functions on classical Lorentz spaces and Hardy’s inequality with weights for nonincreasing functions, Trans. Amer. Math. Soc. 320 (1990), no. 2, 727–735. MR 989570, DOI 10.1090/S0002-9947-1990-0989570-0
- D. W. Boyd, The Hilbert transform on rearrangement-invariant spaces, Canadian J. Math. 19 (1967), 599–616. MR 212512, DOI 10.4153/CJM-1967-053-7
- J. Scott Bradley, Hardy inequalities with mixed norms, Canad. Math. Bull. 21 (1978), no. 4, 405–408. MR 523580, DOI 10.4153/CMB-1978-071-7
- S. G. Kreĭn, Yu. Ī. Petunīn, and E. M. Semënov, Interpolation of linear operators, Translations of Mathematical Monographs, vol. 54, American Mathematical Society, Providence, R.I., 1982. Translated from the Russian by J. Szűcs. MR 649411
- Vladimir G. Maz’ja, Sobolev spaces, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. Translated from the Russian by T. O. Shaposhnikova. MR 817985, DOI 10.1007/978-3-662-09922-3
- Benjamin Muckenhoupt, Hardy’s inequality with weights, Studia Math. 44 (1972), 31–38. MR 311856, DOI 10.4064/sm-44-1-31-38
- C. J. Neugebauer, Weighted norm inequalities for averaging operators of monotone functions, Publ. Mat. 35 (1991), no. 2, 429–447. MR 1201565, DOI 10.5565/PUBLMAT_{3}5291_{0}7
- E. Sawyer, Boundedness of classical operators on classical Lorentz spaces, Studia Math. 96 (1990), no. 2, 145–158. MR 1052631, DOI 10.4064/sm-96-2-145-158
- Gordon Sinnamon, A weighted gradient inequality, Proc. Roy. Soc. Edinburgh Sect. A 111 (1989), no. 3-4, 329–335. MR 1007530, DOI 10.1017/S0308210500018606
- G. J. Sinnamon, Weighted Hardy and Opial-type inequalities, J. Math. Anal. Appl. 160 (1991), no. 2, 434–445. MR 1126128, DOI 10.1016/0022-247X(91)90316-R
- Vladimir D. Stepanov, Weighted inequalities for a class of Volterra convolution operators, J. London Math. Soc. (2) 45 (1992), no. 2, 232–242. MR 1171551, DOI 10.1112/jlms/s2-45.2.232
- Giorgio Talenti, Osservazioni sopra una classe di disuguaglianze, Rend. Sem. Mat. Fis. Milano 39 (1969), 171–185 (Italian, with English summary). MR 280661, DOI 10.1007/BF02924135
- Giuseppe Tomaselli, A class of inequalities, Boll. Un. Mat. Ital. (4) 2 (1969), 622–631. MR 0255751
Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 338 (1993), 173-186
- MSC: Primary 26D15
- DOI: https://doi.org/10.1090/S0002-9947-1993-1097171-4
- MathSciNet review: 1097171