Monotonicity in Banach function spaces

Sinnamon, Gord

Nonlinear Analysis, Function Spaces and Applications, GDML_Books, (2007), p. 205-240 / Harvested from

Résumé

This paper is an informal presentation of material from [28]–[34]. The monotone envelopes of a function, including the level function, are introduced and their properties are studied. Applications to norm inequalities are given. The down space of a Banach function space is defined and connections are made between monotone envelopes and the norms of the down space and its dual. The connection is shown to be particularly close in the case of universally rearrangement invariant spaces. Next, two equivalent norms are given for the down spaces and these are applied to advance a factorization theory for Hardy inequalities and to characterize embeddings of the classes of generalized quasiconcave functions between Lebesgue spaces. This embedding theory is, in turn, applied to find an expression for the dual space of Lorentz $\Gamma $-space and to find necessary and sufficient conditions for the boundedness of the Fourier transform, acting as a map between Lorentz spaces. A new Lorentz space, the $\Theta $-space, is introduced and shown to be the key to extending the characterization of Fourier inequalities to a greater range of Lorentz spaces. Finally, the scale of down spaces of universally rearrangement invariant spaces is completely characterized by means of interpolation theory, when it is shown that the down spaces of $L^1$ and $L^\infty $ (with general measures) form a Calderón couple.

Détail

EUDML-ID : urn:eudml:doc:220888
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Bibliographie

Benedetto J. J.; Heinig H. P. Weighted Fourier inequalities: New proofs and generalizations, J. Fourier Anal. Appl. 9 (2003), no. 1, 1–37. Zbl 1034.42010, MR 2004a:42007. | MR 1953070 | Zbl 1034.42010

Carro M.; Gogatishvili A.; Martín J.; Pick L. Functional properties of rearrangement invariant spaces defined in terms of oscillations, J. Funct. Anal. 229 (2005), no. 2, 375–404. Zbl 1110.46012, MR 2006g:46051. | MR 2182593 | Zbl 1110.46012

Gogatishvili A.; Pick L. Duality principles and reduction theorems, Math. Inequal. Appl. 3 (2000), no. 4, 539–558. Zbl 0985.46013, MR 2002c:46056. | MR 1786395 | Zbl 0985.46013

Gogatishvili A.; Pick L. Discretization and anti-discretization of rearrangement-invariant norms, Publ. Mat. 47 (2003), no. 2, 311–358. Zbl 1066.46023, MR 2005f:46053. | MR 2006487 | Zbl 1066.46023

Kerman R.; Milman M.; Sinnamon G. On the Brudnyi-Krugljak duality theory of spaces formed by the K-method of interpolation, To appear. | MR 2351114

Kufner A.; Persson L.-E. Weighted Inequalities of Hardy Type, World Scientific Publishing Co., London, 2003. Zbl 1065.26018, MR 2004c:42034. | MR 1982932 | Zbl 1065.26018

Sinnamon G. Embeddings of concave functions and duals of Lorentz spaces, Publ. Mat. 46 (2002), no. 2, 489–515. Zbl 1043.46026, MR 2003h:46042. | MR 1934367 | Zbl 1043.46026

Sinnamon G. The Fourier transform in weighted Lorentz spaces, Publ. Mat. 47 (2003), no. 1, 3–29. Zbl 1045.42004, MR 2004a:42032. | MR 1970892 | Zbl 1045.42004

Sinnamon G. Transferring monotonicity in weighted norm inequalities, Collect. Math. 54 (2003), no. 2, 181–216. Zbl 1093.26025, MR 2004m:26031. | MR 1995140 | Zbl 1093.26025

Sinnamon G. Hardy’s inequality and monotonicity, Function Spaces, Differential Operators and Nonlinear Analysis (P. Drábek and J. Rákosník, eds.). Conference Proceedings, Milovy, Czech Republic, May 28-June 2, 2004. Mathematical Institute of the Academy of Sciences of the Czech Republic, Prague, 2005, pp. 292–310.

Sinnamon G.; Mastyło M. A Calderón couple of down spaces, J. Funct. Anal. 240 (2006), no. 1, 192–225. Zbl pre05083447, MR 2007i:46021. | MR 2259895 | Zbl 1116.46015

Hardy G.; Littlewood J. E.; Pólya G. Inequalities, Second Edition. Cambridge University Press, Cambridge, 1952. Zbl 0047.05302, MR 13,727e. (1952) | MR 0046395 | Zbl 0047.05302

Halperin I. Function spaces, Canad. J. Math. 5 (1953), 273–288. Zbl 0052.11303, MR 15,38h. (1953) | MR 0056195 | Zbl 0052.11303

Lorentz G. G. Bernstein Polynomials, Mathematical Expositions, no. 8. University of Toronto Press, Toronto, 1953. Zbl 0051.05001, MR 15,217a. (1953) | MR 0057370 | Zbl 0051.05001

Calderón A. P. Spaces between $L^1$ and $L^\infty $ and the theorem of Marcinkiewicz, Studia Math. 26 (1966), 273–299. Zbl 0149.09203, MR 34 #3295. (1966) | MR 0203444

Zaanen A. C. Integration, Completely revised edition of An introduction to the theory of integration North-Holland Publishing Co., Amsterdam; Interscience Publishers John Wiley & Sons, Inc., New York, 1967. Zbl 0175.05002, MR 36 #5286. (1967) | MR 0222234 | Zbl 0175.05002

Jr M. Jodeit ,. Torchinsky A. Inequalities for Fourier transforms, Studia Math. 37 (1971), 245–276. Zbl 0224.46037, MR 45 #9121. (1971) | MR 0300073

Sinnamon G. The level function in rearrangement invariant spaces, Publ. Mat. 45 (2001), no. 1,175–198. Zbl 0987.46033, MR 2002b:46048. (198.) | MR 1829583

Benedetto J. J.; Heinig H. P. Weighted Hardy spaces and the Laplace transform, Harmonic Analysis Conference, Cortona, Italy. Lecture Notes in Math. vol. 992, Springer Verlag, Berlin, Heidelberg, 1983, pp. 240–277. Zbl 0513.44001, MR 85j:44001. (1983) | MR 0729358 | Zbl 0513.44001

Benedetto J. J.; Heinig H. P.; Johnson R. Weighted Hardy spaces and the Laplace transform II, Math. Nachr. 132 (1987), 29–55. Zbl 0626.44002, MR 88m:44001. (1987) | MR 0910042 | Zbl 0626.44002

Sinnamon G. Operators on Lebesgue Spaces with General Measures, Doctoral Thesis, McMaster University, 1987. (1987) | MR 2635708

Bennett C.; Sharpley R. Interpolation of Operators, Pure and Applied Mathematics, 129. Academic Press, Inc., Boston, MA, 1988. Zbl 0647.46057, MR 89e:46001. (1988) | MR 0928802 | Zbl 0647.46057

Ariño M.; Muckenhoupt B. 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. Zbl 0716.42016, MR 90k:42034. (1990) | MR 0989570 | Zbl 0716.42016

Opic B.; Kufner A. Hardy-type Inequalities, Pitman Research Notes in Mathematics Series, 219. Longman Scientific & Technical, Longman House, Burnt Mill, Harlow, Essex, England, 1990. Zbl 0698.26007, MR 92b:26028. (1990) | MR 1069756 | Zbl 0698.26007

Sawyer E. T. Boundedness of classical operators on classical Lorentz spaces, Studia Math. 96 (1990), no. 2, 145–158. Zbl 0705.42014, MR 91d:26026. (1990) | MR 1052631 | Zbl 0705.42014

Sinnamon G. Interpolation of spaces defined by the level function, Harmonic Analysis (Sendai, 1990), ICM-90 Satell. Conf. Proc.  Springer, Tokyo, 1991, pp. 190–193. Zbl 0783.46018, MR 94k:46063. (1990) | MR 1261440

Brudnyi; Yu. A.; Krugljak N. Ya. Interpolation Functors and Interpolation Spaces, North-Holland Mathematical Library, 47. North-Holland Publishing Co., Amsterdam, 1991. Zbl 0743.46082, MR 93b:46141. (1991) | MR 1107298 | Zbl 0743.46082

Sinnamon G. Weighted Hardy and Opial-type inequalities, J. Math. Anal. Appl. 160 (1991), no. 2, 434-445. Zbl 0756.26011, MR 92f:26037. (1991) | MR 1126128 | Zbl 0756.26011

Sinnamon G. Spaces defined by the level function and their duals, Studia Math. 111 (1994), no. 1, 19–52. Zbl 0805.46027, MR 95k:46043. (1994) | MR 1292851 | Zbl 0805.46027

Heinig H. P.; Maligranda L. Weighted inequalities for monotone and concave functions, Studia Math. 116 (1995), no. 2, 133–165. Zbl 0851.26012, MR 96g:26022. (1995) | MR 1354136 | Zbl 0851.26012

Maligranda L. Weighted inequalities for monotone functions, Fourth International Conference on Function Spaces (Zielona Góra, 1995). Collect. Math. 48 (1997), no. 2–4, 687–700. Zbl 0916.26007, MR 98m:26015). (1995) | MR 1602636

Gol’dman M. L. Heinig H. P.; Stepanov V. D. On the principle of duality in Lorentz spaces, Canad. J. Math. 48 (1996), no. 5, 959–979. Zbl 0874.47011, MR 97h:42008. (1996) | MR 1414066

Heinig H. P.; Kufner A. Hardy operators of monotone functions and sequences in Orlicz spaces, J. London Math. Soc. (2) 53 (1996), no. 2, 256–270. Zbl 0853.42012, MR 96m:26025. (1996) | MR 1373059 | Zbl 0853.42012

Sinnamon G.; Stepanov V. D. The weighted Hardy inequality: New proofs and the case $p=1$ , London Math. Soc. (2) 54 (1996), no. 1, 89–101. Zbl 0856.26012, MR 97e:26021. (1996) | MR 1395069 | Zbl 0856.26012

Maligranda L. Weighted inequalities for quasi-monotone functions, J. London Math. Soc. (2) 57 (1998), no. 2, 363–370. Zbl 0923.26016, MR 99j:26017. (1998) | MR 1644209 | Zbl 0923.26016