# Article

MSC: 26D15, 46B70, 46E30
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Keywords:
Monotone envelope; level function; pushing mass; down space; Hardy inequality; Lorentz pace; rearrangement invariant space; quasiconcave function; Fourier inequality; interpolation; Calderón couple
Summary:
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.
References:
[1] 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. MR 0989570 | Zbl 0716.42016
[2] 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. MR 0729358 | Zbl 0513.44001
[3] 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. DOI 10.1007/s00041-003-0003-3 | MR 1953070 | Zbl 1034.42010
[4] 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. DOI 10.1002/mana.19871320104 | MR 0910042 | Zbl 0626.44002
[5] Bennett C., Sharpley R.: Interpolation of Operators. Pure and Applied Mathematics, 129. Academic Press, Inc., Boston, MA, 1988. Zbl 0647.46057, MR 89e:46001. MR 0928802 | Zbl 0647.46057
[6] 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. MR 1107298 | Zbl 0743.46082
[7] 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. MR 0203444
[8] 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. DOI 10.1016/j.jfa.2005.06.012 | MR 2182593 | Zbl 1110.46012
[9] 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
[10] 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
[11] 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. MR 1414066
[12] Halperin I.: Function spaces. Canad. J. Math. 5 (1953), 273–288. Zbl 0052.11303, MR 15,38h. DOI 10.4153/CJM-1953-031-3 | MR 0056195 | Zbl 0052.11303
[13] Hardy G., Littlewood J. E., Pólya G.: Inequalities. Second Edition. Cambridge University Press, Cambridge, 1952. Zbl 0047.05302, MR 13,727e. MR 0046395 | Zbl 0047.05302
[14] 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. DOI 10.1112/jlms/53.2.256 | MR 1373059 | Zbl 0853.42012
[15] 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. MR 1354136 | Zbl 0851.26012
[16] Jr M. Jodeit ,. Torchinsky A.: Inequalities for Fourier transforms. Studia Math. 37 (1971), 245–276. Zbl 0224.46037, MR 45 #9121. MR 0300073
[17] 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
[18] 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
[19] Lorentz G. G.: Bernstein Polynomials. Mathematical Expositions, no. 8. University of Toronto Press, Toronto, 1953. Zbl 0051.05001, MR 15,217a. MR 0057370 | Zbl 0051.05001
[20] 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). MR 1602636
[21] 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. MR 1644209 | Zbl 0923.26016
[22] 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. MR 1069756 | Zbl 0698.26007
[23] 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. MR 1052631 | Zbl 0705.42014
[24] Sinnamon G.: Operators on Lebesgue Spaces with General Measures. Doctoral Thesis, McMaster University, 1987. MR 2635708
[25] Sinnamon G.: Weighted Hardy and Opial-type inequalities. J. Math. Anal. Appl. 160 (1991), no. 2, 434-445. Zbl 0756.26011, MR 92f:26037. DOI 10.1016/0022-247X(91)90316-R | MR 1126128 | Zbl 0756.26011
[26] 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. MR 1261440
[27] 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. MR 1292851 | Zbl 0805.46027
[28] Sinnamon G.: The level function in rearrangement invariant spaces. Publ. Mat. 45 (2001), no. 1,175–198. Zbl 0987.46033, MR 2002b:46048. MR 1829583
[29] 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
[30] 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
[31] 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
[32] 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.
[33] 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
[34] 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. MR 1395069 | Zbl 0856.26012
[35] 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. MR 0222234 | Zbl 0175.05002

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