The inclusion problem for mixed norm spaces
HTML articles powered by AMS MathViewer
- by Wayne Grey and Gord Sinnamon PDF
- Trans. Amer. Math. Soc. 368 (2016), 8715-8736 Request permission
Abstract:
Given two mixed norm Lebesgue spaces on an $n$-fold product of arbitrary $\sigma$-finite measure spaces, is one contained in the other? If so, what is the norm of the inclusion map? These questions are answered completely for a large range of Lebesgue indices and all measure spaces. When the measure spaces are atomless, both questions are settled for all indices. When the measure spaces are not purely atomic, the first question is settled for all indices. Some complete and some partial results are given in the remaining cases, but a wide variety of behaviour is observed. In particular, the norm problem for purely atomic measure spaces is seen to be intractable for certain ranges of the Lebesgue indices; it is equivalent to an optimization problem that includes a known NP-hard problem as a special case.References
- Robert Algervik and Viktor I. Kolyada, On Fournier-Gagliardo mixed norm spaces, Ann. Acad. Sci. Fenn. Math. 36 (2011), no. 2, 493–508. MR 2865509, DOI 10.5186/aasfm.2011.3624
- Sorina Barza, Anna Kamińska, Lars-Erik Persson, and Javier Soria, Mixed norm and multidimensional Lorentz spaces, Positivity 10 (2006), no. 3, 539–554. MR 2258957, DOI 10.1007/s11117-005-0004-3
- A. Benedek and R. Panzone, The space $L^{p}$, with mixed norm, Duke Math. J. 28 (1961), 301–324. MR 126155, DOI 10.1215/S0012-7094-61-02828-9
- Colin Bennett and Robert Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press, Inc., Boston, MA, 1988. MR 928802
- A. P. Blozinski, Multivariate rearrangements and Banach function spaces with mixed norms, Trans. Amer. Math. Soc. 263 (1981), no. 1, 149–167. MR 590417, DOI 10.1090/S0002-9947-1981-0590417-X
- Antonio Boccuto, Alexander V. Bukhvalov, and Anna Rita Sambucini, Some inequalities in classical spaces with mixed norms, Positivity 6 (2002), no. 4, 393–411. MR 1949750, DOI 10.1023/A:1021353215312
- V. I. Bogachev, Measure theory. Vol. I, II, Springer-Verlag, Berlin, 2007. MR 2267655, DOI 10.1007/978-3-540-34514-5
- Nadia Clavero and Javier Soria, Mixed norm spaces and rearrangement invariant estimates, J. Math. Anal. Appl. 419 (2014), no. 2, 878–903. MR 3225412, DOI 10.1016/j.jmaa.2014.05.030
- Andreas Defant, Dumitru Popa, and Ursula Schwarting, Coordinatewise multiple summing operators in Banach spaces, J. Funct. Anal. 259 (2010), no. 1, 220–242. MR 2610385, DOI 10.1016/j.jfa.2010.01.008
- Andreas Defant, Leonhard Frerick, Joaquim Ortega-Cerdà, Myriam Ounaïes, and Kristian Seip, The Bohnenblust-Hille inequality for homogeneous polynomials is hypercontractive, Ann. of Math. (2) 174 (2011), no. 1, 485–497. MR 2811605, DOI 10.4007/annals.2011.174.1.13
- John J. F. Fournier, Mixed norms and rearrangements: Sobolev’s inequality and Littlewood’s inequality, Ann. Mat. Pura Appl. (4) 148 (1987), 51–76. MR 932758, DOI 10.1007/BF01774283
- V. I. Kolyada, Iterated rearrangements and Gagliardo-Sobolev type inequalities, J. Math. Anal. Appl. 387 (2012), no. 1, 335–348. MR 2845754, DOI 10.1016/j.jmaa.2011.08.077
- W. A. J. Luxemburg, On the measurability of a function which occurs in a paper by A. C. Zaanen, Nederl. Akad. Wetensch. Proc. Ser. A. 61 = Indag. Math. 20 (1958), 259–265. MR 0096765, DOI 10.1016/S1385-7258(58)50034-1
- W. A. J. Luxemburg, Addendum to “On the measurability of a function which occurs in a paper by A. C. Zaanen”, Nederl. Akad. Wetensch. Proc. Ser. A 66=Indag. Math. 25 (1963), 587–590. MR 0158046, DOI 10.1016/S1385-7258(63)50057-2
- Mario Milman, Notes on interpolation of mixed norm spaces and applications, Quart. J. Math. Oxford Ser. (2) 42 (1991), no. 167, 325–334. MR 1120993, DOI 10.1093/qmath/42.1.325
- Dumitru Popa and Gord Sinnamon, Blei’s inequality and coordinatewise multiple summing operators, Publ. Mat. 57 (2013), no. 2, 455–475. MR 3114778, DOI 10.5565/PUBLMAT_{5}7213_{0}9
- U. Schwarting, Vector Valued Bohnenblust-Hille Inequalities, Doctoral Thesis, Carl von Ossietzky University, Oldenburg, Germany, 2013.
- Martin Väth, Some measurability results and applications to spaces with mixed family-norm, Positivity 10 (2006), no. 4, 737–753. MR 2280647, DOI 10.1007/s11117-005-0035-9
- Adriaan Cornelis Zaanen, Integration, North-Holland Publishing Co., Amsterdam; Interscience Publishers John Wiley & Sons, Inc., New York, 1967. Completely revised edition of An introduction to the theory of integration. MR 0222234
Additional Information
- Wayne Grey
- Affiliation: Department of Mathematics, University of Western Ontario, London N6A 5B7, Canada
- Email: wgrey@uwo.ca
- Gord Sinnamon
- Affiliation: Department of Mathematics, University of Western Ontario, London N6A 5B7, Canada
- MR Author ID: 163045
- Email: sinnamon@uwo.ca
- Received by editor(s): October 30, 2014
- Published electronically: January 26, 2016
- Additional Notes: This work was supported by the Natural Sciences and Engineering Research Council of Canada
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 8715-8736
- MSC (2010): Primary 46E30; Secondary 46A45, 26D15
- DOI: https://doi.org/10.1090/tran6665
- MathSciNet review: 3551586