Uniform homeomorphisms between spheres induced by interpolation methods

Corrêa, Willian

doi:10.1090/proc/16730

Uniform homeomorphisms between spheres induced by interpolation methods
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by Willian Corrêa;

Proc. Amer. Math. Soc. 152 (2024), 2157-2167

DOI: https://doi.org/10.1090/proc/16730

Published electronically: March 20, 2024

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Abstract:

M. Daher [Canad. Math. Bull. 38 (1995), pp. 286–294] showed that if $(X_0, X_1)$ is a regular couple of uniformly convex spaces then the unit spheres of the complex interpolation spaces $X_{\theta }$ and $X_{\eta }$ are uniformly homeomorphic for every $0 < \theta , \eta < 1$. We show that this is a rather general phenomenon of the interpolation methods described by the discrete framework of interpolation of Lindemulder and Lorist [A discrete framework for the interpolation of Banach spaces, https://arxiv.org/abs/2105.08373, 2021].

References

Alexandr Andoni, Assaf Naor, Aleksandar Nikolov, Ilya Razenshteyn, and Erik Waingarten, Hölder homeomorphisms and approximate nearest neighbors, 59th Annual IEEE Symposium on Foundations of Computer Science—FOCS 2018, IEEE Computer Soc., Los Alamitos, CA, 2018, pp. 159–169. MR 3899586, DOI 10.1109/FOCS.2018.00024
Yoav Benyamini and Joram Lindenstrauss, Geometric nonlinear functional analysis. Vol. 1, American Mathematical Society Colloquium Publications, vol. 48, American Mathematical Society, Providence, RI, 2000. MR 1727673, DOI 10.1090/coll/048
J. Bergh and J. Löfström, Interpolation spaces: an introduction, Grundlehren der mathematischen Wissenschaften, Springer Berlin Heidelberg, 2012.
A.-P. Calderón, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 113–190. MR 167830, DOI 10.4064/sm-24-2-113-190
Fouad Chaatit, On uniform homeomorphisms of the unit spheres of certain Banach lattices, Pacific J. Math. 168 (1995), no. 1, 11–31. MR 1331992, DOI 10.2140/pjm.1995.168.11
R. Coifman, M. Cwikel, R. Rochberg, Y. Sagher, and G. Weiss, Complex interpolation for families of Banach spaces, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978) Proc. Sympos. Pure Math., XXXV, Part 2, Amer. Math. Soc., Providence, RI, 1979, pp. 269–282. MR 545314
Michael Cwikel, Nigel Kalton, Mario Milman, and Richard Rochberg, A unified theory of commutator estimates for a class of interpolation methods, Adv. Math. 169 (2002), no. 2, 241–312. MR 1926224, DOI 10.1006/aima.2001.2061
Mohamad Daher, Homéomorphismes uniformes entre les sphères unité des espaces d’interpolation, Canad. Math. Bull. 38 (1995), no. 3, 286–294 (French, with French summary). MR 1347300, DOI 10.4153/CMB-1995-042-8
Alon Ivtsan, Stafney’s lemma holds for several “classical” interpolation methods, Proc. Amer. Math. Soc. 140 (2012), no. 3, 881–889. MR 2869072, DOI 10.1090/S0002-9939-2011-10974-2
N. J. Kalton, Differentials of complex interpolation processes for Köthe function spaces, Trans. Amer. Math. Soc. 333 (1992), no. 2, 479–529. MR 1081938, DOI 10.1090/S0002-9947-1992-1081938-1
Nigel Kalton, Peer Kunstmann, and Lutz Weis, Perturbation and interpolation theorems for the $H^\infty$-calculus with applications to differential operators, Math. Ann. 336 (2006), no. 4, 747–801. MR 2255174, DOI 10.1007/s00208-005-0742-3
Nigel J. Kalton, Emiel Lorist, and Lutz Weis, Euclidean structures and operator theory in Banach spaces, Mem. Amer. Math. Soc. 288 (2023), no. 1433, vi+156. MR 4627089, DOI 10.1090/memo/1433
N. Lindemulder and E. Lorist, A discrete framework for the interpolation of Banach spaces, E-prints, arXiv:2105.08373, 2021.
J.-L. Lions and J. Peetre, Sur une classe d’espaces d’interpolation, Inst. Hautes Études Sci. Publ. Math. 19 (1964), 5–68 (French). MR 165343, DOI 10.1007/BF02684796
S. Mazur, Une remarque sur l’homéomorphie des champs fonctionnels (French), Studia Math. 1 (1929), no. 1, 83–85.
Edward Odell and Thomas Schlumprecht, The distortion problem, Acta Math. 173 (1994), no. 2, 259–281. MR 1301394, DOI 10.1007/BF02398436
G. Pisier, Some applications of the complex interpolation method to Banach lattices, J. Analyse Math. 35 (1979), 264–281. MR 555306, DOI 10.1007/BF02791068
Gilles Pisier, Martingales in Banach spaces, Cambridge Studies in Advanced Mathematics, vol. 155, Cambridge University Press, Cambridge, 2016. MR 3617459, DOI 10.1017/CBO9781316480588