Abstract
In this note we study the quantitative side of the famous Enflo-Pisier theorem on the possibility of equivalent uniformly smooth renormings of superreflexive Banach spaces (in particular, uniformly convex and uniformly nonsquare ones). Typical re result: let the modulus of convexity of the space X, which has a locally unconditional structure, satisfy the condition δx(ε) ⩾ C·εp. Then the space X admits an equivalent q-smooth renorming for any q, satisfying the inequality q<1n 2/1n (2× (1−C·2−p/2)).
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Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 135, pp. 120–134, 1984.
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Rakov, S.A. Uniformly smooth renormings of uniformly convex Banach spaces. J Math Sci 31, 2713–2721 (1985). https://doi.org/10.1007/BF02107257
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DOI: https://doi.org/10.1007/BF02107257