Abstract
The Bohnenblust–Hille inequality for complex m-homogeneous polynomials whose monomials have a uniformly bounded number M of variables was first investigated by Carando et al. (Proc Am Math Soc 143(12):5233–5238, 2015). They have proven that the optimal constants were dominated by a polynomial \(2^{M/2}m^{(M+1)/2}.\) In this note we do not only prove that the optimal constants are uniformly bounded, but we even present a uniform bound on the constants, independently of the value of m. We achieve these improvements by combining results resting on a powerful result of Arias and Farmer [Pac J Math 175(1):13–37, 1996, Theorem 1.3] with elementary techniques.
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We are grateful to anonymous referees for helping us to fix some notations and for shedding us light to some imprecisions of the original version.
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M. Maia and T. Nogueira are supported by Capes and D. Pellegrino is supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico - CNPq - Brasil, Grant 302834/2013-3.
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Maia, M., Nogueira, T. & Pellegrino, D. The Bohnenblust–Hille Inequality for Polynomials Whose Monomials have a Uniformly Bounded Number of Variables. Integr. Equ. Oper. Theory 88, 143–149 (2017). https://doi.org/10.1007/s00020-017-2372-z
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DOI: https://doi.org/10.1007/s00020-017-2372-z