A note on the hypercontractivity of the polynomial Bohnenblust--Hille inequality

2012/12/29 by Pellegrino, Daniel
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1212.6672

Abstract

For \mathbbK=ℝ or ℂ and m a positive integer, we remark that there is a constant C so that, for all r∈\lbrack1,\frac 2mm+1], the supremum of the ratio between the ℓ_r norm of the coefficients of any nonzero m-homogeneous polynomial P:ℓ_∞% ⁿ(\mathbbK) →\mathbbK and its supremum norm is dominated by C^m⋅ n^{((m)/(r)-(m+1)/(2))} and, moreover, we prove that the exponent (m)/(r)-(m+1)/(2) is optimal.

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