SECOND DERIVATIVE TEST FOR ISOMETRIC EMBEDDINGS IN Lp

SECOND DERIVATIVE TEST FOR ISOMETRIC EMBEDDINGS IN Lp (1997)

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by Alexander Koldobsky

BibTeX

@MISC{Koldobsky97secondderivative,
    author = {Alexander Koldobsky},
    title = {SECOND DERIVATIVE TEST FOR ISOMETRIC EMBEDDINGS IN Lp},
    year = {1997}
}

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Abstract

Abstract. An old problem of P. Levy is to characterize those Banach spaces which embed isometrically in Lp. We show a new criterion in terms of the second derivative of the norm. As a consequence we show that, if M is a twice differentiable Orlicz function with M ′(0) = M ′′(0) = 0, then the n-dimensional Orlicz space ℓn M, n ≥ 3 does not embed isometrically in Lp with 0 < p ≤ 2. These results generalize and clear up the recent solution to the 1938 Schoenberg’s problem on positive definite functions. 1.

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