## SECOND DERIVATIVE TEST FOR ISOMETRIC EMBEDDINGS IN Lp (1997)

### BibTeX

@MISC{Koldobsky97secondderivative,

author = {Alexander Koldobsky},

title = {SECOND DERIVATIVE TEST FOR ISOMETRIC EMBEDDINGS IN Lp},

year = {1997}

}

### OpenURL

### 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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Citation Context ...nly leads to further generalizations, but also seems to clear things up. To support this ambitious statement, let us show a very simple argument which, unfortunately, is false. It was known to P.Levy =-=[19]-=- that an n-dimensional normed space B = (Rn, ‖ · ‖) embeds isometrically in Lp, p > 0 if and only if there exists a finite Borel measure µ on the unit sphere Ω in Rn so that, for every x ∈ Rn , (1) ‖x... |

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Citation Context ...Banach spaces in Lp and its connections with positive definite functions and isotropic random4 ALEXANDER KOLDOBSKY characterizations of zonoids and related results of convex geometry can be found in =-=[7,9,27]-=-. 2. Proof of the second derivative test We start with notation and simple remarks. As usual, we denote by S(Rn ) the space of rapidly decreasing infinitely differentiable functions (test functions) i... |

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Citation Context ... isometrically in Lp if 0 < p < q ≤ 2. For a long time, the connection with stable random vectors and positive definite functions had been the main source of results on isometric embedding in Lp (see =-=[1,11,15,17,18,23,24]-=-). However, it turns out to be quite difficult to check whether exp(−‖x‖p ) is positive definite for certain norms. For example, the following 1938 Schoenberg’s problem [28] was open for more than fif... |

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Citation Context ...able measure. The equivalence of isometric embedding in Lp with 0 ≤ p ≤ 2 and positive definiteness of the function exp(−‖x‖ p ) was established precisely by Bretagnolle, Dacunha-Castelle and Krivine =-=[2]-=- who used this fact to show that the space Lq embeds isometrically in Lp if 0 < p < q ≤ 2. For a long time, the connection with stable random vectors and positive definite functions had been the main ... |

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Citation Context ... isometrically in Lp if 0 < p < q ≤ 2. For a long time, the connection with stable random vectors and positive definite functions had been the main source of results on isometric embedding in Lp (see =-=[1,11,15,17,18,23,24]-=-). However, it turns out to be quite difficult to check whether exp(−‖x‖p ) is positive definite for certain norms. For example, the following 1938 Schoenberg’s problem [28] was open for more than fif... |

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Citation Context ...lem of how to check whether a given Banach space is isometric to a subspace of Lp was raised by P. Levy [19] in 1937. A well-known fact is that a BanachISOMETRIC EMBEDDINGS IN Lp 3 parallelogram law =-=[6,12]-=-. However, as shown by Neyman [26], subspaces of Lp with p ̸= 2 can not be characterized by a finite number of equations or inequalities. P.Levy [19] pointed out that an n-dimensional normed space B =... |

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Citation Context ... spaces ℓ2 q, 2 < q ≤ ∞ do not embed in Lp with 1 < p ≤ 2 (see [4]). The solution to Schoenberg’s problem in [13] was based on the following Fourier transform characterization of subspaces of Lp (see =-=[14]-=-): for every p > 0 which is not an even integer, an n-dimensional Banach space is isometric to a subspace of Lp if and only if the Fourier transform of the function Γ(−p/2)‖x‖ p is a positive distribu... |

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Citation Context ...r p ∈ (0, 1] the integral in the right-hand side of (2) might diverge because p−2 ≤ −1. This, however, can be fixed to a certain extent using the so-called technique of embedding in L−p introduced in =-=[16]-=-. This technique employs the connection between the Radon and Fourier transforms to define and study the representation (1) in the case of negative p. After relatively short repairments, which clearly... |

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Citation Context ...Banach spaces in Lp and its connections with positive definite functions and isotropic random4 ALEXANDER KOLDOBSKY characterizations of zonoids and related results of convex geometry can be found in =-=[7,9,27]-=-. 2. Proof of the second derivative test We start with notation and simple remarks. As usual, we denote by S(Rn ) the space of rapidly decreasing infinitely differentiable functions (test functions) i... |

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Citation Context ...x1) u(x2, x3) dx = 0. Let ǫ > 0. By Remark (i), the function ‖x2e2 + x3e3‖p−2 is locally integrable in R2 , so ∫ L = ‖x2e2 + x3e3‖ p−2 u(x2, x3) dx2 dx3 < ∞. R 2 Besides, there exists c > 0 so that ∫ =-=(8)-=- ‖x2e2 + x3e3‖ p−2 u(x2, x3) dx2 dx3 < ǫ 3K , {(x2,x3):‖x2e2+x3e3‖<c} where K is the number defined in Remark (iv). By Remark (iv), we have ∫ (9) K ∫ R×{(x2,x3):‖x2e2+x3e3‖<c} R×{(x2,x3):‖x2e2+x3e3‖<c... |

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Citation Context ...ch is not an even integer, an n-dimensional Banach space is isometric to a subspace of Lp if and only if the Fourier transform of the function Γ(−p/2)‖x‖ p is a positive distribution on R n \ {0}. In =-=[3]-=-, this criterion was applied to Lorentz spaces. Not very long after the paper [13] appeared, Zastavny [29,30] proved that a three dimensional space is not isometric to a subspace of Lp with 0 < p ≤ 2 ... |

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(Show Context)
Citation Context ... isometrically in Lp if 0 < p < q ≤ 2. For a long time, the connection with stable random vectors and positive definite functions had been the main source of results on isometric embedding in Lp (see =-=[1,11,15,17,18,23,24]-=-). However, it turns out to be quite difficult to check whether exp(−‖x‖p ) is positive definite for certain norms. For example, the following 1938 Schoenberg’s problem [28] was open for more than fif... |

4 |
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Citation Context ...nger result that there are no non-trivial positive definite functions of the form f(‖x‖q). (For q = ∞ that result was established in [22]; the result for 2 < q < ∞ was shown independently by Lisitsky =-=[21]-=-) We refer the reader to [15,25] for more on isometric embedding of Banach spaces in Lp and its connections with positive definite functions and isotropic random4 ALEXANDER KOLDOBSKY characterization... |

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Citation Context ...lem of how to check whether a given Banach space is isometric to a subspace of Lp was raised by P. Levy [19] in 1937. A well-known fact is that a BanachISOMETRIC EMBEDDINGS IN Lp 3 parallelogram law =-=[6,12]-=-. However, as shown by Neyman [26], subspaces of Lp with p ̸= 2 can not be characterized by a finite number of equations or inequalities. P.Levy [19] pointed out that an n-dimensional normed space B =... |

1 |
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