A function with prescribed initial derivatives in different Banach spaces

Haim Brezis, L. E. Fraenkel

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Abstract

Let X0 ⊂ X1 ⊂ ··· ⊂ Xp be Banach spaces with continuous injection of Xk into Xk + 1 for 0 ≤ k ≤ p - 1, and with X0 dense in Xp. We seek a function u: [0, 1] → X0 such that its kth derivative u(k), k = 0, 1,..., p, is continuous from [0, 1] into xk, and satisfies the initial condition u(k)(0) = ak ε{lunate} Xk. It is shown that such a function exists if and only if the initial values a0, a1, ..., ap satisfy a certain condition reminiscent of interpolation theory. This condition always holds when p = 1; when p ≥ 2, the spaces Xk (k = 0, 1, ..., p) may or may not be such that the desired function exists for any given initial values ak ε{lunate} Xk.

Original languageEnglish (US)
Pages (from-to)328-335
Number of pages8
JournalJournal of Functional Analysis
Volume29
Issue number3
DOIs
StatePublished - Jan 1 1978
Externally publishedYes

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All Science Journal Classification (ASJC) codes

  • Analysis

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