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A phase transition phenomenon between the isometric and isomorphic extension problems for Hölder functions between L p spaces
Author(s) -
Naor Assaf
Publication year - 2001
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300014480
Subject(s) - mathematics , banach space , monotone polygon , extension (predicate logic) , hilbert space , lipschitz continuity , ball (mathematics) , pure mathematics , space (punctuation) , combinatorics , discrete mathematics , mathematical analysis , geometry , computer science , programming language , linguistics , philosophy
It is shown that it is possible to extend α Hölder maps from subsets of L p to L q (1 < p , q ≤ 2) isometrically if and only if α≤ p / q *, and isomorphically if and only if α≤ p /2. It is also proved that the set of αs which allow an isomorphic extension for α Hölder maps from subsets of X to Y is monotone when Y is a dual Banach space. Finally, the isometric and isomorphic extension problems for Hölder functions between L p and L q is studied for general p , q ≥ 1, and a question posed by K. Ball is solved by showing that it is not true that all Lipschitz maps from subsets of Hilbert space into normed spaces extend to the whole of Hilbert space.