Approximate and $L^{p}$ Peano derivatives of nonintegral order

J. Marshall Ash ; Hajrudin Fejzić

Studia Mathematica, Tome 166 (2005), p. 241-258 / Harvested from The Polish Digital Mathematics Library

Résumé

Let n be a nonnegative integer and let u ∈ (n,n+1]. We say that f is u-times Peano bounded in the approximate (resp. $L^{p}$, 1 ≤ p ≤ ∞) sense at $x ∈ ℝ^{m}$ if there are numbers ${f_{α}(x)}$, |α| ≤ n, such that $f(x+h) - ∑_{|α|≤n} f_{α}(x) h^{α}/α!$ is $O(h^{u})$ in the approximate (resp. $L^{p}$) sense as h → 0. Suppose f is u-times Peano bounded in either the approximate or $L^{p}$ sense at each point of a bounded measurable set E. Then for every ε > 0 there is a perfect set Π ⊂ E and a smooth function g such that the Lebesgue measure of E∖Π is less than ε and f = g on Π. The function g may be chosen to be in $C^{u}$ when u is integral, and, in any case, to have for every j of order ≤ n a bounded jth partial derivative that is Lipschitz of order u - |j|. Pointwise boundedness of order u in the $L^{p}$ sense does not imply pointwise boundedness of the same order in the approximate sense. A classical extension theorem of Calderón and Zygmund is confirmed.

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Publié le : 2005-01-01

Zbl 1079.26005

EUDML-ID : urn:eudml:doc:286466