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

J. Marshall Ash¹ ; Hajrudin Fejzić²

¹ Department of Mathematics DePaul University Chicago, IL 60614-3504, U.S.A.
² Department of Mathematics California State University San Bernardino, CA 92407, U.S.A.

Studia Mathematica, Tome 170 (2005) no. 3, pp. 241-258

Résumé

Let $n$ be a nonnegative integer and let $u\in(n,n+1]$. We say that $f$ is $u$-times Peano bounded in the approximate (resp. $L^{p}$, $1\leq p\leq\infty$) sense at $x\in \mathbb{R}^{m}$ if there are numbers $\{ f_{\alpha}(x)\}$, $\vert \alpha\vert \leq n$, such that $f(x+h)-\sum_{\vert \alpha\vert \leq n}f_{\alpha}( x ) h^{\alpha}/\alpha!$ is $O(h^{u})$ in the approximate (resp. $L^{p}$) sense as $h\rightarrow0$. 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 $\varepsilon >0$ there is a perfect set $\varPi \subset E$ and a smooth function $g$ such that the Lebesgue measure of $E\setminus\varPi $ is less than $\varepsilon $ and $f=g$ on $\varPi $. 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 $\leq n$ a bounded $j$th partial derivative that is Lipschitz of order $u-\vert j\vert $.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.

Détail

DOI : 10.4064/sm170-3-3

Keywords: nonnegative integer say u times peano bounded approximate resp leq leq infty sense mathbb there numbers alpha vert alpha vert leq sum vert alpha vert leq alpha alpha alpha approximate resp sense rightarrow suppose u times peano bounded either approximate sense each point bounded measurable set every varepsilon there perfect set varpi subset smooth function lebesgue measure setminus varpi varepsilon varpi function may chosen integral have every order leq bounded jth partial derivative lipschitz order u vert vert pointwise boundedness order sense does imply pointwise boundedness order approximate sense nbsp classical extension theorem calder zygmund confirmed