Approximate and $L^{p}$ Peano derivatives of nonintegral order
Volume 170 / 2005
Abstract
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.
