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

J. Marshall Ash, and Hajrudin Fejzić. "Approximate and $L^{p}$ Peano derivatives of nonintegral order." Studia Mathematica 170.3 (2005): 241-258. <http://eudml.org/doc/286466>.

@article{J2005,
abstract = {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.},
author = {J. Marshall Ash, Hajrudin Fejzić},
journal = {Studia Mathematica},
keywords = {Peano derivative; fractional derivative; nonintegral derivative; function decomposition},
language = {eng},
number = {3},
pages = {241-258},
title = {Approximate and $L^\{p\}$ Peano derivatives of nonintegral order},
url = {http://eudml.org/doc/286466},
volume = {170},
year = {2005},
}

TY - JOUR
AU - J. Marshall Ash
AU - Hajrudin Fejzić
TI - Approximate and $L^{p}$ Peano derivatives of nonintegral order
JO - Studia Mathematica
PY - 2005
VL - 170
IS - 3
SP - 241
EP - 258
AB - 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.
LA - eng
KW - Peano derivative; fractional derivative; nonintegral derivative; function decomposition
UR - http://eudml.org/doc/286466
ER -