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

• Autorzy: J. Marshall Ash , Hajrudin Fejzić
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 26B35: Special properties of functions of several variables, H\"older conditions, etc.
• 26A24: Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems
• 26B05: Continuity and differentiation questions
• 26A16: Lipschitz (H\"older) classes

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2005
• Tom: 170
• Numer: 3
• Strony: 241-258

Daty

wydano

2005

Twórcy

autor

J. Marshall Ash

• Department of Mathematics, DePaul University, Chicago, IL 60614-3504, U.S.A.

autor

Hajrudin Fejzić

• Department of Mathematics, California State University, San Bernardino, CA 92407, U.S.A.

Identyfikatory

DOI

10.4064/sm170-3-3