Approximate and L Peano derivatives of non-integral order

Let n be a nonnegative integer and let u 2 (n; n + 1]. We say that f is u times Peano bounded in the approximate ( resp. L; 1 p 1) sense at x 2 R if there are numbers ff (x)g ; j j n such that f(x+ h) P j j n f (x)h = ! is O(h) in the approximate (resp. L) sense as h ! 0. Suppose f is u times Peano bounded in either the approximate or L sense at each point of a bounded measurable set E: Then for every > 0 there is a perfect set E and a smooth function g such that the Lebesgue measure of E n is less than and f = g on . The function g may be chosen to be in C 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 jjj. Pointwise boundedness of order u in the L sense does not imply pointwise boundedness of the same order in the approximate sense. A classical extension theorem of Calderón and Zygmund is con…rmed. 1. Introduction Throughout this paper n denotes a …xed nonnegative integer, and u a real number in (n; n + 1]. All functions will be de…ned on subsets of m dimensional Euclidean space and will be real valued. Definition 1. We say that f is u-times approximately Peano bounded at x if f is Lebesgue measurable and for each multi-index = ( 1; 2; :::; m) 2000 Mathematics Subject Classi…cation. Primary 26B05, 26B35; Secondary 26A16, 26A24. Key words and phrases. Peano derivative, fractional derivative, non-integral derivative, function decomposition. This research was partially supported by NSF grant DMS 9707011 and a grant from the Faculty and Development Program of the College of Liberal Arts and Sciences, DePaul University. This paper is in …nal form and no version of it will be submitted for publication elsewhere. 1 2 J. MARSHALL ASH AND HAJRUDIN FEJZIĆ all i being nonnegative integers and order of ; j j = n X i=1 i n there is a number f (x) such that