Summary: Approximate and Lp
Peano derivatives of
nonintegral order
J. Marshall Ash and Hajrudin Fejzi´c
Abstract. 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. Lp
; 1
p 1) sense at x 2 Rm
if there are numbers ff (x)g ; j j n such that
f(x + h)
P
j j n f (x) h = ! is O(hu
) in the approximate (resp. Lp
)
sense as h ! 0. Suppose f is u times Peano bounded in either the
approximate or Lp
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 Cu
