 
Summary: TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 360, Number 2, February 2008, Pages 959987
S 00029947(07)042493
Article electronically published on June 25, 2007
QUANTUM SYMMETRIC Lp
DERIVATIVES
J. MARSHALL ASH AND STEFAN CATOIU
Abstract. For 1 p , a oneparameter family of symmetric quantum
derivatives is defined for each order of differentiation as are two families of
Riemann symmetric quantum derivatives. For 1 p , symmetrization
holds, that is, whenever the Lp kth Peano derivative exists at a point, all of
these derivatives of order k also exist at that point. The main result, desym
metrization, is that conversely, for 1 p , each Lp symmetric quantum
derivative is a.e. equivalent to the Lp Peano derivative of the same order. For
k = 1 and 2, each kth Lp symmetric quantum derivative coincides with both
corresponding kth Lp Riemann symmetric quantum derivatives, so, in partic
ular, for k = 1 and 2, both kth Lp Riemann symmetric quantum derivatives
are a.e. equivalent to the Lp Peano derivative.
1. Introduction
