Summary: TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 360, Number 2, February 2008, Pages 959­987
S 0002-9947(07)04249-3
Article electronically published on June 25, 2007
QUANTUM SYMMETRIC Lp
DERIVATIVES
J. MARSHALL ASH AND STEFAN CATOIU
Abstract. For 1 p , a one-parameter 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

Source: Ash, J. Marshall - Department of Mathematical Sciences, DePaul University

Collections: Mathematics