Journal of Inequalities and Applications 
Volume 4 (1999), Issue 4, Pages 301-314
doi:10.1155/S1025583499000417

On weighted positivity of ordinary differential operators

Stefan Eilertsen

 Department of Mathematics, Linköping University, Linköping S-581 83, Sweden
Received 20 October 1998; Revised 14 January 1999

Abstract

Some elliptic differential operators possess a weighted positivity property, where the weight is a fundamental solution of the operator. This property has interesting applications to partial differential operators. The present paper is devoted to the property for ordinarydifferential operators.

It is shown that the operator (1−d2/dx2)m has the positivity property if and only if m=0,1,2,3, while there exist operators of arbitrary even order for which the positivity holds. Some necessary conditions for the property are given.