Equisummability of Eigenfunction Expansions Under Analytic Multipliers

摘要

This paper presents some abstract criteria for equisummability of expansions in eienfunctions of certain pairs of elliptic operators on general domains of R superscript n. The criteria to be developed originate in some cases from general Banach space arguments, and in others from the more specific spatial nature of the differential operators involved. The importance of our analysis of L (superscript infinity) -equisummability of two operators is that the question of convergence of the two summability means is reduced essentially to showing that the difference of the modified resolvent operators is uniformly bounded. These criteria give a simple proof of an equisummability result of Gurarie and Kon for a certain class of elliptic operators whose leading terms are positive and lower order terms have coefficients which are singular on a nowhere dense set. The key technique is to analyze kernels of the resolvents and use L (superscript 1) -radial bounds of the resolvents. (Reprints).