On completeness of the products of harmonic functions

Abstract

Let L L be a partial differential operator in R n {R^n} with constant coefficients. We prove that, under some assumption on L L , the set of products of the elements of the null-space of L L forms a complete set in L p ( D ) {L^p}(D) , p ⩾ 1 p \geqslant 1 , where D D is any bounded domain. In particular, the products of harmonic functions form a complete set in L p ( D ) {L^p}(D) , p ⩾ 1 p \geqslant 1 .