Characterizations of a class of Pilipovic spaces by powers of harmonic oscillator

Abstract

We show that a smooth function integral on R-d belongs to the Pilipovic space H (R-d) or the Pilipovic space H-0 (R-d), if and only if the L-p norm of H-d(N) f for N >= 0, satisfy certain types of estimates. Here H-d = vertical bar x vertical bar(2) - Delta(x) is the harmonic oscillator.