A class of pseudo-differential operators with weight functions and its applications to energy estimates for degenerate elliptic operators (Q1110000)

Pseudo-differential operators with weight functions are very useful tools to deal with partial differential operators with degenerate symbols. These weight functions have some restrictions so that the formal expansions of the products of pseudo-differential operators have their substantial meaning. In this paper, a class of pseudo-differential operators with weight functions is introduced with less restrictions, although their formal expansions have the substantial meaning only for a type of products of pseudo-differential operators. In {\S} 1, a class of pseudo-differential operators is introduced. In {\S} 2, a quasi-parametric is constructed for a pseudo-differential operator belonging to a sub-class, in the same way as by \textit{L. Hörmander} [Proc. Sympos. Pure Math. 10, 138-183 (1967; Zbl 0167.096)]. In {\S} 3 and {\S} 4, some applications on energy estimates are given for a class of elliptic differential operators with a parameter, degenerating on the boundary of a domain.