BERGMAN TYPE OPERATORS ON A CLASS OF WEEKLY PSEUDOCONVEX DOMAINS

A necessary and sufficient condition for the boundedness of the operat or: (T(s,u,v)f)(xi)=h(u+v/a)(xi)integral(Omega a)h(s)(xi')K-s,K-u,K-v( xi,xi')f(xi')dv(xi') on L-p(Omega(a),dv(lambda)),1 < p < infinity, is obtained, where Omega(a)={xi=(z,w)is an element of Cn+m:z is an elemen t of C-n, w is an element of C-m, \z\(2)+\w\(2)/a<1}, h(xi)=(1-\z\(2)) (a)-\w\(2) and K-x,K-u,K-v(xi,xi'). This generalizes the works in lite rature from the unit ball or unit disc to the weakly pseudoconvex doma in Omega(a). As an application, it is proved that f is an element of L -H(p)(Omega(a),dv lambda) implies h(\a\/a+\beta\)(xi)D-2 alpha D-z bet a f is an element of L-p(Omega a,dv lambda),1 less than or equal to p < infinity, for any multi-index alpha=(alpha(1),...,alpha(n)) and beta =(beta(1),...,beta(n)). An interesting question is whether the convers e holds.