Boundedness and compactness of an integral operator in a mixed norm space on the polydisk

We study the following integral type operator [GRAPHICS] in the space of analytic functions on the unit polydisk U(m) complex vector space C(n). We show that the operator is bounded in the mixed norm space [GRAPHICS] with p,q is an element of [1,alpha) and alpha = (alpha(1)..., alpha(n)), such that alpha(j) > -1 for every j = 1...., n, if and only if sup(z is an element of Un) Pi(n)(j=1)(1-vertical bar z(j)vertical bar)vertical bar g(z)vertical bar < alpha. Also, we prove that the operator is compact if and only if limz ->partial derivative Un Pi(n)(j=1)(1-vertical bar z(j)vertical bar)vertical bar g(z)vertical bar = 0.