WEIGHTED COMPOSITION OPERATORS BETWEEN H∞ AND GENERALLY WEIGHTED BLOCH SPACES ON POLYDISKS

Let center dot(n) be the unit polydisk of center dot(n), center dot(center dot) = (center dot(1)(center dot) = (center dot)center dot center dot(2)(center dot)center dot center dot center dot center dot center dot center dot(n)(center dot)) be a holomorphic self- map of center dot(n) and. be center dot holomorphic function on center dot(n). center dot(infinity)(center dot(n)) is the space of all bounded holomorphic functions on center dot(n) and by a generally weighted Bloch space we mean center dot alpha log (n) = {center dot epsilon center dot(center dot(n)) : sup z epsilon Un Sigma(n)(k)=1 vertical bar partial derivative f/partial derivative z(k)(center dot)vertical bar 1-vertical bar center dot(k)vertical bar(2))(alpha)vertical bar log 2/1-z(k vertical bar)(2) }. Wegive necessary and su. cient conditions of the boundedness and compactness of the weighted composition operator.C. between H8( Un) and Ba log( Un).