MULTIDIMENSIONAL ANALOG OF A THEOREM OF PRIVALOV

A criterion is established for the continuity of functions that are conjugate in the sense of Cesari to a given function in the class H(omega(j)(delta), j is an element of B, T-N) = {f is an element of C(T-N) :omega(j) (f,delta) = O[omega(j)(delta)], j is an element of B} (where B subset of or equal to M = {1,..., N}, T-N = (-pi,pi)(N), omega(j) (f,delta) (1 less than or equal to j less than or equal to N) are the partial moduli of continuity of f(($) over bar x), and omega(j)(delta) (j is an element of B) are moduli of continuity). Best possible estimates of the partial modulus of continuity of a function conjugate to f is an element of H(omega(j), j is an element of M, T-N) are obtained in the case when the omega(j)(delta) (j is an element of M) satisfy two specific conditions. These conditions on the modulus of continuity omega(delta) are shown to be necessary and sufficient in order that the conjugation operator violate the invariance of the class H (omega(j) = omega, j is an element of M, T-N) in the same way as it violates that of Lip(alpha, C(T-N)) (O < alpha < 1).