MULTIPLIERS AND INTEGRATION OPERATORS ON DIRICHLET SPACES

For 0 < p < ∞ and α> −1, we let Dpα denote the space of those functions f which are analytic in the unit disc D in C and satisfy D(1−|z|2)α|f ′(z)|p dx dy <∞. Of special interest are the spaces Dpp−1 (0 < p < ∞) which are closely related with Hardy spaces and the analytic Besov spaces Bp = Dpp−2 (1 < p < ∞). A good number of results on the boundedness of integration operators and multipliers from Dpα to Dqβ are known in the case p < q. Here we are mainly concerned with the upper triangle case 0 < q ≤ p. We describe the boundedness of these operators from Dpα to Dqβ in the case 0 < q < p. Among other results we prove that if 0 < q < p and pβ−qα p−q ≤ −1 then the only pointwise multiplier from Dpα to Dqβ is the trivial one. In particular, we have that 0 is the only multiplier from Dpp−1 to Dqq−1 if p 6 = q, and from Bp to Bq if 1 < q < p. Also, we give a number o