MULTIPLIERS AND INTEGRATION OPERATORS ON DIRICHLET SPACES

For 0 < p < infinity and alpha > -1, we let D-alpha(p) denote the space of those functions f which are analytic in the unit disc ID in C and satisfy f(D)(1 - vertical bar z vertical bar(2))(alpha)vertical bar f' (z)vertical bar(p) cis dy < infinity. Of special interest are the spaces D-p-1(p) (0 < p < infinity) which are closely related with Hardy spaces and the analytic Besov spaces B-p = D-p-2(p) (1 < p < infinity). A good number of results on the boundedness of integration operators and multipliers from D-alpha(p) to D-beta(q) 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 D-alpha(p) to D-beta(q) in the case 0 < q < p. Among other results we prove that if 0 < q < p and p beta-q alpha/p-q <= -1 then the only pointwise multiplier from D-alpha(p) to D-beta(q) is the trivial one. In particular, we have that 0 is the only multiplier from to D-p-1(p) to D-q-1(q) if p not equal q, and from B-p to B-q if 1 < q < p. Also, we give a number of explicit examples of multipliers from D-alpha(p) to D-beta(q) in the remaining case p beta-q alpha/p-q > -1. Furthermore, we present a number of results on the self-multipliers of D-alpha(p) < p < infinity, alpha > -1). We prove that 0 is the only compact multiplier from D-p-1(p) to itself (0 < p < infinity) and we give a number of explicit examples of functions which are self-multipliers of D-alpha(p). We also consider the closely related question of characterizing the Carleson measures for the spaces D-alpha(p). In particular, we prove constructively that a result of Arcozzi, Rochberg and Sawyer characterizing the Carleson measures for D-alpha(p) in the range -1 < alpha < p - 1 cannot be extended to cover the case alpha = p - 1 and we find a certain condition on a measure mu which is necessary for mu to be a q-Carleson measure for D-alpha(p) (0 < q < p, alpha > 1). This result plays a basic role in our work concerning integration