Multipliers and integration operators on Dirichlet spaces

Source abstract

For 0 > p > ∞ 0>p>\infty and α > − 1 , \alpha >-1,\, we let D α p \mathcal D^p_{\alpha } denote the space of those functions f f which are analytic in the unit disc D \mathbb {D} in C \mathbb C and satisfy ∫ D ( 1 − | z | 2 ) α | f ′ ( z ) | p d x d y > ∞ \int _{\mathbb {D}}(1-\vert z\vert ^ 2)^ {\alpha }\vert f’(z)\vert ^ p\, dx\, dy >\infty . Of special interest are the spaces D p − 1 p \mathcal D^p_{p-1} ( 0 > p > ∞ 0>p>\infty ) which are closely related with Hardy spaces and the analytic Besov spaces B p = D p − 2 p B^p=\mathcal D^p_{p-2} ( 1 > p > ∞ 1>p>\infty ). A good number of results on the boundedness of integration operators and multipliers from D α p \mathcal D^p_{\alpha } to D β q \mathcal D^q_{\beta } are known in the case p > q p>q . Here we are mainly concerned with the upper triangle case 0 > q ≤ p 0>q\le p . We describe the boundedness of these operators from D α p \mathcal D^p_{\alpha } to D β q \mathcal D^q_{\beta } in the case 0 > q > p 0>q>p . Among other results we prove that if 0 > q > p 0>q>p and p β − q α p − q ≤ − 1 \frac {p\beta -q\alpha }{p-q}\le -1 , then the only pointwise multiplier from D α p \mathcal D^p_{\alpha } to D β q \mathcal D^q_{\beta } is the trivial one. In particular, we have that 0 0 is the only multiplier from D p − 1 p \mathcal D^p_{p-1} to D q − 1 q \mathcal D^q_{q-1} if p ≠ q p\neq q , and from B p B^p to B q B^q if 1 > q > p 1>q>p . Also, we give a number of explicit examples of multipliers from D α p \mathcal D^p_{\alpha } to D β q \mathcal D^q_{\beta } in the remaining case p β − q α p − q > − 1 \frac {p\beta -q\alpha }{p-q}> -1 . Furthermore, we present a number of results on the self-multipliers of D α p \mathcal D^p_{\alpha } ( 0 > p > ∞ 0>p>\infty , α > − 1 \alpha >-1 ). We prove that 0 0 is the only compact multiplier from D p − 1 p \mathcal D^p_{p-1} to itself ( 0 > p > ∞ 0>p>\infty ) and we give a number of explicit examples of functions which are self-multipliers of D α p \mathcal D^p_{\alpha } . We also consider the closely related question of characterizing the Carleson measures for the spaces D α p \mathcal D^p_{\alpha } . In particular, we prove constructively that a result of Arcozzi, Rochberg and Sawyer characterizing the Carleson measures for D α p \mathcal D^p_{\alpha } in the range − 1 > α > p − 1 -1>\alpha >p-1 cannot be extended to cover the case α = p − 1 \alpha =p-1 and we find a certain condition on a measure μ \mu which is necessary for μ \mu to be a q q -Carleson measure for D α p \mathcal D^p_{\alpha } ( 0 > q > p , α > − 1 0>q>p,\, \alpha >-1 ). This result plays a basic role in our work concerning integration operators.