Transactions of the American Mathematical Society

Abstract:

For $0<p<\infty$ and $\alpha >-1,$ we let $\mathcal D^p_{\alpha }$ denote the space of those functions $f$ which are analytic in the unit disc $\mathbb {D}$ in $\mathbb C$ and satisfy $\int _{\mathbb {D}}(1-\vert z\vert ^ 2)^ {\alpha }\vert f’(z)\vert ^ p dx dy <\infty$. Of special interest are the spaces $\mathcal D^p_{p-1}$ ($0<p<\infty$) which are closely related with Hardy spaces and the analytic Besov spaces $B^p=\mathcal D^p_{p-2}$ ($1<p<\infty$). A good number of results on the boundedness of integration operators and multipliers from $\mathcal D^p_{\alpha }$ to $\mathcal D^q_{\beta }$ are known in the case $p<q$. Here we are mainly concerned with the upper triangle case $0<q\le p$. We describe the boundedness of these operators from $\mathcal D^p_{\alpha }$ to $\mathcal D^q_{\beta }$ in the case $0<q<p$. Among other results we prove that if $0<q<p$ and $\frac {p\beta -q\alpha }{p-q}\le -1$, then the only pointwise multiplier from $\mathcal D^p_{\alpha }$ to $\mathcal D^q_{\beta }$ is the trivial one. In particular, we have that $0$ is the only multiplier from $\mathcal D^p_{p-1}$ to $\mathcal D^q_{q-1}$ if $p\neq q$, and from $B^p$ to $B^q$ if $1<q<p$. Also, we give a number of explicit examples of multipliers from $\mathcal D^p_{\alpha }$ to $\mathcal D^q_{\beta }$ in the remaining case $\frac {p\beta -q\alpha }{p-q}> -1$. Furthermore, we present a number of results on the self-multipliers of $\mathcal D^p_{\alpha }$ ($0<p<\infty$, $\alpha >-1$). We prove that $0$ is the only compact multiplier from $\mathcal D^p_{p-1}$ to itself ($0<p<\infty$) and we give a number of explicit examples of functions which are self-multipliers of $\mathcal D^p_{\alpha }$.

We also consider the closely related question of characterizing the Carleson measures for the spaces $\mathcal D^p_{\alpha }$. In particular, we prove constructively that a result of Arcozzi, Rochberg and Sawyer characterizing the Carleson measures for $\mathcal D^p_{\alpha }$ 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 $\mathcal D^p_{\alpha }$ ($0<q<p, \alpha >-1$). This result plays a basic role in our work concerning integration operators.