We study the boundedness and compactness of weighted composition operators acting on weighted Bergman spaces and weighted Dirichlet spaces by using the corresponding Carleson measures. We give an estimate for the norm and the essential norm of weighted composition operators between weighted Bergman spaces as well as the composition operators between weighted Hilbert spaces.

We define the notion of $\unicode[STIX]{x1D6F7}$-Carleson measures, where $\unicode[STIX]{x1D6F7}$ is either a concave growth function or a convex growth function, and provide an equivalent definition. We then characterize $\unicode[STIX]{x1D6F7}$-Carleson measures for Bergman–Orlicz spaces and use them to characterize multipliers between Bergman–Orlicz spaces.

It is well known that functions in the analytic Besov space $B_{1}$ on the unit disk $\mathbb{D}$ admit an integral representation

$$\begin{eqnarray}f(z)=\int _{\mathbb{D}}\frac{z-w}{1-z\overline{w}}\,d{\it\mu}(w),\end{eqnarray}$$

${\it\mu}$

$|{\it\mu}|(\mathbb{D})<\infty$

$B_{p}$

$0<p\leq 1$

${\rm\Lambda}_{t}$

$t>1$

This paper studies the relationship between vector-valued $\text{BMO}$ functions and the Carleson measures defined by their gradients. Let $dA$ and $dm$ denote Lebesgue measures on the unit disc $D$ and the unit circle $\mathbb{T}$, respectively. For $1\,<\,q\,<\,\infty $ and a Banach space $B$, we prove that there exists a positive constant $c$ such that

$$\underset{{{z}_{0}}\in D}{\mathop{\sup }}\,{{\int }_{D}}{{\left( 1-\left| z \right| \right)}^{q-1}}{{\left\| \nabla f\left( z \right) \right\|}^{q}}{{P}_{{{Z}_{0}}}}\left( z \right)dA\left( z \right)\le {{c}^{q}}\underset{{{z}_{0}}\in D}{\mathop{\sup }}\,{{\int }_{\mathbb{T}}}{{\left\| f\left( z \right)-f\left( {{z}_{0}} \right) \right\|}^{q}}{{P}_{{{z}_{0}}}}\left( z \right)dm\left( z \right)$$

holds for all trigonometric polynomials $f$ with coefficients in $B$ if and only if $B$ admits an equivalent norm which is $q$-uniformly convex, where

$${{P}_{{{z}_{0}}}}\left( z \right)=\frac{1-|{{z}_{0}}{{|}^{2}}}{|1-{{{\bar{z}}}_{0}}z{{|}^{2}}}.$$

The validity of the converse inequality is equivalent to the existence of an equivalent $q$-uniformly smooth norm.

Let $\cal D $ denote the collection of dyadic intervals in the unit interval. Let $\tau$ be a rearrangement of the dyadic intervals. We study the induced operator

$$ Th_I = h_{\tau(I)}$$

where $h_I$ is the $L^{\infty}$ normalized Haar function. We find geometric conditions on $\tau$ which are necessary and sufficient for $T$ to be bounded on $BMO$. We also characterize the rearrangements of the $L^p$ normalized Haar system in $L^p.$

1991 Mathematics Subject Classification: 42C20, 43A17, 47B38.