Sharp decay estimates are provided in this paper for spherical averages of a certain multilinear extension operator on $L^{2}(\mathbb{S}^{n-1})\times \ldots\times L^{2}(\mathbb{S}^{n-1})$.

Barry Johnson made major contributions to the theory of Banach algebras, by stimulating research on automatic continuity and cohomology in these algebras. His research on the continuous Hochschild cohomology of Banach and operator algebras led to major developments in these areas, and to a recognition of ‘amenability’ as more than simply a group-theoretic idea, but also one that is widely applicable in modern analysis.

It is known that, if $f$ is a hyperbolic rational function, then the Hausdorff, packing and box dimensions of the Julia set $J(f)$ are equal. It is also known that there is a family of hyperbolic transcendental meromorphic functions with infinitely many poles for which this result fails to be true. In this paper, new methods are used to show that there is a family of hyperbolic transcendental entire functions $f_K$, $K \in {\mathbb N}$, such that the box and packing dimensions of $J(f_K)$ are equal to two, even though as $K \to \infty$ the Hausdorff dimension of $J(f_K)$ tends to one, the lowest possible value for the Hausdorff dimension of the Julia set of a transcendental entire function.

A boundary point of a domain $D$ in $\Bbb{R}^n$ is said to be broadly accessible if it ‘almost lies’ on the boundary of a round ball contained in $D$. If $f$ is a quasiconformal mapping of the unit ball $B^n$ onto $D$, then it is shown that broadly accessible boundary points on $\partial D$ correspond under $f$ to a set of full measure on $\partial B^n$.This research was carried out while R. Näkki was visiting The University of Texas at Austin in 1985–86.