For a normalized analytic function $f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$ in the unit disk $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$, the estimate of the integral means

$$\begin{eqnarray}L_{1}(r,f):=\frac{r^{2}}{2{\it\pi}}\int _{-{\it\pi}}^{{\it\pi}}\frac{d{\it\theta}}{|f(re^{i{\it\theta}})|^{2}}\end{eqnarray}$$

$f(z)$

$\mathbb{D}\setminus \{0\}$

${\rm\Delta}(r,f)$

$\mathbb{D}_{r}:=\{z\in \mathbb{C}:|z|<r\}$

$f$

$0<r\leq 1$

$L_{1}(r,f)$

${\rm\Delta}(r,z/f)$

$r$

$f$

$m$

This paper introduces a family of area-type integrals over cones. These are used to investigate non-tangential boundary behaviour of harmonic functions on a half-space, extending results of Stein and Brossard.

AMS 2000 Mathematics subject classification: Primary 31B25

The invariantly harmonic functions in the unit ball ${\Bbb B}^n$ in ${\Bbb C}^n$ are those annihilated by the Bergman Laplacian $\Delta$. The Poisson-Szeg\"o kernel $P(z,\zeta)$ solves the Dirichlet problem for $\Delta$: if $f\in C(S^n)$, the Poisson-Szeg\"o transform of $f$, $P[f](z)=\int_{S^n}P(z,\zeta) f(\zeta)\,d\sigma(\zeta),$ where $d\sigma$ is the normalized Lebesgue measure on $S^n$, is the unique invariantly harmonic function $u$ in ${\Bbb B}^n$, continuous up to the boundary, such that $u=f$ on $S^n$. The Poisson-Szeg\"o transform establishes, loosely speaking, a one-to-one correspondence between function theory in $S^n$ and invariantly harmonic function theory in ${\Bbb B}^n$. When $n\geq 2$, it is natural to consider on $S^n$ function spaces related to its natural non-isotropic metric, for these are the spaces arising from complex analysis. In the paper, different characterizations of such spaces of smooth functions are given in terms of their invariantly harmonic extensions, using maximal functions and area integrals, as in the corresponding Euclidean theory. Particular attention is given to characterization in terms of purely radial or purely tangential derivatives. The smoothness is measured in two different scales: that of Sobolev spaces and that of Lipschitz spaces, including BMO and Besov spaces.

1991 Mathematics Subject Classification: 32A35, 32A37, 32M15, 42B25.