In 1945 the first author introduced the classes $H^p\left(\alpha \right)$, 1 ≤ p<∞, α > -1, of holomorphic functions in the unit disk with finite integral (1) ∬ |f(ζ)|p (1-|ζ|²)α dξ dη < ∞ (ζ=ξ+iη) and established the following integral formula for $f\in H^p\left(\alpha \right)$: (2) f(z) = (α+1)/π ∬ f(ζ) ((1-|ζ|²)α)/((1-zζ̅)2+α) dξdη, z∈ . We have established that the analogues of the integral representation (2) hold for holomorphic functions in Ω from the classes $L^p(\Omega ;{\left[K\left(w\right)\right]}^{\alpha }dm\left(w\right))$, where: 1) $\Omega =w=(w₁,...,w_n)\in {ℂ}^n:Imw₁>{\sum }_{k=2}^n\left|w_k\right|²$, $K\left(w\right)=Imw₁-{\sum }_{k=2}^n\left|w_k\right|²$; 2) Ω is the matrix domain consisting of those complex m...

For α ≥ 0 let ${ℱ}_{\alpha }$ denote the class of functions defined for |z| < 1 by integrating $1/{(1-xz)}^{\alpha }$ if α > 0, and log(1/(1-xz)) if α = 0, against a complex measure on |x| = 1. We study families of starlike functions where zf’(z)/f(z) ranges over a parabola with given focus and vertex. We prove a number of properties of these functions, among others that they are bounded and that they belong to ${ℱ}_0$. In general, it is only known that bounded starlike functions belong to ${ℱ}_{\alpha }$ for α > 0.

Let $R$ be a hyperbolic Riemann surface, $d_{\chi }$ a harmonic measure supported on the Martin boundary of $R$, and $H^{\infty }\left(d\chi \right)$ the subalgebra of $L^{\infty }\left(d\chi \right)$ consisting of the boundary values of bounded analytic functions on $R$. This paper gives a complete classification of the closed $H^{\infty }\left(d\chi \right)$-submodules of $L^p\left(d\chi \right)$, $1\le p\le \infty$ (weakly${}^{*}$ closed, if $p=\infty$, when $R$ is regular and admits a sufficiently large family of bounded multiplicative analytic functions satisfying an approximation condition. It also gives, as a corollary, a corresponding result for the Hardy...

We considerably improve our earlier results [Ann. Inst. Fourier, 24-4 (1974] concerning Cauchy-Read’s theorems, convergence of Green lines, and the structure of invariant subspaces for a class of hyperbolic Riemann surfaces.