For 0 < γ ≤ 1, let $\Lambda {⁺}_{\gamma }$ be the big Lipschitz algebra of functions analytic on the open unit disc which satisfy a Lipschitz condition of order γ on ̅. For a closed set E on the unit circle and an inner function Q, let $J_{\gamma }(E,Q)$ be the closed ideal in $\Lambda {⁺}_{\gamma }$ consisting of those functions $f\in \Lambda {⁺}_{\gamma }$ for which (i) f = 0 on E, (ii) $|f\left(z\right)-f\left(w\right)|=o\left(\right|z-w{|}^{\gamma })$ as d(z,E),d(w,E) → 0, (iii) $f/Q\in \Lambda {⁺}_{\gamma }$. Also, for a closed ideal I in $\Lambda {⁺}_{\gamma }$, let $E_I$ = z ∈ : f(z) = 0 for every f ∈ I and let $Q_I$ be the greatest common divisor of the inner parts of non-zero functions in I....

The structure of closed ideals of a regular algebra containing the classical A∞ is considered. Several division and approximation results are proved and a characterization of those ideals whose intersection with A∞ is not {0} is obtained. A complete description of the ideals with countable hull is given, with applications to synthesis of hyperfunctions.

We continue an investigation started in a preceding paper. We discuss tha classical result of Carleson connecting Carleson measures with the ∂-equation in a slightly more abstract framework than usual. We also consider a more recent result of Peter Jones which shows the existence of a solution for the ∂-equation, which satisfies simultaneously a good L∞ estimate and a good L1 estimate. This appears as a special case of our main result which can be stated as follows:Let (Ω, A, μ) be any measure space....

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.

Let $\mu$ be a measure on a domain $\Omega$ in ${ℂ}^n$ such that the Bergman space of holomorphic functions in $L^2(\Omega ,\mu )$ possesses a reproducing kernel $K(x,y)$ and $K(x,x)\&gt;0$$\forall x\in \Omega$. The Berezin transform associated to $\mu$ is the integral...