In the paper [B2] Baernstein constructs a simply connected domain Θ in the plane for which the conformal mapping$f$of Θ into the unit disc Δ satisfies $$\int_{R \cap \Omega } {|f'(z)|} ^p |dz| = \infty $$ for some$p$∈(1 2) where R is the real line

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Let$M$be a non-compact connected Riemann surface of a finite type, and$R$⋐$M$be a relatively compact domain such that$H$_{1}($M$,$Z$)=$H$_{1}($R$,$Z$). Let $$\tilde R \to R$$ be a covering. We study the algebra$H$^{∞}($U$) of bounded holomorphic functions defined in certain subdomains $$U \subset \tilde R$$ . Our main result is a Forelli type theorem on projections in$H$^{∞}($D$).

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