In this paper we study the ($L$^{$p$}($u$^{$p$}),$L$^{$q$}($v$^{$q$})) boundedness of the higher order commutators$T$_{$Ω,α,b$}^{$m$}and$M$_{$Ω,α,b$}^{$m$}formed by the fractional integral operator$T$_{$Ω,α$}, the fractional maximal operator$M$_{$Ω,α$}, and a function$b(x)$in BMO($v$), respectively.

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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