On some extension property for BMO functions on Riemann surfaces
Let \(\Omega\) be a plane domain of hyperbolic type. One can consider two types of areal BMO spaces on \(\Omega\), by examining mean oscillation of \(L^ 1(\Omega)\) functions over disks contained in \(\Omega\) (intrinsic BMO), or by pulling functions back to the unit disk by a universal covering map and then computing the areal BMO norm there (conformal BMO). The author's first main result gives several equivalent conditions on \(\Omega\) which insure that the two BMO spaces coincide. Some of these conditions had been found earlier by B. Osgood. Then the author turns his attention to Riemann surfaces R. He defines a space of functions on R which when R is planar reduces to intrinsic BMO and gives, when R is hyperbolic, some relations between intrinsic and conformal BMO(R), along with various properties of his new intrinsic space. Finally, he gives extension theorems for the two types of BMO in the context of R and its double which in the case of intrinsic BMO on plane domains bounded by quasi-circles reduces to a theorem of P. Jones.
scientific article; zbMATH DE number 4118689