The Bloch Space and Bmo Analytic Functions in the Tube over the Spherical Cone

We prove that the Bloch space coincides with the space BMOA in the tube over the spherical cone of R.3; this extends a well-known onedimensional result. Introduction. Let Q be a symmetric Siegel domain of type II contained in Cn. Let V denote the Lebesgue measure in fi and H{Q) the space of holomorphic (or analytic) functions in fi. When n = 1 and fi = tt+ = {z £ C: Imz > 0}, a Bloch function is an element / of H{tt+) which satisfies the estimate 11/11.»= sup {y\f'{z)\} <oo. z=x-\-iytzit+ The Bloch space 3§ of tt+ is then the quotient space of the space of Bloch functions by the subspace of constant functions. It is well known that in ir+, the Bloch space ¿¡8 coincides with the quotient space BMOA of the space of BMO analytic functions by the subspace of constant functions. The definition of BMO in 7r+ is the same as that of (solid) BMO in the unit disk (cf. [6, p. 631]): in tt+, a locally integrable function / is said to be BMO if there exists a constant C such that for any disk D contained in 7r+, there is a constant fo such that mU'-fD\dV <c. In C2, this result can easily be extended to the cartesian product {n+)2 of two upper half-planes. In this case, a Bloch function is an element of H[{ir+)2} which satisfies the estimate sup < y0yi z={z0,zi) = (x0+iyo,x¡+iy¡)€(ir+)'2 I d2 dzodzi < CO. The Bloch space SS of {ir+)2 is then the quotient space of the space of Bloch functions by the subspace JV= {'s*(<*+«:ää'«s0}Received by the editors May 1, 1986 and, in revised form, December 4, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 32M15, 46E99, 47B38.