Pointwise bound of SO(d,C) -invariant holomorphic functions which are square-integrable with respect to a gaussian measure

The Segal-Bargmenn space HL[superscript2] (C[superscript d], u[subscriptt]) has been extensively studied. Here, we consider the space HL[superscript2] (C[superscriptd], u[subscriptt]) [superscrptSO(d,C) consisting of all functions F in HL[superscript2](C[superscript d, u[subscript t]) which are invariant under the action of the complex special orthogonal group SO(d,C). It is a closed subspace of HL[superscript2] (C[superscript d], u[subscript t]), and hence is a Hilbert space. In this work, we establish a pointwise bound for a function in this space.