A canonical decomposition of automorphic forms which vanish on an invariant measurable subset

Abstract

Introduction.Let Γ be a discrete subgroup of the real Mόbius group PSL(2; R).We denote by Ω(Γ) the region of discontinuity of Γ.Let σ be a Γ-invariant closed subset of the extended real line R such that #<j ;> 3 and αs^o, and let D be the component of Ω(Γ)σ containing the upper half-plane U. Then D = U or D = Ω{Γ) -σ according as a -R or not.Let E be a Γ-invariant measurable subset of D, and put V = D -E, where if D Φ U, then E is assumed to be symmetric with respect to R in the sense that zeE whenever zeE.Furthermore, for an integer q ^ 2, let L p , 1 ^ p < <*>, (resp.L°°) be the Banach space consisting of all the p-integrable (resp.bounded) measurable automorphic forms of weight -2q on D for Γ, which are symmetric if D is symmetric (see Section 1 for the precise definition).We denote by A p , 1 <p <; oo, the closed subspace consisting of all the holomorphic elements in ZΛ and set L P (V) = {μeL p ; μ\ E = 0} and A p \ v = {X v φ; φe A p }, where X v is the characteristic function of V.For 1 ^ p < oo and p' satisfying 1/p + 1/p' = 1, L pf is isomorphic to the dual space of ZΛ We denote by (A p ) λ (cL p ') the annihilator of A p .In the present paper, we investigate conditions for E under which (A p ) λ PiL p \V) and A pt \ v are closed and complementary to each other in L P \V), and give two kinds of answers to this question (see Theorems 1 and 3 below).This problem occured in studying extremal quasiconformal mappings with dilatation bound (see, for example, Sakan [10]).Our results can be applied to the study of quasiconformal mappings and Teichmiiller spaces.These applications will be discussed in Ohtake [9].Throughout this paper, as natural assumptions for the problem, we require that V has positive measure and A p Φ {0}.We note that if Έ has (2-dimensional Lebesgue) measure zero, then the spaces (A p ) λ Π

Locations

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