Norms of some singular integral operators and their inverse operators

摘要

Let α and β be bounded measurable functions on the unit circle T. Then the singular integral operator S_(α, β) is defined by S_(α,β) f = αP_+f + βP_f, (f ∈ L~2(T)) where P_+ is an analytic projection and P is a co-analytic projection. In this paper, the norms of S_(α,β) and its inverse operator on the Hilbert space L~2(T) are calculated in general, using α, β and α(β-bar) + H~∞. Moreover, the relations between these and the norms of Hankel operators are established. As an application, in some special case in which α and β are nonconstant functions, the norm of S_(α,β) is calculated in a completely explicit form. If α and β are constant functions, then it is well known that the norm of S_(α,β) on L~2(T) is equal to max {|α|, |β|}. If α and β are nonzero constant functions,then it is also known that S_(α,β) on L~2(T) has an inverse operator S_(α~(-1),β~(-1)) whose norm is equal to max {|α|~(-1) |β|~(-1)}.