Norm-controlled Inversion And Stability Of Operators

Wiener’s lemma is a classic result in harmonic analysis,which states that If the function f(x)has a Fourier series of absolute convergence and is not zero everywhere in the field of real numbers,then 1/f(x)also has a Fourier series of absolute convergence.Wiener’s lemma actually means that the localized memory of the operator is preserved during the inversion process,that is,the invertible matrix or integral operator with some non-diagonal attenuation,the inverse also has the same non-diagonal attenuation.The exact constant problem involved here is fundamental to many applications.The full text is divided into four chapters.In the first chapter,we review the development history and important results of Wiener’s lemma,find the unsolved academic problems with research value,and determine whether the research problems are:whether the inverse closure of singular integral operators in homogeneous spaces and the Lphi-stability of local integral operators in Orlicz spaces can be establishedIn Chapter 2,we introduced the generalized Beurling class IBr,u,α,β,proves on homogeneous spaces(χ,ρ,μ)that if(χ,ρ,μ)exists order θ and Ahlfors d regular measure,μ,u is(α,r)-permissible weight,and generalized Beurling class Banach algebra IBr,u,α,β element A has inverse closure,then A-1 in IBr,u,α,β,A-1 in IBr,u,α,β with polynomial-controlled upper bounds.In Chapter 3,we introduce the Wiener class Wp,uα,prove that Wp,uα has norm control inversion in B2,where mathcalB2 is a Banach algebra on L2 consisting of all operators bounded on the standard norm.In Chapter 4,we show that if there exists a Young function satisfying the Δ2 condition such that the integral operator I+T in an Orlicz space with Luxemburg’s norm has LΦ0-stability,then the integral operator I+T has LΦ-stability for any Young function that satisfies the Δ2 condition.In Chapter 5,we summarize the full text.