F-mild Hyperfunctions and Fuchsian Partial Differential Equations
Toshinori Ôaku · Advanced studies in pure mathematics · 2018
IntroductionNon-characteristic boundary value problems were formulated for hyperfunctions by and Schapira [12].They defined the boundary values of hyperfunction solutions and proved the uniqueness of solutions of the boundary value problem.Solvability of the (local) boundary value problem was proved by Kaneko [2] under the assumption of semi-hyperbolicity.Kataoka [6,8] introduced the notion of mildness on the boundary for hyperfunctions.He studied non-characteristic boundary value problems in detail by using the theory of mild hyperfunctions (see [7,8]).Let P be a linear partial differential operator of order m with analytic coefficients defined on an open subset M of Rn '" x=(xt> x'), and set int M+ ={x EM; xl>O} and N={x EM; Xl =O}.Suppose that N is noncharacteristic with respect to P. Then any hyperfunction u(x) defined on int M+ satisfying Pu(x)=O becomes mild on N, and the boundary value vix' ) = (ajaxl)lu( +0, x') is defined as a hyperfunction on Nfor any integer j :2::0.Moreover if VO(X'), ... , Vm-I(X' ) vanish, then u(x) vanishes near N.However, if N is characteristic with respect to P, then u(x) is not mild in general.In this paper, we define the F-mildness for hyperfunctions defined on int M+.The notion of F-mildness is a generalization of that of mildness.If u(x) is F-mild on N, we can define the boundary value vix' ) = (ajaXI)JU( +0, x') for any integer j>O as a hyperfunction on N in a natural way.Using F-mild hyperfunctions, we formulate boundary value problems for Fuchsian partial differential operators and prove the uniqueness of solutions of the boundary value problem.Let P be a Fuchsian partial differential operator of weight m -k with respect to Xl in the sense of Baouendi-Goulaouic [1] and let u(x) be a hyperfunction on int M+ satisfying Pu(x)=O.Assume that the characteristic exponents of P avoid certain integral values.Under these assumptions, if u(x) is F-mild on N