Complex Linear Differential Equations with Solutions in Dirichlet–Morrey Spaces

Abstract

The nth derivative criterion for functions belonging to the Dirichlet–Morrey space \({\cal D}_p^\lambda \) is given in this paper. Furthermore, two sufficient conditions for coefficients of the complex linear differential equation

$${f^{\left( n \right)}} + {A_{n - 1}}\left( z \right){f^{\left( {n - 1} \right)}} + \cdots + {A_1}\left( z \right){f^\prime } + {A_0}\left( z \right)f = {A_n}\left( z \right)$$

are obtained such that all solutions belong to \({\cal D}_p^\lambda \), where A_j(z) are analytic functions in the unit disc, j = 0,…,n.