Complex linear differential equations with solutions in Bergman-Morrey spaces

In this note, we will show two sufficient conditions such that the solutions of the equation f ( n ) + A n − 1 ( z ) f ( n − 1 ) + ⋯ + A 1 ( z ) f ′ + A 0 ( z ) f = A n ( z ) , lie in the Bergman-Morrey space A p , λ (the space recently introduced by Liu, et al.), where A i ( z ) ( i = 0 , 1 , ⋯ , n ) are analytic functions defined in the unit disc. In addition, we present two sufficient conditions ensuring that all solutions of the complex linear differential equation lie in the Dirichlet-Morrey spaces. This work continues the lines of the previous characterizations by Huusko, et al. and Sun, et al. for growth spaces and Dirichlet-Morrey spaces and so forth.