Azerbaijan Journal of Mathematics V. 10, No 1, 2020, January ISSN 2218-6816 Generalized Morrey Spaces over Unbounded Domains L. Caso, R. D’Ambrosio, L. Softova * Abstract. We study generalized Morrey type spaces M p ω (Ω,d) over unbounded domains. Our goal is to describe the main properties of these spaces and some functional subspaces defined as a closure of the L ∞ and C ∞ 0 functions with respect to the norm in M p ω (Ω,d). Key Words and Phrases: generalized Morrey spaces, vanishing spaces, unbounded domain. 2010 Mathematics Subject Classifications: 46E30, 46E35 1. Introduction In his celebrated work [11] Morrey studied the regularity of the solutions of a kind of elliptic systems. He estimated the L p -norm of the gradient Du of the solution in a ball via a power of the radius of the same ball. That estimate permitted him to obtain local H¨older regularity of u. This result gave rise to the introduction of new functional spaces named after him. The classical Morrey spaces have been formulated and studied in the 60’s by Campanato, Peetre and Brudneii independently, using similar notations. Precisely, a function f ∈ L p loc (R n ) belongs to the Morrey space L p,λ (R n ) with p ≥ 1 and λ ∈ (0,n) if kf k L p,λ (R n ) = sup Br (x)  1 r λ Z Br (x) |f (y)| p dy ! 1 p < +∞ (1) and the supremum is taken over all balls in R n (see [1, 2, 3]). A natural question that arises is what happens if we consider f defined in some domain Ω ⊂ R n bounded or unbounded. In the first case it is enough to take * Corresponding author. http://www.azjm.org 193 c  2010 AZJM All rights reserved.