Some Remarks on Spaces of Morrey Type

and Applied Analysis 3 In literature, several authors have considered different kinds of weighted spaces of Morrey type and their applications to the study of elliptic equations, both in the degenerate case and in the nondegenerate one see e.g., 9–11 . In this paper, given a weight ρ in a class of measurable functions G Ω see § 6 for its definition , we prove that the corresponding weighted space M ρ Ω is a space settled between M o Ω and M̃ Ω . In particular, we provide some conditions on ρ that entail M p,λ o Ω M p,λ ρ Ω . Taking into account the results of this paper, we are now in position to approach the study of some classes of elliptic problems with discontinuous coefficients belonging to the weighted Morrey type space M ρ Ω . 2. Notation and Preliminary Results Let G be a Lebesgue measurable subset of R and Σ G be the σ-algebra of all Lebesgue measurable subsets of G. Given F ∈ Σ G , we denote by |F| its Lebesgue measure and by χF its characteristic function. For every x ∈ F and every t ∈ R ,we set F x, t F∩B x, t ,where B x, t is the open ball with center x and radius t, and in particular, we put F x F x, 1 . The class of restrictions to F of functions ζ ∈ C∞ o R with F∩supp ζ ⊆ F is denoted by D F and, for p ∈ 1, ∞ , Lploc F is the class of all functions g : F → R such that ζ g ∈ L F for any ζ ∈ D F . Let us recall the definition of the classical Morrey space L R . For n ≥ 2, λ ∈ 0, n and p ∈ 1, ∞ , L R is the set of the functions g ∈ Lploc R such that ∥ ∥g ∥ ∥ Lp,λ Rn sup τ>0 x∈Rn τ−λ/p ∥ ∥g ∥ ∥ Lp B x,τ < ∞, 2.1 equipped with the norm defined by 2.1 . IfΩ is an unbounded open subset of R and t is fixed in R , we can consider the space M Ω, t , which is larger than L R whenΩ R. More precisely,M Ω, t is the set of all functions g in Lploc Ω such that ∥ ∥g ∥ ∥ Mp,λ Ω,t sup τ∈ 0,t x∈Ω τ−λ/p ∥ ∥g ∥ ∥ Lp Ω x,τ < ∞, 2.2 endowed with the norm defined in 2.2 . We explicitly observe that a diadic decomposition gives for every t1, t2 ∈ R the existence of c1, c2 ∈ R , depending only on t1, t2, and n, such that c1 ∥ ∥g ∥ ∥ Mp,λ Ω,t1 ≤ ∥∥g∥∥Mp,λ Ω,t2 ≤ c2 ∥ ∥g ∥ ∥ Mp,λ Ω,t1 , ∀g ∈ M Ω, t1 . 2.3 All the norms being equivalent, from now on, we consider the space M Ω M Ω, 1 . 2.4 4 Abstract and Applied Analysis For the reader’s convenience, we briefly recall some properties of functions in L R and M Ω needed in the sequel. The first lemma is a particular case of a more general result proved in 12, Proposition 3 . Lemma 2.1. Let Jh h∈N be a sequence of mollifiers in R . If g ∈ L R and lim y→ 0 ∥ ∥g ( x − y) − g x ∥∥Lp,λ Rn 0, 2.5