REMARKS ON SOBOLEV-MORREY-CAMPANATO SPACES DEFINED ON C_{0;\gamma} DOMAINS

Abstract

We discuss a few old results concerning embedding theorems for Campanato and Sobolev-Morrey spaces adapting the formulations to the case of domains of class C 0,γ , and we present more recent results concerning the extension of functions from Sobolev-Morrey spaces defined on those domains. As a corollary of the extension theorem we obtain an embedding theorem for Sobolev-Morrey spaces on arbitrary C 0,γ domains.

Extension and embedding theorems for Campanato spaces on $C^{0,γ}$ domains

“…and similarly f ϕ C 0,α (U ,δγ ) ≤ C f L λ p,γ (Ω) (here we use the fact that ϕ ∈ C 0,α (U , δ γ ) since γα ≤ 1 by assumption and ϕ is smooth). Thus (17) follows also for λ = 1.…”

“…This implies the existence of a constant c > 0 depending only on N, γ and M such that inequality (8) holds for all x ∈ Ω and r > 0. For a proof we refer to [17].…”

“…These spaces are better suited for the validity of embedding and extension theorems in the spirit of classical results. Indeed, by using the general result of [12] one can prove Campanato's embedding Theorem in the non-critical case λ = 1, see [17]. On the other hand, in the critical case λ = 1 it is possible to use the generalised John-Niremberg inequality from [1] to prove that the spaces L 1 p,γ (Ω) are independent of p, a well-known fact in the Lipschitz case γ = 1.…”

“…As discussed in [17], the pioneering analysis carried out in [2] and [12] suggests that in the case of domains of class C 0,γ the Euclidean metric should be replaced by a metric depending on γ. This metric, denoted by δ γ , is anisotropic and its definition depends on the direction of the possible cusps of the boundary.…”

“…We also quote [21] as a standard reference for BMO spaces and [20] for an extensive study of Morrey spaces. Finally, we quote papers [11], [17], [18] for related recent results concerning the extension problem for Sobolev-Morrey spaces.…”