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Morrey space. (English) Zbl 0612.43003

For 1<p<, Ω an open and bounded subset of n and a non-increasing and non-negative function φ defined in (0,ρ 0 ], ρ 0 =diamΩ, we introduce the space 𝔐 φ,0 p (Ω) of locally integrable functions satisfying

inf c { B(x 0 ,ρ)Ω |f(x)-c| p dx}A|B(x 0 ,ρ)|φ p (ρ)

for every x 0 Ω, 0<ρρ 0 , where |B(x 0 ,ρ)| denotes the volume of the ball centered in x 0 and radius ρ. The constant A>0 does not depend on B(x 0 ,ρ)·

We also define the atomic space H p,φ (Ω) as the set of functions f(x) such that f(x)= iI λ i a i (x) in the sense of distributions where λ i , iI |λ i |<, and a i are atoms satisfying a) supp(a i )B(x i ,ρ i )Ω, b) a i (x)dx=0, c) a i p 1/(|B(x i ,ρ i )| 1/q φ(ρ i )), 1/p+1/q=1·

We have: I) If φ (t) is non-increasing and t n φ q (t) is non-decreasing then H p,φ (Ω) is a Banach space. II)𝔐 φ,0 p (Ω) can be represented as the dual of H q,φ (Ω).

43A15L p -spaces and other function spaces on groups, semigroups, etc.
43A17Analysis on ordered groups, H p -theory
46E30Spaces of measurable functions
26A33Fractional derivatives and integrals (real functions)