Poisson semigroup characterization and the trace theorem of a class of fractional mean oscillation spaces

Abstract

In this paper, we use regular wavelets to study the Poisson extension of the fractional mean oscillation spaces  , (R n ). Via a distributional trace operator , we establish a relation between  , (R n ) and a class of harmonic function spaces  , (R n+1 + ). KEYWORDSdistributional trace, fractional bounded mean oscillation spaces, harmonic function, waveletwhere the supremum is taken over all cubes Q and S ,p,f denotes the set of all polynomials satisfying certain conditions; see Definition 2.4 for the details. The spaces  , (R n ) can be seen as the generalizations of the following function spaces: the bounded mean oscillation space BMO(R n ), the Bloch space B 0,∞ ∞ (R n ), the Hölder spaces  (R n ), 0 < < 1, and the Besov-Q spacesḂ ,0 , (R n ). We give the following table to clarify the relation between these spaces and  , (R n ):