Function spaces of Lorentz-Sobolev type: Atomic decompositions, characterizations in terms of wavelets, interpolation and multiplications

Abstract

We establish atomic decompositions and characterizations in terms of wavelets for Besov-Lorentz spaces Bsq Lp,r (Rn) and for Triebel-Lizorkin-Lorentz spaces Fsq Lp,r (Rn) in the whole range of parameters. As application we obtain new interpolation formulae between spaces of Lorentz-Sobolev type. We also remove the restrictions on the parameters in a result of Peetre on optimal embeddings of Besov spaces. Moreover, we derive results on diffeomorphisms, extension operators and multipliers for Bsq Lp,∞ (Rn). Finally, we describe Bsq Lp,r (Rn) as an approximation space, which allows us to show new sufficient conditions on parameters for Bsq Lp,r (Rn) to be a multiplication algebra.