A trace law for the Hardy-Morrey-Sobolev space

This article shows that if HLp,κ and HLμq,λ are (p,κ)-Hardy-Morrey space based on the standard Lebesgue measure and (q,λ)-Hardy-Morrey space induced by the non-negative Radon measure μ respectively then one has the following trace law for the Hardy-Morrey-Sobolev space Iα(HLp,κ) of Riesz potentials of order α of HLp,κ-functions:sup0<‖f‖HLp,κ<∞⁡‖Iαf‖HLμq,λ‖f‖HLp,κ−1<∞⟺sup(x,r)∈Rn×(0,∞)⁡r−βμ(B(x,r))<∞ under{0<λ≤κ≤n;0<α