Fractional derivatives of holomorphic functions on bounded symmetric domains of Cn

Abstract

Let f∈H(Bn). f|β| denotes the βth fractional derivative of f. If f|β|∈Ap,q,α(Bn), we show that(I) β<α+1p+nq=δ, then f∈As,t,α(Bn), and ‖f‖s,t,α≤C‖f|β|‖p,q,α, s=δpδ−β, t=δqδ−β(II) If β=α+1p+nq, then f∈B(Bn) and ‖f‖B≤C‖f|β|‖p,q,α(III) If β>α+1p+nq, then f∈Λβ−α+1p−nq(Bn) especially If β=1 then ‖f‖Λ1−α+1p−nq≤C‖f|1|‖p,q,α where Bn is the unit ball of Cn.