Interpolation along manifolds in Hardy-Sobolev spaces
Let \(B\) be the unit ball in \(\mathbb{C}^n\) and \(S\) be its boundary. For a holomorphic function \(f\) on \(B\) with a homogeneous expansion \(\Sigma_k f_k\), the fractional derivative of order \(s>0\) is defined by \(R^s(f)= \Sigma_k (k+1)^sf_k\). For \(p\geq 0\) and \(s>0\) the Hardy-Sobolev space \(H^p_s\) consists of those holomorphic functions on \(B\) for which \(\| f\|^p_{p,s} =\sup_{r<1}\int_S | R^sf(r) |^pdm< +\infty\) where \(dm\) denotes the normalised Lebesgue measure on \(S\). The aim of this paper is to study the trace of \(H^p_s\) on closed complex tangential submanifolds \(\Gamma\) of \(S\), that is to study the restriction on \(\Gamma\) not only of the function \(f\in H^p_s\) itself but of its fractional derivatives in direction to given vector field that may be or not be tangential to \(S\) as well. The main result states that each \(f\in H^p_s\) induces on \(\Gamma\) a ``jet'' in Besov type class \(B^p_\beta (\Gamma)\), where the smoothness index \(\beta\) depends on \(p,s\) and the dimension \(d\) of \(\Gamma\).
scientific article; zbMATH DE number 1192502