Some Hölder-logarithmic estimates on Hardy-Sobolev spaces

We prove some optimal estimates of Hölder-logarithmic type in the Hardy-Sobolev spaces H^k,p(G), where k ∈ ℕ*, 1 ⩽ p ⩽ ∞ and G is either the open unit disk ⅅ or the annular domain G_s, 0 < s < 1 of the complex space ℂ. More precisely, we study the behavior on the interior of G of any function f belonging to the unit ball of the Hardy-Sobolev spaces H^k,p(G) from its behavior on any open connected subset I of the boundary ∂G of G with respect to the L¹-norm. Our results can be viewed as an improvement and generalization of those established in S. Chaabane, I. Feki (2009), I. Feki, H. Nfata, F. Wielonsky (2012), I. Feki (2013), I. Feki, H. Nfata (2014). As an application, we establish a logarithmic stability results for the Cauchy problem of the identification of Robin’s coefficient by boundary measurements.