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Hardy-Sobolev space; Hardy-Landau-Littlewood inequality; Hölder regularity; Cauchy problem; inverse problem; logarithmic estimate
We prove some optimal logarithmic estimates in the Hardy space ${H}^{\infty }(G)$ with Hölder regularity, where $G$ is the open unit disk or an annular domain of $\mathbb {C}$. These estimates extend the results established by S. Chaabane and I. Feki in the Hardy-Sobolev space $H^{k,\infty }$ of the unit disk and those of I. Feki in the case of an annular domain. The proofs are based on a variant of Hardy-Landau-Littlewood inequality for Hölder functions. As an application of these estimates, we study the stability of both the Cauchy problem for the Laplace operator and the Robin inverse problem.
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