Estimates in the Hardy-Sobolev space of the annulus and stability result

Feki, Imed

Czechoslovak Mathematical Journal, Tome 63 (2013) no. 2, pp. 481-495

Abstract (VO)

The main purpose of this work is to establish some logarithmic estimates of optimal type in the Hardy-Sobolev space $H^{k,\infty }$; $k \in {\mathbb {N}}^*$ of an annular domain. These results are considered as a continuation of a previous study in the setting of the unit disk by L. Baratchart and M. Zerner, On the recovery of functions from pointwise boundary values in a Hardy-Sobolev class of the disk, J. Comput. Appl. Math. 46 (1993), 255–269 and by S. Chaabane and I. Feki, Optimal logarithmic estimates in Hardy-Sobolev spaces $H^{k,\infty }$, C. R., Math., Acad. Sci. Paris 347 (2009), 1001–1006. As an application, we prove a logarithmic stability result for the inverse problem of identifying a Robin parameter on a part of the boundary of an annular domain starting from its behavior on the complementary boundary part.

Détail

DOI : 10.1007/s10587-013-0032-2

Classification : 30C40, 30H10, 35R30
Keywords: annular domain; Poisson kernel; Hardy-Sobolev space; logarithmic estimate; Robin parameter

Bibliographie

[1] Alessandrini, G., Piere, L. Del, Rondi, L.: Stable determination of corrosion by a single electrostatic boundary measurement. Inverse Probl. 19 (2003), 973-984. | MR

[2] Baratchard, L., Zerner, M.: On the recovery of functions from pointwise boundary values in a Hardy-Sobolev class of the disk. J. Comput. Appl. Math. 46 (1993), 255-269. | DOI | MR | Zbl

[3] Baratchart, L., Leblond, J., Partington, J. R.: Hardy approximation to $L^\infty$ functions on subsets of the circle. Constructive Approximation 12 (1996), 423-435. | MR

[4] Brézis, H.: Analyse fonctionnelle. Théorie et applications. Masson Paris (1983), French. | MR

[5] Chaabane, S., Feki, I.: Optimal logarithmic estimates in Hardy-Sobolev spaces $H^{k,\infty}$. C. R., Math., Acad. Sci. Paris 347 (2009), 1001-1006. | DOI | MR

[6] Chaabane, S., Jaoua, M.: Identification of Robin coefficients by the means of boundary measurements. Inverse Probl. 15 (1999), 1425-1438. | MR | Zbl

[7] Chaabane, S., Fellah, I., Jaoua, M., Leblond, J.: Logarithmic stability estimates for a Robin coefficient in two-dimensional Laplace inverse problems. Inverse Probl. 20 (2004), 47-59. | MR | Zbl

[8] Chaabane, S., Jaoua, M., Leblond, J.: Parameter identification for Laplace equation and approximation in Hardy classes. J. Inverse Ill-Posed Probl. 11 (2003), 33-57. | DOI | MR | Zbl

[9] Chaabane, S., Ferchichi, J., Kunisch, K.: Differentiability properties of the $L^{1}$-tracking functional and application to the Robin inverse problem. Inverse Probl. 20 (2004), 1083-1097. | MR

[10] Chalendar, I., Partington, J. R.: Approximation problems and representations of Hardy spaces in circular domains. Stud. Math. 136 (1999), 255-269. | MR | Zbl

[11] Chevreau, B., Pearcy, C. M., Shields, A. L.: Finitely connected domains $G$, representations of $H^{\infty}(G)$, and invariant subspaces. J. Oper. Theory 6 (1981), 375-405. | MR

[12] Duren, P. L.: Theory of $H^p$ Spaces. Academic Press New York (1970). | MR

[13] Gaier, D., Pommerenke, C.: On the boundary behavior of conformal maps. Mich. Math. J. 14 (1967), 79-82. | DOI | MR | Zbl

[14] Leblond, J., Mahjoub, M., Partington, J. R.: Analytic extensions and Cauchy-type inverse problems on annular domains: stability results. J. Inverse Ill-Posed Probl. 14 (2006), 189-204. | DOI | MR | Zbl

[15] Meftahi, H., Wielonsky, F.: Growth estimates in the Hardy-Sobolev space of an annular domain with applications. J. Math. Anal. Appl. 358 (2009), 98-109. | DOI | MR | Zbl

[16] Nirenberg, L.: An extended interpolation inequality. Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 20 (1966), 733-737. | MR | Zbl

[17] Rudin, W.: Analytic functions of class $H^p$. Trans. Am. Math. Soc. 78 (1955), 46-66. | MR

[18] Sarason, D.: The $H^p$ Spaces of An Annulus. Mem. Am. Math. Soc. 56 (1965), Providence, RI. | MR

[19] Wang, H.-C.: Real Hardy spaces of an annulus. Bull. Austral. Math. Soc. 27 (1983), 91-105. | DOI | MR | Zbl