Article
Full entry | Fulltext not available
(moving wall
24 months)
Feedback
Keywords:
Hardy-Sobolev space; annular domain; Kernel function
Hardy-Sobolev space; annular domain; Kernel function
Summary:
We prove some optimal estimates of Hölder-logarithmic type in the Hardy-Sobolev spaces $H^{k,p}(G)$, where $k \in {\mathbb N}^*$, $1\leq p\leq \infty $ and $G$ is either the open unit disk ${\mathbb D}$ or the annular domain $G_s$, $0<s<1$ of the complex space ${\mathbb C}$. 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 $\partial G$ of $G$ with respect to the $L^1$-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.
References:
[1] Alessandrini, G., Piere, L. Del, Rondi, L.: Stable determination of corrosion by a single electrostatic boundary measurement. Inverse Probl. 19 (2003), 973-984. DOI 10.1088/0266-5611/19/4/312 | MR 2005313 | Zbl 1050.35134
[2] Baratchart, L., Mandréa, F., Saff, E. B., Wielonsky, F.: 2D inverse problems for the Laplacian: A meromorphic approximation approach. J. Math. Pures Appl. (9) 86 (2006), 1-41. DOI 10.1016/j.matpur.2005.12.001 | MR 2246355 | Zbl 1106.35129
[3] Baratchart, 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 10.1016/0377-0427(93)90300-Z | MR 1222486 | Zbl 0818.65017
[4] 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 10.1016/j.crma.2009.07.018 | MR 2554565 | Zbl 1181.46023
[5] Chalendar, I., Partington, J. R.: Approximation problems and representations of Hardy spaces in circular domains. Stud. Math. 136 (1999), 255-269. DOI 10.4064/sm-136-3-255-269 | MR 1724247 | Zbl 0952.30033
[6] 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 0643698 | Zbl 0525.47004
[7] Duren, P. L.: Theory of $H^p$ Spaces. Pure and Applied Mathematics 38. Academic Press, New York (1970). DOI 10.1016/s0079-8169(08)x6157-4 | MR 0268655 | Zbl 0215.20203
[8] Feki, I.: Estimates in the Hardy-Sobolev space of the annulus and stability result. Czech. Math. J. 63 (2013), 481-495. DOI 10.1007/s10587-013-0032-2 | MR 3073973 | Zbl 1289.30231
[9] Feki, I., Nfata, H.: On $L^p-L^1$ estimates of logarithmic-type in Hardy-Sobolev spaces of the disk and the annulus. J. Math. Anal. Appl. 419 (2014), 1248-1260. DOI 10.1016/j.jmaa.2014.05.042 | MR 3225432 | Zbl 1293.30073
[10] Feki, I., Nfata, H., Wielonsky, F.: Optimal logarithmic estimates in the Hardy-Sobolev space of the disk and stability results. J. Math. Anal. Appl. 395 (2012), 366-375. DOI 10.1016/j.jmaa.2012.05.055 | MR 2943628 | Zbl 1250.30051
[11] Hardy, G. H.: The mean value of the modulus of an analytic function. Proc. Lond. Math. Soc. (2) 14 (1915), 269-277 \99999JFM99999 45.1331.03. DOI 10.1112/plms/s2_14.1.269
[12] 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 10.1515/156939406777571049 | MR 2242304 | Zbl 1111.35121
[13] 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 10.1016/j.jmaa.2009.04.040 | MR 2527584 | Zbl 1176.46029
[14] Miller, P. D.: Applied Asymptotic Analysis. Graduate Studies in Mathematics 75. AMS, Providence (2006). DOI 10.1090/gsm/075 | MR 2238098 | Zbl 1101.41031
[15] Nirenberg, L.: An extended interpolation inequality. Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 20 (1966), 733-737. MR 0208360 | Zbl 0163.29905
[16] Rudin, W.: Analytic functions of class $H_p$. Trans. Am. Math. Soc. 78 (1955), 46-66. DOI 10.1090/S0002-9947-1955-0067993-3 | MR 0067993 | Zbl 0064.31203
[17] Sarason, D.: The $H^p$ spaces of an annulus. Mem. Am. Math. Soc. 56 (1965), 78 pages. DOI 10.1090/memo/0056 | MR 0188824 | Zbl 0127.07002
[18] Wang, H.-C.: Real Hardy spaces of an annulus. Bull. Austr. Math. Soc. 27 (1983), 91-105. DOI 10.1017/S0004972700011515 | MR 0696647 | Zbl 0512.42023
Search
Partner of
