Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
annular domain; Poisson kernel; Hardy-Sobolev space; logarithmic estimate
annular domain; Poisson kernel; Hardy-Sobolev space; logarithmic estimate
Summary:
The main purpose of this article is to give a generalization of the logarithmic-type estimate in the Hardy-Sobolev spaces $H^{k,p}(G)$; $k \in {\mathbb N}^*$, $1 \leq p \leq \infty $ and $G$ is the open unit disk or the annulus of the complex space $\mathbb C$.
References:
[1] Alessandrini, G., Piero, 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] Balinskii, A. A., Novikov, S. P.: Poisson brackets of hydrodynamic type, Frobenius algebras and Lie algebras. Sov. Math., Dokl. 32 (1985), 228-231. English. Russian original translation from Dokl. Akad. Nauk SSSR 283 1985 1036-1039. MR 0802121 | Zbl 0606.58018
[3] Babenko, V. F., Kofanov, V. A., Pichugov, S. A.: Inequalities of Kolmogrov type and some their applications in approximation theory. Proceedings of the 3rd International Conference on Functional Analysis and Approximation Theory, Palermo: Circolo Matemàtico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 52 (1998), 223-237. MR 1644551 | Zbl 0909.41008
[4] 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
[5] 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
[6] Brézis, H.: Analyse fonctionnelle. Théorie et applications. Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris (1983), French. MR 0697382 | Zbl 0511.46001
[7] 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
[8] 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. DOI 10.1088/0266-5611/20/1/003 | MR 2044605 | Zbl 1055.35135
[9] 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
[10] 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
[11] 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
[12] 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
[13] 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
[14] 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
[15] Gaier, D., Pommerenke, C.: On the boundary behavior of conformal maps. Mich. Math. J. 14 (1967), 79-82. DOI 10.1307/mmj/1028999660 | MR 0204631 | Zbl 0182.10204
[16] Hardy, G. H.: The mean value of the modulus of an analytic function. Lond. M. S. Proc. (2) 14 (1915), 269-277. DOI 10.1112/plms/s2_14.1.269 | Zbl 45.1331.03
[17] Hardy, G. H., Landau, E., Littlewood, J. E.: Some inequalities satisfied by the integrals or derivatives of real or analytic functions. Math. Z. 39 (1935), 677-695. DOI 10.1007/BF01201386 | MR 1545530 | Zbl 0011.06102
[18] Klots, B. E.: Approximation of differentiable functions by functions of large smoothness. Math. Notes 21 (1977), 12-19. English. Russian original translation from Mat. Zametki 21 1977 21-32. DOI 10.1007/BF02317028 | MR 0481797 | Zbl 0363.41015
[19] 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
[20] 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
[21] Miller, P. D.: Applied Aymptotic Analysis. Graduate Studies in Mathematics 75, American Mathematical Society, Providence (2006). DOI 10.1090/gsm/075 | MR 2238098 | Zbl 1101.41031
[22] 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
[23] Rudin, W.: Analytic functions of class $H_p$. Trans. Am. Math. Soc. 78 (1955), 46-66. DOI 10.2307/1992948 | MR 0067993 | Zbl 0064.31203
[24] Sarason, D.: The $H^p$ spaces of an annulus. Mem. Amer. Math. Soc. 56 (1965), 78 pages. DOI 10.1090/memo/0056 | MR 0188824 | Zbl 0127.07002
[25] Wang, H.-C.: Real Hardy spaces of an annulus. Bull. Aust. Math. Soc. 27 (1983), 91-105. DOI 10.1017/S0004972700011515 | MR 0696647 | Zbl 0512.42023
Search
Partner of
