Hardy--Steklov operators and Sobolev-type embedding inequalities

M. G. Nasyrova ; E. P. Ushakova

Informatics and Automation, Function spaces, approximation theory, and related problems of mathematical analysis, Tome 293 (2016), pp. 236-262.

Résumé

We characterize weighted inequalities corresponding to the embedding of a class of absolutely continuous functions into a fractional-order Sobolev space. As auxiliary results of the paper, which are also of independent interest, we obtain several new types of necessary and sufficient conditions for the boundedness of the Hardy–Steklov operator (integral operator with two variable limits) in weighted Lebesgue spaces.

Bibliographie

[1] Aronszajn N., “Boundary values of functions with finite Dirichlet integral”, Proc. Conf. on partial differential equations (Univ. Kansas, 1954), Studies in eigenvalue problems, 14, Univ. Kansas, Lawrence, KS, 1955, 77–93

[2] Batuev E.N., Stepanov V.D., “O vesovykh neravenstvakh tipa Khardi”, Sib. mat. zhurn., 30:1 (1989), 13–22 | MR | Zbl

[3] Besov O.V., Ilin V.P., Nikolskii S.M., Integralnye predstavleniya funktsii i teoremy vlozheniya, Nauka, M., 1975 | MR

[4] Chen T., Sinnamon G., “Generalized Hardy operators and normalizing measures”, J. Inequal. Appl., 7:6 (2002), 829–866 | MR | Zbl

[5] Gagliardo E., “Proprietà di alcune classi di funzioni in più variabili”, Ric. Mat., 7 (1958), 102–137 | MR | Zbl

[6] Gogatishvili A., Lang J., “The generalized Hardy operator with kernel and variable integral limits in Banach function spaces”, J. Inequal. Appl., 4:1 (1999), 1–16 | MR | Zbl

[7] Grisvard P., “Espaces intermèdiaires entre espaces de Sobolev avec poids”, Ann. Sc. Norm. Super. Pisa. Sci. Fis. Mat. Sér. 3., 17 (1963), 255–296 | MR | Zbl

[8] Harper Z., “Weighted norm inequalities for convolution operators and links with the Weiss conjecture”, J. Evol. Eqns., 5 (2005), 387–405 | DOI | MR | Zbl

[9] Heinig H.P., Sinnamon G., “Mapping properties of integral averaging operators”, Stud. math., 129:2 (1998), 157–177 | MR | Zbl

[10] Kantorovich L.V., Akilov G.P., Funktsionalnyi analiz, Nauka, M., 1977 | MR

[11] Kufner A., Maligranda L., Persson L.-E., The Hardy inequality: About its history and some related results, Vydavatelský Servis, Plzeň, 2007 | MR | Zbl

[12] Kufner A., Persson L.-E., Weighted inequalities of Hardy type, World Sci., Singapore, 2003 | MR | Zbl

[13] Lions J.L., Magenes E., Non-homogeneous boundary value problems and applications, V. 1. (Grundl. Math. Wiss.; Bd. 181)., 1, Springer, Berlin, 1972

[14] Lomakina E.N., “Otsenki approksimativnykh chisel odnogo klassa integralnykh operatorov. I, II”, Sib. mat. zhurn., 44:1 (2003), 178–192 ; 2, 372–388 | MR | Zbl | MR | Zbl

[15] Nasyrova M., “Overdetermined weighted Hardy inequalities on semiaxis”, Function spaces and applications, Proc. Conf., New Delhi, 1997, Narosa Publ., New Delhi, 2000, 201–231 | MR | Zbl

[16] Nasyrova M., Stepanov V., “On weighted Hardy inequalities on semiaxis for functions vanishing at the endpoints”, J. Inequal. Appl., 1:3 (1997), 223–238 | MR | Zbl

[17] Nasyrova M., Stepanov V., “On maximal overdetermined Hardy's inequality of second order on a finite interval”, Math. Bohem., 124:2–3 (1999), 293–302 | MR | Zbl

[18] Oinarov R., “Boundedness of integral operators from weighted Sobolev space to weighted Lebesgue space”, Complex Var. Elliptic Eqns., 56:10–11 (2011), 1021–1038 | DOI | MR | Zbl

[19] Opic O., Kufner A., Hardy-type inequalities, Pitman Res. Notes Math., 219, Longman Sci. Tech., Harlow, 1990 | MR | Zbl

[20] Prokhorov D.V., Stepanov V.D., “Vesovye otsenki operatorov Rimana–Liuvillya i prilozheniya”, Tr. MIAN, 243 (2003), 289–312 | MR | Zbl

[21] Slobodetskii L.N., “Obobschennye prostranstva S.L. Soboleva i ikh prilozhenie k kraevym zadacham dlya differentsialnykh uravnenii v chastnykh proizvodnykh”, Uchen. zap. LGPI, 197 (1958), 54–112 | MR

[22] Stepanov V.D., “O vesovykh neravenstvakh tipa Khardi dlya drobnykh integralov Rimana–Liuvillya”, Sib. mat. zhurn., 31:3 (1990), 186–197 | MR | Zbl

[23] Stepanov V.D., “Weighted norm inequalities for integral operators and related topics”, Nonlinear analysis, function spaces and applications, 5, Prometheus, Prague, 1994, 139–175 | MR | Zbl

[24] Stepanov V.D., Ushakova E.P., “Ob integralnykh operatorakh s peremennymi predelami integrirovaniya”, Tr. MIAN, 232 (2001), 298–317 | MR | Zbl

[25] Stepanov V.D., Ushakova E.P., “Hardy operator with variable limits on monotone functions”, J. Funct. Spaces Appl., 1:1 (2003), 1–15 | DOI | MR | Zbl

[26] Stepanov V.D., Ushakova E.P., “Ob operatore geometricheskogo srednego s peremennymi predelami integrirovaniya”, Tr. MIAN, 260 (2008), 264–288 | MR | Zbl

[27] Stepanov V.D., Ushakova E.P., “Kernel operators with variable intervals of integration in Lebesgue spaces and applications”, Math. Inequal. Appl., 13:3 (2010), 449–510 | MR | Zbl

[28] Stepanov V.D., Ushakova E.P., “On boundedness of a certain class of Hardy–Steklov type operators in Lebesgue spaces”, Banach J. Math. Anal., 4:1 (2010), 28–52 | DOI | MR | Zbl

[29] Tribel G., “Interpolyatsionnye svoistva $\varepsilon $-entropii i poperechnikov. Geometricheskie kharakteristiki vlozheniya prostranstv funktsii tipa Soboleva–Besova”, Mat. sb., 98:1 (1975), 27–41 | MR | Zbl

[30] Ushakova E.P., “On boundedness and compactness of a certain class of kernel operators”, J. Funct. Spaces Appl., 9:1 (2011), 67–107 | DOI | MR | Zbl

[31] Ushakova E.P., “Estimates for Schatten–von Neumann norms of Hardy–Steklov operators”, J. Approx. Theory, 173 (2013), 158–175 | DOI | MR | Zbl

[32] Yakovlev G.N., “Granichnye svoistva funktsii klassa $W_p^{(l)}$ na oblastyakh s uglovymi tochkami”, DAN SSSR., 140:1 (1961), 73–76 | MR | Zbl