Continuity of embeddings of weighted Sobolev spaces in Lebesgue spaces on anisotropically irregular domains

B. V. Trushin

Informatics and Automation, Function theory and differential equations, Tome 269 (2010), pp. 271-289.

Résumé

In our earlier publications, the domains satisfying the flexible $\sigma$-cone condition were classified with respect to an anisotropy parameter $\lambda$. In the present paper we establish the continuity of embeddings of weighted Sobolev spaces in Lebesgue spaces in these classes of domains. For each class of domains with parameter $\lambda\ne(1,\dots,1)$, the theorems obtained are stronger than those in the general case of domains satisfying the flexible $\sigma$-cone condition.

Bibliographie

[1] Sobolev S.L., Nekotorye primeneniya funktsionalnogo analiza v matematicheskoi fizike, Izd-vo LGU, L., 1950, 2-e izd. Novosibirsk: Izd-vo SO AN SSSR, 1962; 3-e izd. M.: Nauka, 1988. | MR

[2] Sobolev S.L., “O nekotorykh otsenkakh, otnosyaschikhsya k semeistvam funktsii, imeyuschikh proizvodnye, integriruemye s kvadratom”, DAN SSSR, 1:7 (1936), 267–270

[3] Sobolev S., “Ob odnoi teoreme funktsionalnogo analiza”, Mat. sb., 4:3 (1938), 471–497 | Zbl

[4] Mazya V.G., Prostranstva S.L. Soboleva, Izd-vo LGU, L., 1985 | MR | Zbl

[5] Reshetnyak Yu.G., “Integralnye predstavleniya differentsiruemykh funktsii v oblastyakh s negladkoi granitsei”, Sib. mat. zhurn., 21:6 (1980), 108–116 | MR | Zbl

[6] Goldshtein V.M., Reshetnyak Yu.G., Vvedenie v teoriyu funktsii s obobschennymi proizvodnymi i kvazikonformnye otobrazheniya, Nauka, M., 1983 | MR

[7] Besov O.V., Ilin V.P., Nikolskii S.M., Integralnye predstavleniya funktsii i teoremy vlozheniya, Nauka, M., 1996 | MR

[8] Buckley S., Koskela P., “Sobolev–Poincaré implies John”, Math. Res. Lett., 2 (1995), 577–593 | DOI | MR | Zbl

[9] Smith W., Stegenga D.A., “Hölder domains and Poincaré domains”, Trans. Amer. Math. Soc., 319 (1990), 67–100 | DOI | MR | Zbl

[10] Hajlasz P., Koskela P., “Isoperimetric inequalities and imbedding theorems in irregular domains”, J. London Math. Soc. Ser. 2, 58 (1998), 425–450 | DOI | MR | Zbl

[11] Kilpeläinen T., Malý J., “Sobolev inequalities on sets with irregular boundaries”, Ztschr. Anal. und Anwend., 19:2 (2000), 369–380 | DOI | MR | Zbl

[12] Labutin D.A., “Integralnoe predstavlenie funktsii i vlozhenie prostranstv Soboleva na oblastyakh s nulevymi uglami”, Mat. zametki, 61:2 (1997), 201–219 | DOI | MR | Zbl

[13] Labutin D.A., “Vlozhenie prostranstv Soboleva na gëlderovykh oblastyakh”, Tr. MIAN, 227, 1999, 170–179 | MR | Zbl

[14] Besov O.V., “Teorema vlozheniya Soboleva dlya oblasti s neregulyarnoi granitsei”, Mat. sb., 192:3 (2001), 3–26 | DOI | MR | Zbl

[15] Labutin D.A., “Neuluchshaemost neravenstv Soboleva dlya klassa neregulyarnykh oblastei”, Tr. MIAN, 232, 2001, 218–222 | MR | Zbl

[16] Mazya V.G., Poborchii S.V., Teoremy vlozheniya i prodolzheniya dlya funktsii v nelipshitsevykh oblastyakh, Izd-vo S.-Peterb. un-ta, SPb., 2006

[17] Trushin B.V., “Teoremy vlozheniya Soboleva dlya nekotorogo klassa anizotropnykh neregulyarnykh oblastei”, Tr. MIAN, 260, 2008, 297–319 | MR | Zbl

[18] Trushin B.V., “O neuluchshaemosti vesovykh teorem vlozheniya Soboleva dlya anizotropno neregulyarnykh oblastei”, Sovremennye problemy fundamentalnoi i prikladnoi matematiki, Sb. nauch. trudov, Izd-vo MFTI, M., 2009, 166–190

[19] Trushin B.V., “Vesovye teoremy vlozheniya Soboleva dlya nekotorogo klassa anizotropnykh neregulyarnykh oblastei”, Differentsialnye uravneniya, funktsionalnye prostranstva, teoriya priblizhenii, Tez. dokl. Mezhdunar. konf., posv. 100-letiyu so dnya rozhd. S.L. Soboleva, Novosibirsk, 2008, Izd-vo In-ta matematiki SO RAN, Novosibirsk, 2008, 212

[20] Kokilashvili V.M., Gabidzashvili M.A., “O vesovykh neravenstvakh dlya anizotropnykh potentsialov i maksimalnykh funktsii”, DAN SSSR, 282:6 (1985), 1304–1306 | MR | Zbl

[21] Gabidzashvili M.A., “Vesovye neravenstva dlya anizotropnykh potentsialov”, Tr. Tbil. mat. in-ta, 82 (1986), 25–36 | MR | Zbl

[22] Besov O.V., “Vlozhenie prostranstv differentsiruemykh funktsii peremennoi gladkosti”, Tr. MIAN, 214, 1997, 25–58 | Zbl

[23] de Guzman M., “A covering lemma with applications to differentiability of measures and singular integral operators”, Stud. math., 34 (1970), 299–317 | MR | Zbl

[24] Gusman M., Differentsirovanie integralov v $\mathbb R^n$, Mir, M., 1978 | MR

[25] Besov O.V., “O kompaktnosti vlozhenii vesovykh prostranstv Soboleva na oblasti s neregulyarnoi granitsei”, Tr. MIAN, 232, 2001, 72–93 | MR | Zbl