Fractional multianisotropic spaces and embedding theorems. (Russian) Zbl 1492.46036
Summary: In the present article, we introduce multianisotropic function spaces with fractional derivatives. We also study multianisotropic Bessel potentials and prove embedding theorems for them.
MSC:
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
Keywords:
fractional multianisotropic spaces; integral representation; embedding theorems; Bessel potentialsReferences:
| [1] | Besov O. V., “Issledovanie odnogo semeistva funktsionalnykh prostranstv v svyazi s teoremami vlozheniya i prodolzheniya”, Tr. Matem. in-ta im. V. A. Steklova, 60, 1961, 42-81 · Zbl 0116.31701 |
| [2] | Besov O. V., Ilin V. P., Nikolskii S. M., Integralnye predstavleniya funktsii i teoremy vlozheniya, Nauka, M., 1975 |
| [3] | Ilin V. P., “Obobschenie odnogo integralnogo neravenstva”, Uspekhi mat. nauk, 11:4 (70) (1956), 131-138 · Zbl 0073.32302 |
| [4] | Kazaryan G. G., “Otsenki differentsialnykh operatorov i gipoellipticheskie operatory”, Tr. Matem. in-ta im. V. A. Steklova, 140, 1976, 130-161 · Zbl 0399.35025 |
| [5] | Karapetyan G. A., “Integralnoe predstavlenie i teoremy vlozheniya dlya \(n\)-mernykh multianizotropnykh prostranstv s odnoi vershinoi anizotropnosti”, Sib. matem. zhurn., 58:3 (2017), 573-590 · Zbl 1381.46033 |
| [6] | Karapetyan G. A., Arakelyan M. K., “Teoremy vlozheniya dlya obschikh multianizotropnykh prostranstv”, Matem. zametki, 104:3 (2018), 422-438 · Zbl 1416.46037 · doi:10.4213/mzm12115 |
| [7] | Karapetyan G. A., Khachaturyan M. A., “Predelnye teoremy vlozheniya dlya multianizotropnykh funktsionalnykh prostranstv”, Izv. NAN Armenii, 54:2 (2019), 54-64 |
| [8] | Kondrashov V. I., “O nekotorykh svoistvakh funktsii iz prostranstv \(L_p\)”, DAN SSSR, 48:8 (1945), 563-566 |
| [9] | Lizorkin P. I., “Teoremy vlozheniya dlya funktsii iz prostranstv \(L_p^r\)”, DAN SSSR, 143:5 (1962), 1042-1045 · Zbl 0117.33402 |
| [10] | Lizorkin P. I., “Obobschennoe liuvillevskoe differentsirovanie i funktsionalnye prostranstva \(L_p^r (E_n)\). Teoremy vlozheniya”, Matem. sb., 60 (102):3 (1963), 325-353 · Zbl 0134.31703 |
| [11] | Lizorkin P. I., “Obobschennoe liuvillevskoe differentsirovanie i metod multiplikatorov v teorii vlozhenii klassov differentsiruemykh funktsii”, Tr. Matem. in-ta im. V. A. Steklova, 105, 1969, 89-167 · Zbl 0204.43705 |
| [12] | Mikhailov V. P., “O povedenii na beskonechnosti odnogo klassa mnogochlenov”, Tr. Matem. in-ta im. V. A. Steklova, 91, 1967, 58-80 |
| [13] | Nikolskii S. M., “Ob odnom semeistve funktsionalnykh prostranstv”, Uspekhi mat. nauk, 11:6 (72) (1956), 203-212 · Zbl 0074.09306 |
| [14] | Nikolskii S. M., “Ob ustoichivykh granichnykh znacheniyakh differentsiruemoi funktsii mnogikh peremennykh”, Matem. sb., 61 (103):2 (1963), 224-252 · Zbl 0166.39701 |
| [15] | Nikolskii S. M., Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya, Nauka, M., 1969 |
| [16] | Samko S. G., Kilbas A. A., Marichev O. I., Integraly i proizvodnye drobnogo poryadka i nekotorye ikh primeneniya, Nauka i tekhnika, Minsk, 1987 |
| [17] | Semyanistyi V. I., “O nekotorykh integralnykh preobrazovaniyakh v evklidovom prostranstve”, DAN SSSR, 134:3 (1960), 536-539 |
| [18] | Slobodetskii L. N., “Obobschennye prostranstva S. L. Soboleva i ikh prilozhenie k kraevym zadacham dlya differentsialnykh uravnenii v chastnykh proizvodnykh”, Uch. zap. Leningr. gos. ped. in-ta, 197 (1958), 54-112 |
| [19] | Sobolev S. L., “Ob odnoi teoreme funktsionalnogo analiza”, Matem. sb., 4 (34):3 (1938), 471-497 · Zbl 0022.14803 |
| [20] | Sobolev S. L., Nekotorye primeneniya funktsionalnogo analiza v matematicheskoi fizike, Izd-vo Leningr. un-ta, L., 1950 |
| [21] | Uspenskii S. V., “O teoremakh vlozheniya dlya obobschennykh klassov \(W_p^r\) Soboleva”, Sib. matem. zhurn., 3:3 (1962), 418-445 · Zbl 0152.12801 |
| [22] | Aronszajn N., Smith K. T., “Theory of Bessel potentials (I)”, Ann. Inst. Fourier (Grenoble), 11 (1961), 385-475 · Zbl 0102.32401 · doi:10.5802/aif.116 |
| [23] | Calderon A. P., Zygmund A., “Local properties of solutions of elliptic partial differential equations”, Studia Math., 20:2 (1961), 171-225 · Zbl 0099.30103 · doi:10.4064/sm-20-2-181-225 |
| [24] | Gindikin S., Volevich L. R., The Method of Newton’s Polyhedron in the Theory of Partial Differential Equations, Mathematics and its Applications, 86, Kluwer Acad. Publ., Boston-London-Dordrecht, 1992 · Zbl 0779.35001 |
| [25] | Hörmander L., “Integral inequalities for partial differential operators and functions satisfying general boundary conditions”, Comm. Pure. Appl. Math., 12 (1959), 37-66 · Zbl 0093.29401 · doi:10.1002/cpa.3160120104 |
| [26] | Karapetyan G. A., “Integral representation of functions and embedding theorems for multianisotropic spaces for the three-dimensional case”, Eurasian Math. J., 7:4 (2016), 19-39 · Zbl 1463.46059 |
| [27] | Liouville I., “Sure le calcul des differentielles a indice quelconques”, Jorn. de I’Ecole Polytechn., 13 (1832), 1-69 |
| [28] | Riemann B., “Versuch einer allgemeinen Auffassung der Integration und Differentiation”, Ges. Werke, Teubner, Leipzig, 1876, 331-344 |
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