Multipliers of Fourier integrals and bounds of convolution in spaces with mixed norms. Applications

P. I. Lizorkin

Izvestiya. Mathematics , Tome 4 (1970) no. 1, pp. 225-255.

Résumé

Multipliers of Fourier integrals and estimates of integrals of potential type in spaces with mixed norms $L_{\vec{p}}$ are considered in this paper. As applications we prove theorems of imbedding type for spaces of Bessel potentials with density from $L_{\vec{p}}$.

Bibliographie

[1] Khërmander L., Otsenki operatorov, invariantnykh otnositelno sdviga, IL, M., 1962

[2] Benedek A., Panzone R., “The spaces $L^p$ with mixed norm”, Duke Math. J., 28:3 (1961), 301–324 | DOI | MR | Zbl

[3] Benedek A., Calderon A. P., Panzone R., “Convolutions operators on Banach space valued functions”, Proc. Nat. Akad. Sci. USA, 48:3 (1962), 356–365 | DOI | MR | Zbl

[4] Lizorkin P. I., “Teorema tipa Littlvuda–Paleya dlya kratnykh integralov Fure”, Tr. Matem. in-ta im. V. A. Steklova AN SSSR, 89, 1967, 214–230 | MR

[5] Lizorkin P. I., “O multiplikatorakh integralov Fure v prostranstvakh $L_{p,\theta}$”, Tr. Matem. in-ta im. V. A. Steklova AN SSSR, 89, 1967, 231–248 | MR | Zbl

[6] Gudiev A. X., “Ob integralakh tipa potentsiala i teoremakh vlozheniya v prostranstvakh so smeshannoi normoi”, Differentsialnye uravneniya, 2:1 (1966), 83–106 | MR | Zbl

[7] Sobolev S. L., “Ob odnoi teoreme funktsionalnogo analiza”, Matem. sb., 4(46) (1938), 471–498

[8] Ilin V. P., “Svoistva nekotorykh klassov differentsiruemykh funktsii mnogikh peremennykh, zadannykh v $n$-mernoi oblasti”, Tr. Matem. in-ta im. V. A. Steklova AN SSSR, 66, 1962, 227–363 | MR | Zbl

[9] Lizorkin P. I., “Neizotropnye besselevy potentsialy”, Dokl. AN SSSR, 170:3 (1966), 508–511 | MR | Zbl

[10] Lizorkin P. I., “Obobschennoe liuvillevskoe differentsirovanie i metod multiplikatorov v teorii vlozhenii klassov funktsii”, Tr. Matem. in-ta im. V. A. Steklova AN SSSR, 105, 1969, 89–167 | MR | Zbl

[11] Gudiev A. X., “Nekotorye obobscheniya teorem vlozheniya”, Sib. matem. zh., 6:4 (1965), 775–797 | MR | Zbl

[12] Danford N., Shvarts Dzh., Lineinye operatory. II, IL, M., 1967

[13] Schwartz I., “A remark on inequalities of Calderon–Zygmund type”, Comm. Pure and Appl. Math., 14:4 (1961), 758–799

[14] Krée P., “Proprietes de continuite dans $L^p$ de certains noyaux”, Boll. Unione mat. ital., 22:3 (1967), 330–344 | MR | Zbl

[15] Xapdi G. G, Littlvud D. E., Polia G., Neravenstva, IL, M., 1948

[16] Lizorkin P. I., Nikolskii S. M., “Klassifikatsiya differentsiruemykh funktsii na osnove prostranstv s dominiruyuschei smeshannoi proizvodnoi”, Tr. Matem. in-ta im. V. A. Steklova AN SSSR, 77, 1965, 143–167 | MR | Zbl

[17] Besov O. V., “Issledovanie odnogo semeistva funktsionalnykh prostranstv v svyazi s teoremami vlozheniya i prodolzheniya”, Tr. Matem. in-ta im. V. A. Steklova AN SSSR, 60, 1961, 42–81 | MR

[18] Nikolskii S. M., “Neravenstva dlya tselykh funktsii konechnoi stepeni i ikh primenenie v teorii differentsiruemykh funktsii mnogikh peremennykh”, Tr. Matem. in-ta im. V. A. Steklova AN SSSR, 38, 1951, 244–278