Weighted inequalities for families of operators generated by truncated Ces\`aro kernels

K. S. Kazarian

Izvestiya. Mathematics , Tome 32 (1989) no. 2, pp. 289-311.

Résumé

For subsystems of the trigonometric system of the form $\{e^{inx}\}_{|n|\geqslant k}$ and $\{e^{inx}\}_{\substack{|n|\geqslant k\\n\ne -k}}$ ($k=1,2,\dots$) this paper establishes necessary and sufficient conditions on a weight function $\psi$ for the given system to be a $(C,\alpha)$-basis ($\alpha>0$) in the space $L^p_{[-\pi,\pi]}(\psi)$, $1\leqslant p\infty$. Bibliography: 23 titles.

Bibliographie

[1] Boas R., Pollard H., “The multiplicative completion of sets functions”, Bull. Amer. Math. Soc., 176 (1948), 499–508 | MR

[2] Coifman R., Fefferman G., “Weighted norm inequalities for maximal functions and singular integrals”, Stud. Math., 51 (1974), 241–256 | MR

[3] Gaposhkin V. F., “Multiplikativnoe popolnenie trigonometricheskoi sistemy do bazisov Chezaro”, Soob. AN GSSR, 96 (1979), 541–543 | MR | Zbl

[4] Gapoškin V. F., “Trigonometric Cesaro bases in the spaces of functions integrable with power weighty”, Analysis Math., 8 (1982), 103–124 | DOI | MR | Zbl

[5] Gundy R. F., Wheeded R. L., “Weighted integral inequalities for the nontangential maximal function, Lusin area integral, and Walsh–Paley series”, Stud. Math., 49 (1974), 107–124 | MR | Zbl

[6] Hunt R., Muckenhoupt B., Wheeden R. L., “Weighted norm inequalities for the conjugate function and Hilbert transform”, Trans. Amer. Math. Soc., 176 (1973), 227–251 | DOI | MR | Zbl

[7] Zigmund L., Trigonometricheskie ryady, Mir, M., 1965

[8] Kazaryan K. S., “O multiplikativnom dopolnenii bazisnykh posledovatelnostei v $L_p$, $1\leqslant p\infty$, do bazisov v $L_p$”, Dokl. AN ArmSSR, 52 (1976), 203–209

[9] Kazaryan K. S., “O multiplikativnom dopolnenii nekotorykh nepolnykh ortonormirovannykh sistem do bazisov v $L_p$, $1\leqslant p\infty$”, Analysis Math., 4 (1978), 37–52 | DOI | Zbl

[10] Kazaryan K. S., “O bazisnosti podsistem trigonometricheskoi sistemy v prostranstvakh $L^p(d\mu)$, $1\leqslant p\infty$”, Dokl. AN ArmSSR, 69 (1979), 257–259 | MR | Zbl

[11] Kazaryan K. S., “Summirovanie po Abelyu podsistem trigonometricheskoi sistemy v prostranstvakh $L^p(d\mu)$, $1\leqslant p\infty$”, Dokl. AN ArmSSR, 71 (1980), 257–262 | MR | Zbl

[12] Kazarian K. S., “On bases and unconditional bases in the spaces $L^p(d\mu)$, $1\leqslant p\infty$”, Stud. Math., 71 (1982), 227–249 | MR | Zbl

[13] Kazaryan K. S., “Ravnomernaya nepreryvnost v vesovykh prostranstvakh semeistv operatorov, porozhdennykh usechennymi yadrami”, Dokl. AN SSSR, 272, 1048–1052 | MR | Zbl

[14] Kazaryan K. S., “Summiruemost obobschennykh ryadov Fure v vesovoi metrike i pochti vsyudu”, Dokl. AN SSSR, 287, 543–546 | MR | Zbl

[15] Kazarian K. S., “Weighted norm inequalities for some classes of singular integrals”, Stud. Math., 87 (1987), 97–130 | MR

[16] Kazaryan K. S., “Summiruemost obobschennykh ryadov Fure v $L^p(d\mu)$, zadacha Dirikhle v vesovoi metrike i vesovye prostranstva $H^p$, $p>1$”, Analysis Math., 13 (1987), 173–197 | DOI | MR | Zbl

[17] Muckenhoupt B., “Weighted norm inequalities for the Hardy maximal function”, Trans. Amer. Math. Soc., 165 (1972), 207–226 | DOI | MR | Zbl

[18] Muckenhoupt B., “Hardys inequality with weights”, Stud. Math., 64 (1972), 31–38 | MR

[19] Rosenblum M., “Summability of Fourier series in $L^p(d\mu)$”, Trans. Amer. Math. Soc., 105 (1962), 32–42 | DOI | MR | Zbl

[20] Talenti G., “Osservazioni sopra una classe di disuguaglianze”, Rend. Semin. Math. Fis. Milario, 39 (1969), 171–185 | DOI | MR | Zbl

[21] Tomaselli G., “A class of inequalities”, Bull. Unione Mat. Ital., 2 (1969), 622–631 | MR | Zbl

[22] Zygmund A., “Sur un theoreme de M. Gronwall”, Bull. de l'Acad. Polon., 1925, 207–217 | Zbl

[23] Grunwall T. H., “Über eine Summationsmethode und ihre Anwendung auf die Fouriersche Reine”, für die Reine und Ang. Math., 147 (1916), 16–35