Weighted extrapolation in Iwaniec--Sbordone spaces. Applications to integral operators and approximation theory

V. M. Kokilashvili ; A. N. Meskhi

Informatics and Automation, Function spaces, approximation theory, and related problems of mathematical analysis, Tome 293 (2016), pp. 167-192.

Résumé

We prove extrapolation theorems in weighted Iwaniec–Sbordone spaces and apply them to one-weight inequalities for several integral operators of harmonic analysis. In addition, in weighted grand Lebesgue spaces, we establish Bernstein and Nikol'skii type inequalities and prove direct and inverse theorems on the approximation of functions.

Bibliographie

[1] Bernardis A., Hartzstein S., Pradolini G., “Weighted inequalities for commutators of fractional integrals on spaces of homogeneous type”, J. Math. Anal. Appl., 322 (2006), 825–846 | DOI | MR | Zbl

[2] Bernshtein S.N., Sobranie sochinenii, T. 1., 1, Izd-vo AN SSSR, M., 1952

[3] Capone C., Fiorenza A., “On small Lebesgue spaces”, J. Funct. Spaces Appl., 3 (2005), 73–89 | DOI | MR | Zbl

[4] Coifman R.R., Weiss G., Analyse harmonique non-commutative sur certains espaces homogènes: Etude de certaines intégrales singulières, Lect. Notes Math., 242, Springer, Berlin, 1971 | DOI | MR

[5] Dragičević O., Grafakos L., Pereyra M.C., Petermichl S., “Extrapolation and sharp norm estimates for classical operators on weighted Lebesgue spaces”, Publ. Mat., 49:1 (2005), 73–91 | MR | Zbl

[6] Duoandikoetxea J., “Extrapolation of weights revisited: New proofs and sharp bounds”, J. Funct. Anal., 260:6 (2011), 1886–1901 | DOI | MR | Zbl

[7] Fiorenza A., “Duality and reflexivity in grand Lebesgue spaces”, Collect. math., 51:2 (2000), 131–148 | MR | Zbl

[8] Fiorenza A., Gupta B., Jain P., “The maximal theorem for weighted grand Lebesgue spaces”, Stud. math., 188:2 (2008), 123–133 | DOI | MR | Zbl

[9] Grafakos L., Classical Fourier analysis, Grad. Texts Math., 249, 3rd ed., Springer, New York, 2014 | MR | Zbl

[10] Greco L., Iwaniec T., Sbordone C., “Inverting the $p$-harmonic operator”, Manuscr. math., 92 (1997), 249–258 | DOI | MR | Zbl

[11] Harboure E., Macías R.A., Segovia C., “Extrapolation results for classes of weights”, Amer. J. Math., 110 (1988), 383–397 | DOI | MR | Zbl

[12] Hytönen T. P., Pérez C., Rela E., “Sharp reverse Hölder property for $A_\infty $ weights on spaces of homogeneous type”, J. Funct. Anal., 263:12 (2012), 3883–3899 | DOI | MR | Zbl

[13] Iwaniec T., Sbordone C., “On the integrability of the Jacobian under minimal hypotheses”, Arch. Ration. Mech. Anal., 119 (1992), 129–143 | DOI | MR | Zbl

[14] Kokilashvili V., “Boundedness criterion for the Cauchy singular integral operator in weighted grand Lebesgue spaces and application to the Riemann problem”, Proc. A. Razmadze Math. Inst., 151 (2009), 129–133 | MR | Zbl

[15] Kokilashvili V., “Boundedness criteria for singular integrals in weighted grand Lebesgue spaces”, J. Math. Sci., 170:1 (2010), 20–33 | DOI | MR | Zbl

[16] Kokilashvili V., “Singular integrals and strong maximal functions in weighted grand Lebesgue spaces”, Nonlinear analysis, function spaces and applications, 9, Czech Acad. Sci., Inst. Math., Prague, 2011, 261–269 | MR | Zbl

[17] Kokilashvili V., Meskhi A., “A note on the boundedness of the Hilbert transform in weighted grand Lebesgue spaces”, Georgian Math. J., 16:3 (2009), 547–551 | MR | Zbl

[18] Kokilashvili V., Meskhi A., “Trace inequalities for fractional integrals in grand Lebesgue spaces”, Stud. math., 210:2 (2012), 159–176 | DOI | MR | Zbl

[19] Kokilashvili V., Meskhi A., “Potentials with product kernels in grand Lebesgue spaces: One-weight criteria”, Lith. Math. J., 53:1 (2013), 27–39 | DOI | MR | Zbl

[20] Kokilashvili V., Meskhi A., Rafeiro H., “Boundedness of commutators of singular and potential operators in generalized grand Morrey spaces and some applications”, Stud. math., 217:2 (2013), 159–178 | DOI | MR | Zbl

[21] Lacey M.T., Moen K., Pérez C., Torres R.H., “Sharp weighted bounds for fractional integral operators”, J. Funct. Anal., 259:5 (2010), 1073–1097 | DOI | MR | Zbl

[22] Lizorkin P.I., “Otsenki trigonometricheskikh integralov i neravenstvo Bernshteina dlya drobnykh proizvodnykh”, Izv. AN SSSR. Ser. mat., 29:1 (1965), 109–126 | MR | Zbl

[23] Macías R.A., Segovia C., “Lipschitz functions on spaces of homogeneous type”, Adv. Math., 33 (1979), 257–270 | DOI | MR | Zbl

[24] Meskhi A., Criteria for the boundedness of potential operators in grand Lebesgue spaces, E-print, 2010, arXiv: 1007.1185v1 [math.FA] | MR

[25] Nikolskii S.M., “Neravenstva dlya tselykh funktsii konechnoi stepeni i ikh primenenie v teorii differentsiruemykh funktsii mnogikh peremennykh”, Tr. MIAN., 38 (1951), 244–278

[26] Pradolini G., Salinas O., “Commutators of singular integrals on spaces of homogeneous type”, Czech. Math. J., 57:1 (2007), 75–93 | DOI | MR | Zbl

[27] Rubio de Francia J.L., “Factorization and extrapolation of weights”, Bull. Amer. Math. Soc., 7 (1982), 393–395 | DOI | MR | Zbl

[28] Samko S.G., Kilbas A.A., Marichev O.I., Integraly i proizvodnye drobnogo poryadka i nekotorye ikh prilozheniya, Nauka i tekhnika, Minsk, 1987 | MR

[29] Samko S.G., Umarkhadzhiev S.M., “On Iwaniec–Sbordone spaces on sets which may have infinite measure”, Azerb. J. Math., 1:1 (2011), 67–84 | MR | Zbl

[30] Strömberg J.-O., Torchinsky A., Weighted Hardy spaces, Lect. Notes Math., 1381, Springer, Berlin, 1989 | DOI | MR | Zbl

[31] Zigmund A., Trigonometricheskie ryady, T. 2., 2, Mir, M., 1965 | MR