Extrapolation in Grand Lebesgue Spaces with $A_{\infty}$ Weights

V. M. Kokilashvili ; A. N. Meskhi

Matematičeskie zametki, Tome 104 (2018) no. 4, pp. 539-551.

Résumé

Results on extrapolation with $A_{\infty}$ weights in grand Lebesgue spaces are obtained. Generally, these spaces are defined with respect to the product measure $\mu_1\times \dotsb\times \mu_n$ on $X_1\times \dotsb\times X_n$, where $(X_i,d_i,\mu_i)$, $i=1,\dots,n$, are spaces of homogeneous type. As applications of the obtained results, new one-weight estimates with $A_{\infty}$ weights for operators of harmonic analysis are derived.

Détail

Keywords: weighted extrapolation, strong maximal operators, multiple integral operators, Calderón–Zygmund operators with product kernels, fractional integrals with product kernels.
Mots-clés : grand Lebesgue spaces

Bibliographie

[1] T. Iwaniec, C. Sbordone, “On the integrability of the Jacobian under minimal hypotheses”, Arch. Ration. Mech. Anal., 119:2 (1992), 129–143 | DOI | MR

[2] L. Greco, T. Iwaniec, C. Sbordone, “Inverting the $p$-harmonic operator”, Manuscripta Math., 92:2 (1997), 249–258 | DOI | MR

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

[4] C. Capone, A. Fiorenza, “On small Lebesgue spaces”, J. Funct. Spaces Appl., 3:1 (2005), 73–89 | DOI | MR

[5] A. Fiorenza, B. Gupta, P. Jain, “The maximal theorem in weighted grand Lebesgue spaces”, Studia Math., 188:2 (2008), 123–133 | DOI | MR

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

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

[8] V. Kokilashvili, “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

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

[10] V. M. Kokilashvili, A. N. Meskhi, “Vesovaya ekstrapolyatsiya v prostranstvakh Ivanetsa–Sbordone. Prilozheniya k integralnym operatoram i teorii priblizhenii”, Funktsionalnye prostranstva, teoriya priblizhenii, smezhnye razdely matematicheskogo analiza, Tr. MIAN, 293, MAIK «Nauka/Interperiodika», M., 2016, 167–192 | DOI | MR

[11] V. Kokilashvili, A. Meskhi, “Extrapolation results in grand Lebesgue spaces defined on product sets”, Positivity, 2018 | DOI

[12] R. A. Macías, C. Segovia, “Lipschitz functions on spaces of homogeneous type”, Adv. in Math., 33:3 (1979), 257–270 | DOI | MR

[13] J. O. Strömberg, A. Torchinsky, Weighted Hardy Spaces, Lecture Notes in Math., 1381, Springer-Verlag, Berlin, 1989 | DOI | MR

[14] R. R. Coifman, G. Weiss, Analyse harmonique non-commutative sur certains espaces homogénes, Lecture Notes in Math., 242, Springer-Verlag, Berlin, 1971 | DOI | MR

[15] J. García-Cuerva, J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland Math. Stud., 116, North Holland Publ., Amsterdam, 1985 | MR

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

[17] D. Cruz-Uribe, J. M. Martell, C. Pérez, “Extrapolation from $A_\infty$ weights and applications”, J. Func. Anal., 213:2 (2004), 412–439 | DOI | MR

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

[19] D. E. Edmunds, V. Kokilashvili, A. Meskhi, Bounded and Compact Integral Operators, Math. Appl., 543, Kluwer Acad. Publ., Dordrecht, 2002 | MR

[20] G. Pradolini, O. Salinas, “Commutators of singular integrals on spaces of homogeneous type”, Czechoslovak Math. J., 57 (132):1 (2007), 75–93 | DOI | MR

[21] V. Kokilashvili, “Singular integrals and strong fractional maximal functions in weighted grand Lebesgue spaces”, Nonlinear Analysis, Function Spaces and Application, Vol. 9, Acad. Sci. Czech Repub. Inst. Math., Prague, 2010, 261–269 | MR

[22] S. Hruščev, “A description of weights satisfying the $A_\infty$ condition of Muckenhoupt”, Proc. Amer. Math. Soc., 90:2 (1984), 253–257 | MR

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

[24] V. Kokilashvili, A. Meskhi, H. Rafeiro, S. Samko, Integral Operators in Non-Standard Function Spaces. Vol. 2. Variable Exponent Hölder, Morrey–Campanato and Grand Spaces, Oper. Theory Adv. Appl., 249, Birkhäuser, Cham, 2016 | MR

[25] V. M. Kokilashvili, “Vesovye prostranstva Lizorkina–Tribelya. Singulyarnye integraly, multiplikatory, teoremy vlozheniya”, Issledovaniya po teorii differentsiruemykh funktsii mnogikh peremennykh i ee prilozheniyam. Chast 9, Tr. MIAN SSSR, 161, 1983, 125–149 | MR | Zbl

[26] R. Fefferman, E. M. Stein, “Singular integrals on product spaces”, Adv. in Math., 45:2 (1982), 117–143 | DOI | MR

[27] V. Kokilashvili, A. Meskhi, M. A. Zaighum, “Sharp weighted bounds for multiple integral operators”, Trans. A. Razmadze Math. Inst., 170:1 (2016), 75–90 | DOI | MR

[28] A. Bernardis, S. Hartzstein, G. Pradolini, “Weighted inequalities for commutators of fractional integrals on spaces of homogeneous type”, J. Math. Anal. Appl., 322:2 (2006), 825–846 | DOI | MR