Abstract
Necessary and sufficient condition governing the boundedness of the multilinear fractional integral operator \(T_{\gamma , \mu }\) defined with respect to a measure \(\mu \) on a \(\sigma \)-algebra of Borel sets of quasi-metric space X from the product \(L^{p_1}(X, \mu )\times \cdots \times L^{p_m}(X, \mu )\) to \(L^q(X, \mu )\) is established. The related weak type inequality is also obtained. The derived results are used to get appropriate boundedness of \(T_{\gamma , \mu }\) in Morrey spaces defined with respect to a measure \(\mu \).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chen, X., Xue, Q.: Weighted estimates for a class of multilinear fractional type operators. J. Math. Anal. Appl. 362, 355–373 (2010)
Edmunds, D.E., Kokilashvili, V., Meskhi, A.: Bounded and Compact Integral Operators. Kluwer Academic Publishers, Dordrecht (2002)
Eridani, A., Kokilashvili, V., Meskhi, A.: Morrey spaces and fractional integral operators. Expo. Math. 27, 227–239 (2009)
Garcia-Cuerva, J., Gatto, A.E.: Boundedness properties of fractional integral operators associated to non-doubling measures. Stud. Math. 162, 245–261 (2004)
Giaquinta, M.: Multiple Integrals in the Calculus of Variations and Non-linear Elliptic Systems. Princeton University Press, Princeton (1983)
Grafakos, L.: On multilinear fractional integrals. Stud. Math. 102, 49–56 (1992)
Grafakos, L.: Classical Fourier Analysis. Graduate Texts in Mathematics, 3rd edn. Springer, New York (2014)
Grafakos, L., Kalton, N.: Some remarks on multilinear maps and interpolation. Math. Ann. 319(1), 151–180 (2001)
Grafakos, L., Liu, L., Maldonado, D., Yang, D.: Multilinear analysis on metric spaces. Dissertationes Math. (Rozprawy Mat.) 497, 121 (2014)
Hedberg, L.: On certain convolution inequalities. Proc. Amer. Math. Soc. 36, 505–510 (1972)
Hytönen, T.: A framework for non-homogeneous analysis on metric spaces, and the RBMO space of Tolsa. Publ. Mat. 54(2), 485–504 (2010)
Iida, T., Sato, E., Sawano, Y., Tanaka, H.: Sharp bounds for multilinear fractional integral operators on Morrey type spaces. Positivity 16, 339–358 (2012)
Kenig, C., Stein, E.: Multilinear estimates and fractional integration. Math. Res. Lett. 6(1), 1–15 (1999)
Kokilashvili, V.: Weighted estimates for classical integral operators. Nonlinear Analysis, Function Spaces and Applications. Vol. 119 Teubner-Texte zur Mathematik, vol. 4, pp. 86–103. Vieweg+Teubner, Wiesbaden (1990)
Kokilashvili, V., Mastyło, M., Meskhi, A.: On the boundedness of the multilinear fractional integral operators. Nonlinear Anal. 94, 142–147 (2014)
Kokilashvili, V., Mastyło, M., Meskhi, A.: Two-weight norm estimates for multilinear fractional integrals in classical Lebesgue spaces. Fract. Calc. Appl. Anal. 18(5), 1146–1163 (2015)
Kokilashvili, V., Meskhi, A.: Fractional integrals on measure spaces. Fract. Calc. Appl. Anal. 4(1), 1–24 (2001)
Kufner, A., John, O., Fučik, S.: Function spaces, Monographs and Textbooks on Mechanics of Solids and Fluids; Mechanics: Analysis. Noordhoff International Publishing, Leyden (1977)
Lerner, A.K., Ombrosi, S., Pérez, C., Torres, R.H., Trujillo-González, R.: New maximal functions and multiple weights for the multilinear Calderón–Zygmund theory. Adv. Math. 220(4), 1222–1264 (2009)
Moen, K.: Weighted inequalities for multilinear fractional integral operators. Collect. Math. 60, 213–238 (2009)
Morrey, C.B.: On the solutions of quasi-linear elliptic partial differential equations. Trans. Amer. Math. Soc. 43(1), 126–166 (1938)
Nazarov, F., Treil, S., Volberg, A.: Cauchy integral and Calderon–Zygmund operators on nonhomogeneous spaces. Int. Math. Res. Notices 15, 703–726 (1997)
Pradolini, G.: Weighted inequalities and pointwise estimates for the multilinear fractional integral and maximal operators. J. Math. Anal. Appl. 367, 640–656 (2010)
Rafeiro, H., Samko, N., Samko, S.: Morrey–Campanato spaces: an overview. Operator Theory, Pseudo-Differential Equations, and Mathematical Physics. Operator Theory: Advances and Applications, vol. 228, pp. 293–323. Birkhäuser, Basel (2013)
Sawano, Y.: Generalized Morrey spaces for non-doubling measures. Nonlinear Differ. Equ. Appl. 15(4), 413–425 (2008)
Sawano, Y., Tanaka, H.: Morrey spaces for nondoubling measures. Acta Math. Sin. 21(6), 1535–1544 (2005)
Shi, Y., Tao, X.: Weighted \(L^p\) boundedness for multilinear fractional integral on product spaces. Analysis in Theory and Applications 24(3), 280–291 (2008)
Tao, X., He, S.: The boundedness of multilinear operators on generalized Morrey spaces over the quasi-metric space of non-homogeneous type. J. Inequal. Appl. 330, 15 (2013)
Tolsa, X.: Cotlar’s inequality without the doubling condition and existence of principal values for the Cauchy integral of measures. J. Reine Angew. Math. 502, 199–235 (1998)
Acknowledgements
The authors are grateful to the anonymous referee for remarks which led to an improvement of the manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
M. Mastyło was supported by the National Science Centre, Poland, Project No. 2015/17/B/ST1/00064.
Rights and permissions
About this article
Cite this article
Kokilashvili, V., Mastyło, M. & Meskhi, A. On the Boundedness of Multilinear Fractional Integral Operators. J Geom Anal 30, 667–679 (2020). https://doi.org/10.1007/s12220-019-00159-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12220-019-00159-6
Keywords
- Multilinear fractional integrals
- Spaces defined with respect to measure
- Quasi-metric measure spaces
- Morrey spaces