Abstract: Necessary and sufficient conditions on a measure μ guaranteeing the boundedness of the
multilinear fractional integral operator Tγ,μ (m) (defined with respect to a measure μ) from the product of Lorentz spaces ∏ Lμ m rk,sk k=1 to the Lorentz space Lμ p,q (X) are derived. The results are new even for linear fractional integrals Tγ,μ (i.e., for m = 1). From the general results we obtain a criterion for the validity of the Sobolev inequality for Tγ,μ(m) in Lorentz spaces defined with respect to μ. We investigate the same problem for Morrey-Lorentz spaces.

Sharp form for the Olsen’s inequality in multilinear setting is obtained. Criteria for the bound-edness of multilinear Riesz potential operator from Lebesgue space to a Lebesgue space with weight will be psesented.Finally, weighted criteria for the boundedness of m− linear Riemann-Liouville operators will be also discussed.Talk is based on the papers [1]–[5].
References:
[1] L. Grafakos and A. Meskhi, On sharp Olsen’s and trace inequalities for multilinear fractional
integrals, Potential Analysis 59 (2023), 1039-1050.
[2] V. Kokilashvili, M. Mastylo and A. Meskhi, On the Boundedness of Multilinear Fractional
Integral Operators, J. Geome. Anal. 30 (2020), 667-679.
[3] V. Kokilashvili and A. Meskhi, Fractional integrals on measure spaces, Fract. Calc. Appl.
Anal. 4 (2001), No.1, 1–24.
[4] A. Meskhi and L. Natelashvili, Boundedness criteria for linear and multilinear fractional
integral operators in Lorentz spaces, Trans. A. Razmadze Math. Inst. 178 (2024), No. 2, 331-333.
[5] A. Meskhi and L. Natelashvili, Boundedness criteria for linear and multilinear fractional
integral operators in Lorentz spaces (to appear).